trajectory_id stringlengths 39 45 | problem_id stringlengths 12 18 | domain stringclasses 6
values | turn int64 0 0 | accumulated_directives listlengths 0 0 | prompt stringlengths 298 2.32k | target_output stringlengths 220 2.83k | source_candidate_ids listlengths 1 1 | dataset_tier stringclasses 1
value | source stringclasses 2
values | instruction stringlengths 298 2.32k | input stringclasses 1
value | output stringlengths 220 2.83k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
E1_solver_manual_det:physgym_0039:turn0 | physgym_0039 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A rectangular wooden block of height $H$ and density $\rho_{1}$ ... | Taking the top at equilibrium as the origin, establish the coordinate system as shown.
Let the position of the top of the block be $y$. The net force on the block is $-\rho_{1}H S g + \rho_{2}(H-h_{0}-y)S g = -\rho_{2}S g y$. The net force on the block is a linear restoring force, and its motion is simple harmonic mot... | [
"physgym_0039_sol1"
] | E1_solver_manual_det | physgym | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A rectangular wooden block of height $H$ and density $\rho_{1}$ ... | Taking the top at equilibrium as the origin, establish the coordinate system as shown.
Let the position of the top of the block be $y$. The net force on the block is $-\rho_{1}H S g + \rho_{2}(H-h_{0}-y)S g = -\rho_{2}S g y$. The net force on the block is a linear restoring force, and its motion is simple harmonic mot... | |
E1_solver_manual_det:physgym_0041:turn0 | physgym_0041 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The rocket ascends vertically from the Earth's surface with a co... | At the moment of takeoff, $N_1=(1+k_1)mg_0$, where $m$ is the mass of the tester.
When ascending to a height $h$, the gravitational acceleration changes to $g=g_0(\dfrac{R}{R+h})^2$
The pressure at this time is $N_2=mg_0(\dfrac{R}{R+h})^2+k_1mg_0$
Since $N_2=k_2N_1$, solving the equations together gives
$h=R(\dfrac{1}{... | [
"physgym_0041_sol1"
] | E1_solver_manual_det | physgym | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The rocket ascends vertically from the Earth's surface with a co... | At the moment of takeoff, $N_1=(1+k_1)mg_0$, where $m$ is the mass of the tester.
When ascending to a height $h$, the gravitational acceleration changes to $g=g_0(\dfrac{R}{R+h})^2$
The pressure at this time is $N_2=mg_0(\dfrac{R}{R+h})^2+k_1mg_0$
Since $N_2=k_2N_1$, solving the equations together gives
$h=R(\dfrac{1}{... | |
E1_solver_manual_det:physgym_0072:turn0 | physgym_0072 | THERMODYNAMICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At time $t=0$, an adiabatic cylinder with a cross-sectional area... | From the proportional relationship between work done and heat absorbed:
$$
\frac{1}{2}\left(p_{1}\mathrm{d}V_{1}+p_{2}\mathrm{d}V_{2}\right)=T_{1}\mathrm{d}S_{1}=T_{2}\mathrm{d}S_{2}
$$
From the first law of thermodynamics:
$$
\mathrm{d}U_{i}=T_{i}\mathrm{d}S_{i}-p_{i}\mathrm{d}V_{i}
$$
And the equation of state:
... | [
"physgym_0072_sol1"
] | E1_solver_manual_det | physgym | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At time $t=0$, an adiabatic cylinder with a cross-sectional area... | From the proportional relationship between work done and heat absorbed:
$$
\frac{1}{2}\left(p_{1}\mathrm{d}V_{1}+p_{2}\mathrm{d}V_{2}\right)=T_{1}\mathrm{d}S_{1}=T_{2}\mathrm{d}S_{2}
$$
From the first law of thermodynamics:
$$
\mathrm{d}U_{i}=T_{i}\mathrm{d}S_{i}-p_{i}\mathrm{d}V_{i}
$$
And the equation of state:
... | |
E1_solver_manual_det:physgym_0401:turn0 | physgym_0401 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A certain planet has a smooth and elastic surface, perfect for p... | The two components of the initial throw speed are:
$$
v_{r}=k v_{0}\sin\theta\quad,\quad v_{\theta}=k v_{0}\cos\theta
$$
Based on the symmetry of the starting and landing points with respect to the major axis of the elliptical orbit, and combining the definition of the restitution coefficient under the conditions of ... | [
"physgym_0401_sol1"
] | E1_solver_manual_det | physgym | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A certain planet has a smooth and elastic surface, perfect for p... | The two components of the initial throw speed are:
$$
v_{r}=k v_{0}\sin\theta\quad,\quad v_{\theta}=k v_{0}\cos\theta
$$
Based on the symmetry of the starting and landing points with respect to the major axis of the elliptical orbit, and combining the definition of the restitution coefficient under the conditions of ... | |
E1_solver_manual_det:physgym_0653:turn0 | physgym_0653 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A massless insulating string of length $a$ is fixed at one end, ... | If the ring is a superconducting ring, when it moves in a magnetic field, due to the properties of the superconductor, the magnetic flux in the ring remains unchanged, i.e.: $$ L I + B S = B \cdot \pi r^2 $$ Thus, the current in the ring is: $$ \begin{array}{l}{I = \displaystyle\frac{B}{L}(\pi r^{2} - S)}\ = \displayst... | [
"physgym_0653_sol1"
] | E1_solver_manual_det | physgym | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A massless insulating string of length $a$ is fixed at one end, ... | If the ring is a superconducting ring, when it moves in a magnetic field, due to the properties of the superconductor, the magnetic flux in the ring remains unchanged, i.e.: $$ L I + B S = B \cdot \pi r^2 $$ Thus, the current in the ring is: $$ \begin{array}{l}{I = \displaystyle\frac{B}{L}(\pi r^{2} - S)}\ = \displayst... | |
E1_solver_manual_det:physgym_0674:turn0 | physgym_0674 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coin is placed at rest on the edge of a smooth tabletop, with ... | The moment of inertia of the coin about the axis is \[J=\frac14mr^2\] According to the impulse and impulse-momentum theorem, about the center of mass we have \[I+I_N=m v_0, Ir-I_Nr=J\omega_0\] There is also the velocity relation \[v_0=\omega_0 r\] Therefore, we have \[v_0=\frac{8I}{5m}, \omega_0=\frac{8I}{5mr}\] After ... | [
"physgym_0674_sol1"
] | E1_solver_manual_det | physgym | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coin is placed at rest on the edge of a smooth tabletop, with ... | The moment of inertia of the coin about the axis is \[J=\frac14mr^2\] According to the impulse and impulse-momentum theorem, about the center of mass we have \[I+I_N=m v_0, Ir-I_Nr=J\omega_0\] There is also the velocity relation \[v_0=\omega_0 r\] Therefore, we have \[v_0=\frac{8I}{5m}, \omega_0=\frac{8I}{5mr}\] After ... | |
E1_solver_manual_det:book_mech_0003:turn0 | book_mech_0003 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The speed of light is 3.0 × 108 m/s. Convert this to furlongs
pe... | 3.0× 108 m/s
×
(100 cm
1 m
)( 1 inch
2.54 cm
)( 1 foot
12 inches
)
×
(1 yard
3 feet
)( 1 furlong
220 yards
)
×
( 60 s
1 min
)( 60 min
1 hr
)( 24 hr
1 day
)( 14 days
1 forthnight
)
= 1.8× 1012 furlongs/fortnight
ANSWER: 1.8× 1012 furlongs/fortnight | [
"book_mech_0003_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The speed of light is 3.0 × 108 m/s. Convert this to furlongs
pe... | 3.0× 108 m/s
×
(100 cm
1 m
)( 1 inch
2.54 cm
)( 1 foot
12 inches
)
×
(1 yard
3 feet
)( 1 furlong
220 yards
)
×
( 60 s
1 min
)( 60 min
1 hr
)( 24 hr
1 day
)( 14 days
1 forthnight
)
= 1.8× 1012 furlongs/fortnight
ANSWER: 1.8× 1012 furlongs/fortnight | |
E1_solver_manual_det:book_mech_0006:turn0 | book_mech_0006 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
You jump up straight up in the air. When do you have the
greates... | Gravitational energy depends on how far
apart the masses are. When you’re at the top
of your jump, your mass is as far as it’s ever going to be from the mass of the planet, so that’s
when your gravitational energy is the greatest.
Energy is conserved, so your greatest kinetic energy must be when your gravitational ener... | [
"book_mech_0006_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
You jump up straight up in the air. When do you have the
greates... | Gravitational energy depends on how far
apart the masses are. When you’re at the top
of your jump, your mass is as far as it’s ever going to be from the mass of the planet, so that’s
when your gravitational energy is the greatest.
Energy is conserved, so your greatest kinetic energy must be when your gravitational ener... | |
E1_solver_manual_det:book_mech_0008_a:turn0 | book_mech_0008_a | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If weight B moves down by a certain amount, how much
does weight... | The piece of string connected to B has
three vertical parts: one connecting to the ceiling, one between the two pulleys, and one going
down to B. If the part connecting to B lengthens
by, say 2 cm, then this extra 2 cm has to come
out of the other two segments. In other words,
1 cm of string is lost from the part conne... | [
"book_mech_0008_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If weight B moves down by a certain amount, how much
does weight... | The piece of string connected to B has
three vertical parts: one connecting to the ceiling, one between the two pulleys, and one going
down to B. If the part connecting to B lengthens
by, say 2 cm, then this extra 2 cm has to come
out of the other two segments. In other words,
1 cm of string is lost from the part conne... | |
E1_solver_manual_det:book_mech_0008_b:turn0 | book_mech_0008_b | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
What should the ratio of the two weights be if they are to balan... | If they move, one will lose gravitational energy while the other gains gravitational energy.
If A was a boulder and B was a feather, then A
could release more gravitational energy by dropping than B would consume by rising. Likewise if
B was ridiculously heavy compared to A, you’d
release gravitational energy by lettin... | [
"book_mech_0008_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
What should the ratio of the two weights be if they are to balan... | If they move, one will lose gravitational energy while the other gains gravitational energy.
If A was a boulder and B was a feather, then A
could release more gravitational energy by dropping than B would consume by rising. Likewise if
B was ridiculously heavy compared to A, you’d
release gravitational energy by lettin... | |
E1_solver_manual_det:book_mech_0010:turn0 | book_mech_0010 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
For an astronaut sealed inside a space suit, getting rid of body... | To raise her body temperature by six degrees, the amount of energy required will be
6× 200 kJ, or 1200 kJ. Converting to the basic units of joules, this is 1 .2× 106 J. A watt
is a joule per second, so every second, the heat
energy added to her body will be 200 J. That
means that the time it will take for her to die
wi... | [
"book_mech_0010_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
For an astronaut sealed inside a space suit, getting rid of body... | To raise her body temperature by six degrees, the amount of energy required will be
6× 200 kJ, or 1200 kJ. Converting to the basic units of joules, this is 1 .2× 106 J. A watt
is a joule per second, so every second, the heat
energy added to her body will be 200 J. That
means that the time it will take for her to die
wi... | |
E1_solver_manual_det:book_mech_0014:turn0 | book_mech_0014 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A typical balance like the ones used in school classes can be
re... | The mass change in that example came
out to be so small because we were dividing by
c2, which is big. To make it an easily measurable mass change, we’d have to make the decimal
325
place move over by about 8 places on the final result, i.e., make c2 about 108 times smaller. That
means making c about 10 4 times smaller.... | [
"book_mech_0014_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A typical balance like the ones used in school classes can be
re... | The mass change in that example came
out to be so small because we were dividing by
c2, which is big. To make it an easily measurable mass change, we’d have to make the decimal
325
place move over by about 8 places on the final result, i.e., make c2 about 108 times smaller. That
means making c about 10 4 times smaller.... | |
E1_solver_manual_det:book_mech_0015_a:turn0 | book_mech_0015_a | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A free neutron (as opposed to a neutron bound into an
atomic nuc... | In units of 10 −27 kg, the amount of
mass lost is
1.67495− 1.67265− 0.00091 = 0.00139 ,
i.e., 0.00139× 10−27 kg. This mass has been
converted into energy:
E =mc2
= (0.00139× 10−27)(3× 108)2
= 1.25× 10−13 joules .
(In the meter-kilogram-second system of units,
kilograms multiplied velocity squared give units
of joules.)... | [
"book_mech_0015_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A free neutron (as opposed to a neutron bound into an
atomic nuc... | In units of 10 −27 kg, the amount of
mass lost is
1.67495− 1.67265− 0.00091 = 0.00139 ,
i.e., 0.00139× 10−27 kg. This mass has been
converted into energy:
E =mc2
= (0.00139× 10−27)(3× 108)2
= 1.25× 10−13 joules .
(In the meter-kilogram-second system of units,
kilograms multiplied velocity squared give units
of joules.)... | |
E1_solver_manual_det:book_mech_0017:turn0 | book_mech_0017 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
You’re a passenger in the open basket hanging under a hotair bal... | If the balloon is being carried along by the
wind at constant velocity, then it’s at rest with
respect to the air. Only relative motion can
cause a physical effect — absolute motion isn’t
even a well-defined concept. Since there is no
relative motion between the balloon and the air,
there can’t be any physical effect. ... | [
"book_mech_0017_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
You’re a passenger in the open basket hanging under a hotair bal... | If the balloon is being carried along by the
wind at constant velocity, then it’s at rest with
respect to the air. Only relative motion can
cause a physical effect — absolute motion isn’t
even a well-defined concept. Since there is no
relative motion between the balloon and the air,
there can’t be any physical effect. ... | |
E1_solver_manual_det:book_mech_0018:turn0 | book_mech_0018 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Driving along in your car, you take your foot off the gas, and
y... | Suppose you’re being observed by a cop in
a car that was initially next to you and matching
your speed. Let’s say that as you take your foot
off the gas, the cop car continues at the same
speed, so you fall behind. In the cop’s frame
of reference, you were initially at rest, and then
started moving backward, so you wer... | [
"book_mech_0018_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Driving along in your car, you take your foot off the gas, and
y... | Suppose you’re being observed by a cop in
a car that was initially next to you and matching
your speed. Let’s say that as you take your foot
off the gas, the cop car continues at the same
speed, so you fall behind. In the cop’s frame
of reference, you were initially at rest, and then
started moving backward, so you wer... | |
E1_solver_manual_det:book_mech_0020:turn0 | book_mech_0020 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A bullet flies through the air, passes through a paperback
book,... | Force is a transfer of momentum from one
object to another. Equivalently, we can describe force as an interaction between two objects. Whichever way we say it, it takes two to
tango — you can’t have a force unless two objects are involved. There is no force when the
bullet is in the air. There is a force when the
bulle... | [
"book_mech_0020_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A bullet flies through the air, passes through a paperback
book,... | Force is a transfer of momentum from one
object to another. Equivalently, we can describe force as an interaction between two objects. Whichever way we say it, it takes two to
tango — you can’t have a force unless two objects are involved. There is no force when the
bullet is in the air. There is a force when the
bulle... | |
E1_solver_manual_det:book_mech_0024_b:turn0 | book_mech_0024_b | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Two cars with different masses each have the same kinetic
energy... | Since the stopping distances are equal, if you
watched videos of the two boats stopping, the
videos would look exactly the same except for a
rescaling of the time. That is, the slower boat’s
motion would look like the faster one’s, but in
slow motion. Therefore the time for the slower
boat to stop will be greater.
ANSW... | [
"book_mech_0024_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Two cars with different masses each have the same kinetic
energy... | Since the stopping distances are equal, if you
watched videos of the two boats stopping, the
videos would look exactly the same except for a
rescaling of the time. That is, the slower boat’s
motion would look like the faster one’s, but in
slow motion. Therefore the time for the slower
boat to stop will be greater.
ANSW... | |
E1_solver_manual_det:book_mech_0028:turn0 | book_mech_0028 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A physical therapist wants her patient to rehabilitate his injur... | The total torque required is the torque
needed to lift his arm plus the torque needed
to lift the weight. Since his arm is not changing,
she needs to tell him to set things up so that the
torque needed to lift the weight is also the same
as before. Torque is rF sinθ, and since he is decreasing r by a factor of 17/33, s... | [
"book_mech_0028_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A physical therapist wants her patient to rehabilitate his injur... | The total torque required is the torque
needed to lift his arm plus the torque needed
to lift the weight. Since his arm is not changing,
she needs to tell him to set things up so that the
torque needed to lift the weight is also the same
as before. Torque is rF sinθ, and since he is decreasing r by a factor of 17/33, s... | |
E1_solver_manual_det:book_mech_0029:turn0 | book_mech_0029 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object is observed to have constant angular momentum.
Can you... | No, it just means that any torques that do
act are adding up to produce zero total torque.
For instance, a merry-go-round rotating with
constant angular velocity has torques acting on
it, but the motor’s torque and the torque due
to friction are just canceling out to produce zero
total torque. Of course it is also poss... | [
"book_mech_0029_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object is observed to have constant angular momentum.
Can you... | No, it just means that any torques that do
act are adding up to produce zero total torque.
For instance, a merry-go-round rotating with
constant angular velocity has torques acting on
it, but the motor’s torque and the torque due
to friction are just canceling out to produce zero
total torque. Of course it is also poss... | |
E1_solver_manual_det:book_mech_0030:turn0 | book_mech_0030 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Two horizontal tree branches on the same tree have equal
diamete... | For the purpose of calculating gravitational torques, we can treat each branch as if its
weight were concentrated at its center of mass.
The longer branch’s center of mass is twice as far
from the trunk, so the r is doubled. The gravitational force is also twice as much, so τ =rF is
four times greater.
ANSWER: rF is fo... | [
"book_mech_0030_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Two horizontal tree branches on the same tree have equal
diamete... | For the purpose of calculating gravitational torques, we can treat each branch as if its
weight were concentrated at its center of mass.
The longer branch’s center of mass is twice as far
from the trunk, so the r is doubled. The gravitational force is also twice as much, so τ =rF is
four times greater.
ANSWER: rF is fo... | |
E1_solver_manual_det:book_mech_0031_a:turn0 | book_mech_0031_a | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Alice says Cathy’s body has zero momentum, but Bob
says Cathy’s ... | Alice could be in a frame of reference in
which Cathy isn’t moving. For instance, if Alice
and Cathy were in an airplane together, it would
be natural for both of them to adopt the frame
of reference of the plane. Bob could be using the
frame of reference of the dirt.
ANSWER: frame of reference of the dirt.
ANSWER: fra... | [
"book_mech_0031_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Alice says Cathy’s body has zero momentum, but Bob
says Cathy’s ... | Alice could be in a frame of reference in
which Cathy isn’t moving. For instance, if Alice
and Cathy were in an airplane together, it would
be natural for both of them to adopt the frame
of reference of the plane. Bob could be using the
frame of reference of the dirt.
ANSWER: frame of reference of the dirt.
ANSWER: fra... | |
E1_solver_manual_det:book_mech_0031_b:turn0 | book_mech_0031_b | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Alice and Bob agree that Dong’s body has nonzero momentum,
but d... | Your opinions about angular momentum will
depend on what point in space you arbitrarily
chose as the axis. Let’s say Dong is sitting on
some grass. Bob might be using the center of
the spinning Earth is his axis, while Alice uses
the spot Dong is sitting on.
ANSWER: the spot Dong is sitting on.
ANSWER: the spot Dong is... | [
"book_mech_0031_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Alice and Bob agree that Dong’s body has nonzero momentum,
but d... | Your opinions about angular momentum will
depend on what point in space you arbitrarily
chose as the axis. Let’s say Dong is sitting on
some grass. Bob might be using the center of
the spinning Earth is his axis, while Alice uses
the spot Dong is sitting on.
ANSWER: the spot Dong is sitting on.
ANSWER: the spot Dong is... | |
E1_solver_manual_det:book_mech_0032:turn0 | book_mech_0032 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A person of weight W stands on the ball of one foot. Find
the te... | Basically we have a situation with two
equations in two unknowns. There is an equation that says the total torque on the foot is
zero, and another equation that says the total
force on the foot is zero. The two unknowns
are the tension in the calf muscle, T , and the
force exerted by the shinbone, Fs. We can however si... | [
"book_mech_0032_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A person of weight W stands on the ball of one foot. Find
the te... | Basically we have a situation with two
equations in two unknowns. There is an equation that says the total torque on the foot is
zero, and another equation that says the total
force on the foot is zero. The two unknowns
are the tension in the calf muscle, T , and the
force exerted by the shinbone, Fs. We can however si... | |
E1_solver_manual_det:book_rel_0034_a:turn0 | book_rel_0034_a | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Astronauts in three different spaceships are communicating
with ... | She says her own velocity is zero.
Time dilation happens because of relative motion. We can conclude that A and B are not in
motion relative to one another, but that there is
relative motion between them and C.
ANSWER: relative motion between them and C.
ANSWER: relative motion between them and C | [
"book_rel_0034_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Astronauts in three different spaceships are communicating
with ... | She says her own velocity is zero.
Time dilation happens because of relative motion. We can conclude that A and B are not in
motion relative to one another, but that there is
relative motion between them and C.
ANSWER: relative motion between them and C.
ANSWER: relative motion between them and C | |
E1_solver_manual_det:book_em_0041:turn0 | book_em_0041 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Physicist Richard Feynman originated a new way of thinking
about... | Replacing either charge in p/1 with a lightcolored charge moving backward in time gives
a diagram like p/2, which is still an attraction.
Doing something similar to p/4 results in p/3,
which is still a repulsion. It also works the other
way around, e.g., you can do it to the bottom
charge in p/2, which gives something ... | [
"book_em_0041_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Physicist Richard Feynman originated a new way of thinking
about... | Replacing either charge in p/1 with a lightcolored charge moving backward in time gives
a diagram like p/2, which is still an attraction.
Doing something similar to p/4 results in p/3,
which is still a repulsion. It also works the other
way around, e.g., you can do it to the bottom
charge in p/2, which gives something ... | |
E1_solver_manual_det:book_em_0045:turn0 | book_em_0045 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
This book begins and ends with the topic of light. Give an
examp... | The correspondence principle says that
new models have to be backward-compatible
with old ones, i.e., the new model has to explain
successfully all the same observations that the
old model was able to deal with. The beginning of the book used the ray model, but now
we’ve learned about the wave model. The ray
model was ... | [
"book_em_0045_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
This book begins and ends with the topic of light. Give an
examp... | The correspondence principle says that
new models have to be backward-compatible
with old ones, i.e., the new model has to explain
successfully all the same observations that the
old model was able to deal with. The beginning of the book used the ray model, but now
we’ve learned about the wave model. The ray
model was ... | |
E1_solver_manual_det:book_optics_0051:turn0 | book_optics_0051 | OPTICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A man is walking at 1.0 m/s directly towards a flat mirror.
At w... | The image is at an equal distance behind
the mirror, so the distance between the man and
his image is twice his distance from the mirror,
and his separation from his image is reducing at
a rate of 2.0 m/s.
ANSWER: a rate of 2.0 m/s.
ANSWER: a rate of 2.0 m/s | [
"book_optics_0051_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A man is walking at 1.0 m/s directly towards a flat mirror.
At w... | The image is at an equal distance behind
the mirror, so the distance between the man and
his image is twice his distance from the mirror,
and his separation from his image is reducing at
a rate of 2.0 m/s.
ANSWER: a rate of 2.0 m/s.
ANSWER: a rate of 2.0 m/s | |
E1_solver_manual_det:book_optics_0055:turn0 | book_optics_0055 | OPTICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If the user of an astronomical telescope moves her head
closer t... | The magnification is the ratio of the image’s size to the object’s size. It has nothing
to do with the person’s location. The angular
magnification, however, does depend on the person’s location, because things farther away subtend smaller angles. The distance to the actual
object is not changed significantly, since it... | [
"book_optics_0055_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If the user of an astronomical telescope moves her head
closer t... | The magnification is the ratio of the image’s size to the object’s size. It has nothing
to do with the person’s location. The angular
magnification, however, does depend on the person’s location, because things farther away subtend smaller angles. The distance to the actual
object is not changed significantly, since it... | |
E1_solver_manual_det:book_mech_0061_a:turn0 | book_mech_0061_a | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Pendulum 2 has a string twice as long as pendulum 1. If
we defin... | Pendulum 2 has then same total force
on it when it is at the same angle, but when the
333
Problem 7-8.
two pendula are at the same angle, pendulum 2
is at twice the distance from equilibrium. Pendulum 2’s F−x graph will therefore look like
pendulum 1’s, but stretched out twice as wide.
Since k is the slope of this grap... | [
"book_mech_0061_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Pendulum 2 has a string twice as long as pendulum 1. If
we defin... | Pendulum 2 has then same total force
on it when it is at the same angle, but when the
333
Problem 7-8.
two pendula are at the same angle, pendulum 2
is at twice the distance from equilibrium. Pendulum 2’s F−x graph will therefore look like
pendulum 1’s, but stretched out twice as wide.
Since k is the slope of this grap... | |
E1_solver_manual_det:book_mech_0062_c:turn0 | book_mech_0062_c | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following is a graph of the height of a water wave as a
func... | The information about the velocity is irrelevant. We’re
given the wavelength and amplitude directly, so
that’s all we need to show. (The change in velocity implies an additional change in frequency
for the given wavelength, but the wavelength is
1/3 the original, not 1/6.)
original
ANSWER: original | [
"book_mech_0062_c_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following is a graph of the height of a water wave as a
func... | The information about the velocity is irrelevant. We’re
given the wavelength and amplitude directly, so
that’s all we need to show. (The change in velocity implies an additional change in frequency
for the given wavelength, but the wavelength is
1/3 the original, not 1/6.)
original
ANSWER: original | |
E1_solver_manual_det:book_em_0068:turn0 | book_em_0068 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If you add a magnetic field with magnitude 1 T to a field of
mag... | The maximum is 3 T, which is achieved
when the fields are in the same direction. The
minimum is 1 T, when they are in opposite direction. By varying the angle between the vectors
from 0 to 180 degrees, all values in between can
be achieved as well.
ANSWER: be achieved as well.
ANSWER: be achieved as well | [
"book_em_0068_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If you add a magnetic field with magnitude 1 T to a field of
mag... | The maximum is 3 T, which is achieved
when the fields are in the same direction. The
minimum is 1 T, when they are in opposite direction. By varying the angle between the vectors
from 0 to 180 degrees, all values in between can
be achieved as well.
ANSWER: be achieved as well.
ANSWER: be achieved as well | |
E1_solver_manual_det:book_em_0072:turn0 | book_em_0072 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find a vector that is perpendicular to both of the following
two... | The easiest way to find such a vector is to
take the cross product.
(A× B)x =AyBz−ByAz =−3
(A× B)y =AzBx−BzAx = 6
(A× B)z =AxBy−BxAy =−3
335
So our vector is −3ˆx + 6ˆy− 3ˆz. We could also
use any other vector lying along the same line,
i.e., any scalar multiple of this one. I checked my
result by computing its dot pro... | [
"book_em_0072_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find a vector that is perpendicular to both of the following
two... | The easiest way to find such a vector is to
take the cross product.
(A× B)x =AyBz−ByAz =−3
(A× B)y =AzBx−BzAx = 6
(A× B)z =AxBy−BxAy =−3
335
So our vector is −3ˆx + 6ˆy− 3ˆz. We could also
use any other vector lying along the same line,
i.e., any scalar multiple of this one. I checked my
result by computing its dot pro... | |
E1_solver_manual_det:book_em_0074:turn0 | book_em_0074 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The nuclei 3H (hydrogen-3, or tritium) and 3He (helium-3)
have a... | The energy density is proportional to the
square of the field, so if the field patterns are
identical except for a scaling up by a factor of 2,
the energy in 3He is greater by a factor of 4.
ANSWER: the energy in 3He is greater by a factor of 4.
ANSWER: the energy in 3He is greater by a factor of 4 | [
"book_em_0074_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The nuclei 3H (hydrogen-3, or tritium) and 3He (helium-3)
have a... | The energy density is proportional to the
square of the field, so if the field patterns are
identical except for a scaling up by a factor of 2,
the energy in 3He is greater by a factor of 4.
ANSWER: the energy in 3He is greater by a factor of 4.
ANSWER: the energy in 3He is greater by a factor of 4 | |
E1_solver_manual_det:book_em_0077_a:turn0 | book_em_0077_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an electrical storm, an electric field builds up in the space... | The energy per unit volume is
E
V = 18πk|E|2 .
(E is energy, E is electric field.) At the moment
when the lightning strikes, the field equals Ec.
The volume of the air in which the field exists is
L2h.
E = 18πkE2cL2h
ANSWER: 18πkE2cL2h | [
"book_em_0077_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an electrical storm, an electric field builds up in the space... | The energy per unit volume is
E
V = 18πk|E|2 .
(E is energy, E is electric field.) At the moment
when the lightning strikes, the field equals Ec.
The volume of the air in which the field exists is
L2h.
E = 18πkE2cL2h
ANSWER: 18πkE2cL2h | |
E1_solver_manual_det:book_em_0080:turn0 | book_em_0080 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Prove the anticommutative property of the vector cross product, ... | The definition of the vector cross product
in terms of the components is (A×B)x =AyBz−
ByAz, and similarly for y and z. Reversing the
roles of A and B interchanges the positive and
negative terms on the right, so the sign flips.
ANSWER: negative terms on the right, so the sign flips.
ANSWER: negative terms on the right... | [
"book_em_0080_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Prove the anticommutative property of the vector cross product, ... | The definition of the vector cross product
in terms of the components is (A×B)x =AyBz−
ByAz, and similarly for y and z. Reversing the
roles of A and B interchanges the positive and
negative terms on the right, so the sign flips.
ANSWER: negative terms on the right, so the sign flips.
ANSWER: negative terms on the right... | |
E1_solver_manual_det:book_em_0082:turn0 | book_em_0082 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Let a and b be any two numbers (not both zero), and let
u = aˆx ... | Let v = cˆx +dˆy, where c and d are to
be found. We don’t expect to be able to determine both c and d uniquely, because for any v
that works, so will any nonzero scalar multiple
of it. That is, we only expect to be able to find
the ratio c/d, which expresses how big c is in
proportion to d.
We want v to be perpendicula... | [
"book_em_0082_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Let a and b be any two numbers (not both zero), and let
u = aˆx ... | Let v = cˆx +dˆy, where c and d are to
be found. We don’t expect to be able to determine both c and d uniquely, because for any v
that works, so will any nonzero scalar multiple
of it. That is, we only expect to be able to find
the ratio c/d, which expresses how big c is in
proportion to d.
We want v to be perpendicula... | |
E1_solver_manual_det:book_em_0086_b:turn0 | book_em_0086_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The gravitational field g of a typical asteroid is so weak that ... | Plugging in gives 83 m. This is comparable to the asteroid’s “football-field” size, so the
approximation ofg as a constant may not be correct. Our result should probably be taken as a
lower bound.
ANSWER: lower bound.
ANSWER: lower bound | [
"book_em_0086_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The gravitational field g of a typical asteroid is so weak that ... | Plugging in gives 83 m. This is comparable to the asteroid’s “football-field” size, so the
approximation ofg as a constant may not be correct. Our result should probably be taken as a
lower bound.
ANSWER: lower bound.
ANSWER: lower bound | |
E1_solver_manual_det:book_em_0087_a:turn0 | book_em_0087_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an inkjet printer, a series of droplets of ink, each with
mas... | Let thex axis be in the initial direction
of motion andy the direction of the electric field.
Newton’s laws tell us that thex andy motion are
independent, so the x velocity is constant, and
the time taken to pass through the capacitor is
t = w/u. The final velocity in the y diection is
v = at = (q/m)Et = qEw/mu. The de... | [
"book_em_0087_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In an inkjet printer, a series of droplets of ink, each with
mas... | Let thex axis be in the initial direction
of motion andy the direction of the electric field.
Newton’s laws tell us that thex andy motion are
independent, so the x velocity is constant, and
the time taken to pass through the capacitor is
t = w/u. The final velocity in the y diection is
v = at = (q/m)Et = qEw/mu. The de... | |
E1_solver_manual_det:book_em_0088_b:turn0 | book_em_0088_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If this is done with two different objects and they are observed... | You can conclude that they have the same
ratio of q/m, but you can’t tell whether they
have the sameq and the samem. One could have
twice the charge and twice the mass compared to
the other, for instance.
ANSWER: the other, for instance.
ANSWER: the other, for instance | [
"book_em_0088_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
If this is done with two different objects and they are observed... | You can conclude that they have the same
ratio of q/m, but you can’t tell whether they
have the sameq and the samem. One could have
twice the charge and twice the mass compared to
the other, for instance.
ANSWER: the other, for instance.
ANSWER: the other, for instance | |
E1_solver_manual_det:book_em_0092:turn0 | book_em_0092 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Three charges are arranged on a square as shown. All three
charg... | Let the square’s sides be of length a. The
field at the center is the vector sum of the fields
that would have been produced individually by
the three charges. Each of these individual fields
iskq/r2, wherer1 =a/√2 for the two chargesq1,
andr2 =a/2 forq2. Vector addition can be done
by adding components. Letx be horizo... | [
"book_em_0092_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Three charges are arranged on a square as shown. All three
charg... | Let the square’s sides be of length a. The
field at the center is the vector sum of the fields
that would have been produced individually by
the three charges. Each of these individual fields
iskq/r2, wherer1 =a/√2 for the two chargesq1,
andr2 =a/2 forq2. Vector addition can be done
by adding components. Letx be horizo... | |
E1_solver_manual_det:book_em_0093_a:turn0 | book_em_0093_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Use Gauss’ law to find the field inside an infinite cylinder
wit... | The setup is similar to the preceding problem. We again choose as our Gaussian
surface a finite cylinder of length 𝓁 and radius
R. Gauss’s law gives 2 πR𝓁E = 4πkqin, where
qin = πR2𝓁ρ. Solving for the field, we have
E = 2πkRρ.
ANSWER: 2πkRρ | [
"book_em_0093_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Use Gauss’ law to find the field inside an infinite cylinder
wit... | The setup is similar to the preceding problem. We again choose as our Gaussian
surface a finite cylinder of length 𝓁 and radius
R. Gauss’s law gives 2 πR𝓁E = 4πkqin, where
qin = πR2𝓁ρ. Solving for the field, we have
E = 2πkRρ.
ANSWER: 2πkRρ | |
E1_solver_manual_det:book_modern_0097_a:turn0 | book_modern_0097_a | MODERN_PHYSICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A carbon dioxide has a symmetrical, straight shape, with
the car... | The total force on the carbon is zero by
symmetry. Let positive be to the right, and consider the forces acting on the oxygen on the right.
The total force is kqQ/𝓁2 +kQ2/(2𝓁)2, where 𝓁
is the spacing between the atoms. Setting this
to zero gives qQ +Q2/4 = 0, or q/Q =−1/4.
ANSWER: −1/4 | [
"book_modern_0097_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A carbon dioxide has a symmetrical, straight shape, with
the car... | The total force on the carbon is zero by
symmetry. Let positive be to the right, and consider the forces acting on the oxygen on the right.
The total force is kqQ/𝓁2 +kQ2/(2𝓁)2, where 𝓁
is the spacing between the atoms. Setting this
to zero gives qQ +Q2/4 = 0, or q/Q =−1/4.
ANSWER: −1/4 | |
E1_solver_manual_det:book_modern_0097_c:turn0 | book_modern_0097_c | MODERN_PHYSICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A carbon dioxide has a symmetrical, straight shape, with
the car... | Counting all three pairs, the total electrical potential energy is kQ2/2𝓁 +kqQ/𝓁 +kqQ/𝓁,
which comes out to be zero. This makes sense
because if we moved the oxygen atoms outward
symmetrically at the same time, the force on
them would always be zero, and therefore we
could bring them to infinity with zero work. But
... | [
"book_modern_0097_c_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A carbon dioxide has a symmetrical, straight shape, with
the car... | Counting all three pairs, the total electrical potential energy is kQ2/2𝓁 +kqQ/𝓁 +kqQ/𝓁,
which comes out to be zero. This makes sense
because if we moved the oxygen atoms outward
symmetrically at the same time, the force on
them would always be zero, and therefore we
could bring them to infinity with zero work. But
... | |
E1_solver_manual_det:book_modern_0100:turn0 | book_modern_0100 | MODERN_PHYSICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A helium atom finds itself momentarily in this arrangement.
Find... | Let’s use positive numbers for forces to
the right, and negative numbers for forces to
the left. There are two forces on the righthand electron: a repulsive force from the other
electron, and an attractive force from the nucleus. These will be represented as positive and
negative numbers, respectively. The charges of
t... | [
"book_modern_0100_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A helium atom finds itself momentarily in this arrangement.
Find... | Let’s use positive numbers for forces to
the right, and negative numbers for forces to
the left. There are two forces on the righthand electron: a repulsive force from the other
electron, and an attractive force from the nucleus. These will be represented as positive and
negative numbers, respectively. The charges of
t... | |
E1_solver_manual_det:book_em_0104:turn0 | book_em_0104 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the following four potentials, which exist in some
regi... | In each case, the electric field is minus the
derivative of the potential.
Potentials 1 and 2 give the same field,E =−a.
These are uniform fields. The constant a is the
strength of the field, and b is just an arbitrary
constant with no physical significance.
Potentials 3 and 4 both give E = a/x2. This
looks like the fi... | [
"book_em_0104_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the following four potentials, which exist in some
regi... | In each case, the electric field is minus the
derivative of the potential.
Potentials 1 and 2 give the same field,E =−a.
These are uniform fields. The constant a is the
strength of the field, and b is just an arbitrary
constant with no physical significance.
Potentials 3 and 4 both give E = a/x2. This
looks like the fi... | |
E1_solver_manual_det:book_em_0106:turn0 | book_em_0106 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The voltage difference between the ends of a AAA battery is
1.5 ... | When a function has a constant derivative, we don’t really need calculus and can find
∆.../ ∆... , which is the same as d.../... Ignoring signs, the field is
E = (ignored−)∆φ∆x
= 1.5 V4.2× 10−2 m
= 36 V/m.
ANSWER: 36 V/m | [
"book_em_0106_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The voltage difference between the ends of a AAA battery is
1.5 ... | When a function has a constant derivative, we don’t really need calculus and can find
∆.../ ∆... , which is the same as d.../... Ignoring signs, the field is
E = (ignored−)∆φ∆x
= 1.5 V4.2× 10−2 m
= 36 V/m.
ANSWER: 36 V/m | |
E1_solver_manual_det:book_em_0112:turn0 | book_em_0112 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 8, p. 59, we found that the field of a uniform,
strai... | V =−
E dr
=−2kλ lnr + const.
This is a little bit of a strange-looking expression,
because normally we can’t evaluate a transcendental function on an input that has units. However, we can get away with this in the case of a
log, because ln(a/b) = lna−lnb, so if we wanted
to make the units not look so wrong, we could
wr... | [
"book_em_0112_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 8, p. 59, we found that the field of a uniform,
strai... | V =−
E dr
=−2kλ lnr + const.
This is a little bit of a strange-looking expression,
because normally we can’t evaluate a transcendental function on an input that has units. However, we can get away with this in the case of a
log, because ln(a/b) = lna−lnb, so if we wanted
to make the units not look so wrong, we could
wr... | |
E1_solver_manual_det:book_em_0113_d:turn0 | book_em_0113_d | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that the potential had instead been
φ = a(x2 + y2 + z2) ... | It would have had no effect. Mathematically,
the gradient operator ∇ is a kind of derivative,
and derivatives are linear and have the property
that the derivative of a constant is zero. Therefore adding a constant onto an expression never
has any effect on its derivative. Physically, the
potentialφ is a measure of pote... | [
"book_em_0113_d_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose that the potential had instead been
φ = a(x2 + y2 + z2) ... | It would have had no effect. Mathematically,
the gradient operator ∇ is a kind of derivative,
and derivatives are linear and have the property
that the derivative of a constant is zero. Therefore adding a constant onto an expression never
has any effect on its derivative. Physically, the
potentialφ is a measure of pote... | |
E1_solver_manual_det:book_em_0116:turn0 | book_em_0116 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Three charges, each of strength Q (Q > 0) form a fixed
equilater... | The basic idea is to find the voltage V
at the center of the triangle (defining the voltage infinitely far away to be zero, as usual). We
then use conservation of energy to find the minimum initial kinetic energy, which corresponds to
some minimum initial speed. A little trig shows
that the distance r from the center t... | [
"book_em_0116_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Three charges, each of strength Q (Q > 0) form a fixed
equilater... | The basic idea is to find the voltage V
at the center of the triangle (defining the voltage infinitely far away to be zero, as usual). We
then use conservation of energy to find the minimum initial kinetic energy, which corresponds to
some minimum initial speed. A little trig shows
that the distance r from the center t... | |
E1_solver_manual_det:book_em_0118_a:turn0 | book_em_0118_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coaxial cable consists of a cylindrical inner conductor
with r... | This is similar to the example of the
spherical capacitor in section 3.6. The electric
field in this geometry was found in section 2.8 to
be proportional to 1/r, by applying Gauss’s law
to a cylindrical Gaussian surface: as the cylinder’s radius grows, its area grows like r to the
first power, so the field must fall of... | [
"book_em_0118_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A coaxial cable consists of a cylindrical inner conductor
with r... | This is similar to the example of the
spherical capacitor in section 3.6. The electric
field in this geometry was found in section 2.8 to
be proportional to 1/r, by applying Gauss’s law
to a cylindrical Gaussian surface: as the cylinder’s radius grows, its area grows like r to the
first power, so the field must fall of... | |
E1_solver_manual_det:book_em_0118_b:turn0 | book_em_0118_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that your answer makes sense when a = b and when b < a.
| When b = a, we have ln( b/a) = 0, and
E =∞. This makes sense because the only way
to have a finite voltage change over an infinitesimal distance is if the field is infinite. When
b < a, the log is negative, which flips the sign
of the field, and that makes sense, because it’s
like we’ve switched the electrodes around.
... | [
"book_em_0118_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Show that your answer makes sense when a = b and when b < a.
| When b = a, we have ln( b/a) = 0, and
E =∞. This makes sense because the only way
to have a finite voltage change over an infinitesimal distance is if the field is infinite. When
b < a, the log is negative, which flips the sign
of the field, and that makes sense, because it’s
like we’ve switched the electrodes around.
... | |
E1_solver_manual_det:book_em_0124:turn0 | book_em_0124 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particle with a charge of 1.0 C and a mass of 1.0 kg is
observ... | The electric force vector is
FE =qE
= (1 N)ˆy .
The magnitude of the magnetic force is
|FB| =|qv× B|
=q|v× B|
=q|v||B| sinθvB
= 2 N ,
and its direction is determined by the right hand
rule to be along the z axis, so
FB = (2 N)ˆz .
The total force on the particle is
F = (1 N)ˆy + (2 N)ˆz .
ANSWER: (1 N)ˆy + (2 N)ˆz | [
"book_em_0124_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particle with a charge of 1.0 C and a mass of 1.0 kg is
observ... | The electric force vector is
FE =qE
= (1 N)ˆy .
The magnitude of the magnetic force is
|FB| =|qv× B|
=q|v× B|
=q|v||B| sinθvB
= 2 N ,
and its direction is determined by the right hand
rule to be along the z axis, so
FB = (2 N)ˆz .
The total force on the particle is
F = (1 N)ˆy + (2 N)ˆz .
ANSWER: (1 N)ˆy + (2 N)ˆz | |
E1_solver_manual_det:book_em_0125:turn0 | book_em_0125 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose a charged particle is moving through a region of
space i... | The electric force is either in the same direction as the electric field or in the opposite direction, depending on whether the particles are
positively or negatively charged. Since the magnetic force is perpendicular to both v and B
and the question states that the electric field is
perpendicular to both v and B, the ... | [
"book_em_0125_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Suppose a charged particle is moving through a region of
space i... | The electric force is either in the same direction as the electric field or in the opposite direction, depending on whether the particles are
positively or negatively charged. Since the magnetic force is perpendicular to both v and B
and the question states that the electric field is
perpendicular to both v and B, the ... | |
E1_solver_manual_det:book_em_0126:turn0 | book_em_0126 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle is released from rest. We see it start to
mov... | There must be either E or B or both,
since some force caused the particle to accelerate. It can’t be a pure magnetic field, because
F = qv× B is zero for v = 0, so the particle
would not have accelerated from rest. It can’t
be a pure electric field, because a uniform electric field would just have caused the particle t... | [
"book_em_0126_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A charged particle is released from rest. We see it start to
mov... | There must be either E or B or both,
since some force caused the particle to accelerate. It can’t be a pure magnetic field, because
F = qv× B is zero for v = 0, so the particle
would not have accelerated from rest. It can’t
be a pure electric field, because a uniform electric field would just have caused the particle t... | |
E1_solver_manual_det:book_em_0128:turn0 | book_em_0128 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | The second data-point isn’t very surprising, because multiplying charge by −2 and the
velocity by−1 multiplies the magnetic force by
(−2)(−1) = 2. The only thing the second datapoint really tells us is that the field is purely
magnetic. If the F, v, and B vectors had to
be perpendicular, there would only be one possibi... | [
"book_em_0128_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | The second data-point isn’t very surprising, because multiplying charge by −2 and the
velocity by−1 multiplies the magnetic force by
(−2)(−1) = 2. The only thing the second datapoint really tells us is that the field is purely
magnetic. If the F, v, and B vectors had to
be perpendicular, there would only be one possibi... | |
E1_solver_manual_det:book_em_0129:turn0 | book_em_0129 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | It can’t be a purely electric field, because
then the velocity would have no effect on the
345
force. It can’t be purely magnetic, either, because tripling the velocity didn’t simply triple
the force, as it would if the only force was
FB =qv× B.
ANSWER: qv× B | [
"book_em_0129_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The following data give the results of two experiments in which
... | It can’t be a purely electric field, because
then the velocity would have no effect on the
345
force. It can’t be purely magnetic, either, because tripling the velocity didn’t simply triple
the force, as it would if the only force was
FB =qv× B.
ANSWER: qv× B | |
E1_solver_manual_det:book_em_0135_a:turn0 | book_em_0135_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electric dipole is composed of charges ±q (q >0), each
with m... | The maximum kinetic energy will occur when the electrical potential energy of the
dipole is at a minimum, which occurs when the
dipole is pointed in the same direction as the
field, i.e., at 90 degrees compared to its initial
orientation. Conservation of energy gives
Ui +Ki =Uf +Kf
0 =−q𝓁E + 12Iω 2.
The moment of iner... | [
"book_em_0135_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electric dipole is composed of charges ±q (q >0), each
with m... | The maximum kinetic energy will occur when the electrical potential energy of the
dipole is at a minimum, which occurs when the
dipole is pointed in the same direction as the
field, i.e., at 90 degrees compared to its initial
orientation. Conservation of energy gives
Ui +Ki =Uf +Kf
0 =−q𝓁E + 12Iω 2.
The moment of iner... | |
E1_solver_manual_det:book_em_0135_b:turn0 | book_em_0135_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electric dipole is composed of charges ±q (q >0), each
with m... | The square root can have either sign. This
represents the physical fact that the dipole will
oscillate indefinitely back and forth, achieving
the maximum|ω| over and over again, but with
opposite signs.
ANSWER: opposite signs.
ANSWER: opposite signs | [
"book_em_0135_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electric dipole is composed of charges ±q (q >0), each
with m... | The square root can have either sign. This
represents the physical fact that the dipole will
oscillate indefinitely back and forth, achieving
the maximum|ω| over and over again, but with
opposite signs.
ANSWER: opposite signs.
ANSWER: opposite signs | |
E1_solver_manual_det:book_em_0136:turn0 | book_em_0136 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electron has a nonzero magnetic dipole moment, which is
align... | The distant field of a dipole goes liker−3.
Since B∝r−3, we have
B(10r)
B(r) =
(10r
r
)−3
= 10−3.
Since we’re notating B(r) as B, the field at the
more distant point is 10−3B.
ANSWER: more distant point is 10−3B.
ANSWER: more distant point is 10−3B | [
"book_em_0136_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An electron has a nonzero magnetic dipole moment, which is
align... | The distant field of a dipole goes liker−3.
Since B∝r−3, we have
B(10r)
B(r) =
(10r
r
)−3
= 10−3.
Since we’re notating B(r) as B, the field at the
more distant point is 10−3B.
ANSWER: more distant point is 10−3B.
ANSWER: more distant point is 10−3B | |
E1_solver_manual_det:book_em_0137_a:turn0 | book_em_0137_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A free neutron (as opposed to a neutron bound into an
atomic nuc... | In units of 10 −27 kg, the amount of
mass lost is
1.67495− 1.67265− 0.00091 = 0.00139 ,
i.e., 0.00139× 10−27 kg. This mass has been
converted into energy:
E =mc2
= (0.00139× 10−27)(3× 108)2
= 1.25× 10−13 joules .
(In the meter-kilogram-second system of units,
kilograms multiplied velocity squared give units
of joules.)... | [
"book_em_0137_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A free neutron (as opposed to a neutron bound into an
atomic nuc... | In units of 10 −27 kg, the amount of
mass lost is
1.67495− 1.67265− 0.00091 = 0.00139 ,
i.e., 0.00139× 10−27 kg. This mass has been
converted into energy:
E =mc2
= (0.00139× 10−27)(3× 108)2
= 1.25× 10−13 joules .
(In the meter-kilogram-second system of units,
kilograms multiplied velocity squared give units
of joules.)... | |
E1_solver_manual_det:book_em_0139_b:turn0 | book_em_0139_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Verify, as claimed on p. 132, that if the only unitful quantitie... | We already know that the energy density is
equal to E2/k times a unitless constant. Therefore E2/k also has the right units to be a pressure. We could worry about whether there is
some other proportionality we could form from
these quantities, with different exponents, that
would also have units of pressure. But this c... | [
"book_em_0139_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Verify, as claimed on p. 132, that if the only unitful quantitie... | We already know that the energy density is
equal to E2/k times a unitless constant. Therefore E2/k also has the right units to be a pressure. We could worry about whether there is
some other proportionality we could form from
these quantities, with different exponents, that
would also have units of pressure. But this c... | |
E1_solver_manual_det:book_em_0141:turn0 | book_em_0141 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a particular point in time and space, an electromagnetic
plan... | Since it’s a plane wave, the direction of
the magnetic field is perpendicular to the electric field and the direction of propagation, so it
must be in the ±y direction. What gives E× B
in the positive z direction is if the magnetic field
is in the +y direction.
The Poynting vector is S = (c2/4πk)E× B.
Because E and B a... | [
"book_em_0141_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a particular point in time and space, an electromagnetic
plan... | Since it’s a plane wave, the direction of
the magnetic field is perpendicular to the electric field and the direction of propagation, so it
must be in the ±y direction. What gives E× B
in the positive z direction is if the magnetic field
is in the +y direction.
The Poynting vector is S = (c2/4πk)E× B.
Because E and B a... | |
E1_solver_manual_det:book_em_0142:turn0 | book_em_0142 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The solar constant is defined as the average flux of electromagn... | The given information is the Poynting vector, and we need to find the momentum. Comparing the expressions for the Poynting vector
and the momentum density, we find
p
v =c−2S.
The result is p = 15 kg·m/s.
ANSWER: 15 kg·m/s | [
"book_em_0142_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The solar constant is defined as the average flux of electromagn... | The given information is the Poynting vector, and we need to find the momentum. Comparing the expressions for the Poynting vector
and the momentum density, we find
p
v =c−2S.
The result is p = 15 kg·m/s.
ANSWER: 15 kg·m/s | |
E1_solver_manual_det:book_em_0143:turn0 | book_em_0143 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Our intuition is that if we combine two beams of light into a
si... | The Poynting vector of the superimposed
waves is (E1+E2)×(B1+B2). If we multiply this
expression out, we get four terms. Two of these
are just S1 and S2, the Poynting vectors of the
separate waves. Now suppose the polarization
of the two waves differs by an angle θ. Because
they are propagating in the same direction, t... | [
"book_em_0143_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Our intuition is that if we combine two beams of light into a
si... | The Poynting vector of the superimposed
waves is (E1+E2)×(B1+B2). If we multiply this
expression out, we get four terms. Two of these
are just S1 and S2, the Poynting vectors of the
separate waves. Now suppose the polarization
of the two waves differs by an angle θ. Because
they are propagating in the same direction, t... | |
E1_solver_manual_det:book_rel_0145:turn0 | book_rel_0145 | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
What happens in the equation for G when you put in a negative
nu... | The equation for gamma isγ = 1/
1−v2.
Ifv is negative,v2 still comes out positive. Negative and positive values ofv give the same results
for gamma.
This makes sense, because positive and negative values ofv indicate motion in opposite directions. Relativistic effects like time dilation and
length contraction shouldn’t... | [
"book_rel_0145_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
What happens in the equation for G when you put in a negative
nu... | The equation for gamma isγ = 1/
1−v2.
Ifv is negative,v2 still comes out positive. Negative and positive values ofv give the same results
for gamma.
This makes sense, because positive and negative values ofv indicate motion in opposite directions. Relativistic effects like time dilation and
length contraction shouldn’t... | |
E1_solver_manual_det:book_rel_0146_b:turn0 | book_rel_0146_b | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The Voyager 1 space probe, launched in 1977, is moving faster
re... | Gamma gives us a comparison of the rate
at which time flows on earth and on the probe. A
gamma of one would indicate equal rates. Since
the gamma is a tiny bit greater than one, the
ratio is a tiny bit different from a one-to-one ratio. Gamma differs from one by 1 .6× 10−9, i.e.,
1.6 parts per billion. The disagreement... | [
"book_rel_0146_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The Voyager 1 space probe, launched in 1977, is moving faster
re... | Gamma gives us a comparison of the rate
at which time flows on earth and on the probe. A
gamma of one would indicate equal rates. Since
the gamma is a tiny bit greater than one, the
ratio is a tiny bit different from a one-to-one ratio. Gamma differs from one by 1 .6× 10−9, i.e.,
1.6 parts per billion. The disagreement... | |
E1_solver_manual_det:book_rel_0147:turn0 | book_rel_0147 | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The earth is orbiting the sun, and therefore is contracted
relat... | Let r = 6.4× 106 m be the radius of the
earth, R = 1.5× 1011 m the radius of its orbit,
and T = 1 year = 3 .2× 107 s the period of its
orbit. Then the velocity is v = (2πR)/T , and we
have γ = 1/
1−v2/c2 = 1 + 4.8× 10−9. The
resulting contraction of the diameter is about
(4.8× 10−9)(2r) = 6 cm.
ANSWER: 6 cm | [
"book_rel_0147_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The earth is orbiting the sun, and therefore is contracted
relat... | Let r = 6.4× 106 m be the radius of the
earth, R = 1.5× 1011 m the radius of its orbit,
and T = 1 year = 3 .2× 107 s the period of its
orbit. Then the velocity is v = (2πR)/T , and we
have γ = 1/
1−v2/c2 = 1 + 4.8× 10−9. The
resulting contraction of the diameter is about
(4.8× 10−9)(2r) = 6 cm.
ANSWER: 6 cm | |
E1_solver_manual_det:book_rel_0148_b:turn0 | book_rel_0148_b | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find another value of v for which γ is a simple fraction.
| The (3/5)c example came out simple because 3, 4, and 5 are integers that have the relationship 32 + 42 = 52. This is just like a 3-4-5
right triangle, but there’s no triangle involved
here! We need to find another set of numbers
like this. The simplest way is to reverse the role
of the 3 and 4, in which case we get
v =... | [
"book_rel_0148_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find another value of v for which γ is a simple fraction.
| The (3/5)c example came out simple because 3, 4, and 5 are integers that have the relationship 32 + 42 = 52. This is just like a 3-4-5
right triangle, but there’s no triangle involved
here! We need to find another set of numbers
like this. The simplest way is to reverse the role
of the 3 and 4, in which case we get
v =... | |
E1_solver_manual_det:book_em_0157:turn0 | book_em_0157 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
We have referred to resistors dissipating heat, i.e., we have
as... | To get negative values ofP , you’d have to
have opposite signs forI and ∆V . But you can’t
control the signs of I and ∆V independently —
switching the wires reverses both. The current
flows in response to the electric force, whose direction is related to the direction of the voltage
drop. Another way of looking at it i... | [
"book_em_0157_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
We have referred to resistors dissipating heat, i.e., we have
as... | To get negative values ofP , you’d have to
have opposite signs forI and ∆V . But you can’t
control the signs of I and ∆V independently —
switching the wires reverses both. The current
flows in response to the electric force, whose direction is related to the direction of the voltage
drop. Another way of looking at it i... | |
E1_solver_manual_det:book_em_0165_a:turn0 | book_em_0165_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A student in a biology lab is given the following instructions: ... | They have correctly set up the CE and ND
in parallel. The ammeter is measuring the total
current out of the power supply, however, rather
than just the current that is going to go through
the ND. The voltmeter is hooked up in series, but
it should be in parallel. A voltmeter has a very
high resistance, so the circuit w... | [
"book_em_0165_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A student in a biology lab is given the following instructions: ... | They have correctly set up the CE and ND
in parallel. The ammeter is measuring the total
current out of the power supply, however, rather
than just the current that is going to go through
the ND. The voltmeter is hooked up in series, but
it should be in parallel. A voltmeter has a very
high resistance, so the circuit w... | |
E1_solver_manual_det:book_em_0166_b:turn0 | book_em_0166_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A 1.0 Ω toaster and a 2.0 Ω lamp are connected in parallel with
... | The E-shaped piece of wire across the top
is one big piece of conducting material, so it
must have the same voltage throughout it. Similarly, the whole E-shaped piece on the bottom
is at the same voltage. Therefore, the voltage difference across each of the three components is the same. Since we know the supply
is 110 ... | [
"book_em_0166_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A 1.0 Ω toaster and a 2.0 Ω lamp are connected in parallel with
... | The E-shaped piece of wire across the top
is one big piece of conducting material, so it
must have the same voltage throughout it. Similarly, the whole E-shaped piece on the bottom
is at the same voltage. Therefore, the voltage difference across each of the three components is the same. Since we know the supply
is 110 ... | |
E1_solver_manual_det:book_em_0166_c:turn0 | book_em_0166_c | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A 1.0 Ω toaster and a 2.0 Ω lamp are connected in parallel with
... | In this setup, which is not what’s really used
in your house, it’s not the voltage drops that are
equal but the currents — whatever current goes
through one component also has to go through
the others. The two resistances in series add together to make a total resistance of 3 Ohms, so
Itoaster = Ilamp = V/3 Ω ≈ 40 A. O... | [
"book_em_0166_c_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A 1.0 Ω toaster and a 2.0 Ω lamp are connected in parallel with
... | In this setup, which is not what’s really used
in your house, it’s not the voltage drops that are
equal but the currents — whatever current goes
through one component also has to go through
the others. The two resistances in series add together to make a total resistance of 3 Ohms, so
Itoaster = Ilamp = V/3 Ω ≈ 40 A. O... | |
E1_solver_manual_det:book_em_0172:turn0 | book_em_0172 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the current drawn from the battery.
| The resistor on the left looks different
from the others, but actually it’s connected to
exactly the same things they are. The whole circuit is just four resistors in parallel, but drawn in
a funny way. By the loop rule, the voltage across
each of the resistors is the same as the battery’s
voltageV , and therefore each... | [
"book_em_0172_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the current drawn from the battery.
| The resistor on the left looks different
from the others, but actually it’s connected to
exactly the same things they are. The whole circuit is just four resistors in parallel, but drawn in
a funny way. By the loop rule, the voltage across
each of the resistors is the same as the battery’s
voltageV , and therefore each... | |
E1_solver_manual_det:book_em_0173:turn0 | book_em_0173 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The bulbs are all identical. Which one doesn’t light up?
| This is a problem that has a trick to it.
If you don’t think of the trick, it’s very difficult,
because the circuit can’t be broken down into
parallel and series parts. The trick is to realize that although the circuit has been drawn in
an asymmetric way, it’s actually very symmetric. The two rightmost bulbs play ident... | [
"book_em_0173_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The bulbs are all identical. Which one doesn’t light up?
| This is a problem that has a trick to it.
If you don’t think of the trick, it’s very difficult,
because the circuit can’t be broken down into
parallel and series parts. The trick is to realize that although the circuit has been drawn in
an asymmetric way, it’s actually very symmetric. The two rightmost bulbs play ident... | |
E1_solver_manual_det:book_em_0174:turn0 | book_em_0174 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The heating element of an electric stove is connected in series
... | This setup could be made to work just
fine at the extreme settings. Let Rk be the (variable) resistance of the knob, and Rh the resistance of the heater.
The “off” position would have to have Rk =
∞, i.e., it would be like an open circuit. The
total resistance R = Rk +Rh would be infinite
as well, zero current would fl... | [
"book_em_0174_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The heating element of an electric stove is connected in series
... | This setup could be made to work just
fine at the extreme settings. Let Rk be the (variable) resistance of the knob, and Rh the resistance of the heater.
The “off” position would have to have Rk =
∞, i.e., it would be like an open circuit. The
total resistance R = Rk +Rh would be infinite
as well, zero current would fl... | |
E1_solver_manual_det:book_em_0181:turn0 | book_em_0181 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A Helmholtz coil is defined as a pair of identical circular coil... | The field at any point on the axis can be
found by adding the contributions to the field
from the two loops. The center of the entire apparatus does not, however, coincide with the center of either loop: in each case it is out of that
plane by a distance h/2. By symmetry, both
these fields are along the axis, so we can... | [
"book_em_0181_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A Helmholtz coil is defined as a pair of identical circular coil... | The field at any point on the axis can be
found by adding the contributions to the field
from the two loops. The center of the entire apparatus does not, however, coincide with the center of either loop: in each case it is out of that
plane by a distance h/2. By symmetry, both
these fields are along the axis, so we can... | |
E1_solver_manual_det:book_em_0183:turn0 | book_em_0183 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Four long wires are arranged, as shown, so that their crosssecti... | The hard part is visualizing the directions of the four fields that are adding together.
Looking along the length of the apparatus, two
of the fields point at a 45-degree angle down
and to the left, while the other two point at
45 degrees down and to the right. By vector
addition, this means that the total field is 2
2... | [
"book_em_0183_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Four long wires are arranged, as shown, so that their crosssecti... | The hard part is visualizing the directions of the four fields that are adding together.
Looking along the length of the apparatus, two
of the fields point at a 45-degree angle down
and to the left, while the other two point at
45 degrees down and to the right. By vector
addition, this means that the total field is 2
2... | |
E1_solver_manual_det:book_em_0184_a:turn0 | book_em_0184_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
This problem will lead you through the steps of applying the
Bio... | The Biot-Savart law gives
B = kIc2
∫ d𝓁× r
r3
Here r is the vector from a point on the wire
to the point at which we’re calculating the field.
Let the wire lie on thex axis, and let the point of
interest be on the y axis at y =R. Let θ be the
angle between the r vector and the wire, measured on the right side so that ... | [
"book_em_0184_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
This problem will lead you through the steps of applying the
Bio... | The Biot-Savart law gives
B = kIc2
∫ d𝓁× r
r3
Here r is the vector from a point on the wire
to the point at which we’re calculating the field.
Let the wire lie on thex axis, and let the point of
interest be on the y axis at y =R. Let θ be the
angle between the r vector and the wire, measured on the right side so that ... | |
E1_solver_manual_det:book_em_0192_a:turn0 | book_em_0192_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 6, p. 101, we analyzed a spherical capacitor.
Buildin... | Doing this using the relation U =
q2/2C would be fairly complicated, since the energy density is not constant. We would have to
find the energy stored in a spherical shell betweenr andr +dr, and then integrate to findU.
It’s much easier to do this using C =q/V , since
we already know how to relate q andV from the
analy... | [
"book_em_0192_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In example 6, p. 101, we analyzed a spherical capacitor.
Buildin... | Doing this using the relation U =
q2/2C would be fairly complicated, since the energy density is not constant. We would have to
find the energy stored in a spherical shell betweenr andr +dr, and then integrate to findU.
It’s much easier to do this using C =q/V , since
we already know how to relate q andV from the
analy... | |
E1_solver_manual_det:book_em_0192_b:turn0 | book_em_0192_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In the limit where the gap between a and b is very small, show
t... | In the notation of the parallel-plate capacitor
example, we have A = 4πa2 (or 5πb2, it doesn’t
matter becausea andb are nearly the same) and
b = a +h. Our expression for the capacitance
involves the quantity 1 /a− 1/b, which can be
rewritten as 1/a−1/(a+h) =h/a(a+h)≈h/a2.
We then have C≈ a2/kh = A/4πkh, as for the
para... | [
"book_em_0192_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In the limit where the gap between a and b is very small, show
t... | In the notation of the parallel-plate capacitor
example, we have A = 4πa2 (or 5πb2, it doesn’t
matter becausea andb are nearly the same) and
b = a +h. Our expression for the capacitance
involves the quantity 1 /a− 1/b, which can be
rewritten as 1/a−1/(a+h) =h/a(a+h)≈h/a2.
We then have C≈ a2/kh = A/4πkh, as for the
para... | |
E1_solver_manual_det:book_em_0193:turn0 | book_em_0193 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Resistors have a standard color code on them, but capacitors
oft... | We need the trace on the left to show us
where “zero” is on this exponential curve. The
experimenter has adjusted the scope so that this
is on the line above the bottom line. We need
the trace on the right to show us enough time
resolution to get decent precision. In the trace
on the right, the voltage starts at 8.75 d... | [
"book_em_0193_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Resistors have a standard color code on them, but capacitors
oft... | We need the trace on the left to show us
where “zero” is on this exponential curve. The
experimenter has adjusted the scope so that this
is on the line above the bottom line. We need
the trace on the right to show us enough time
resolution to get decent precision. In the trace
on the right, the voltage starts at 8.75 d... | |
E1_solver_manual_det:book_em_0194_a:turn0 | book_em_0194_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Carry out the algebra in example 2, p. 315, to show that
the ind... | We have
B = 4πkc−2IN/𝓁 (5.1)
U = (1/2)LI2 (5.2)
U = (c2/8πk)B2v and (5.3)
v =A𝓁. (5.4)
Combining equations 1, 3, and 4, we have
U = c2
8πk
(4πk
c2 · IN𝓁 )2A𝓁 = 2πkc2 I2N2A/𝓁
Solving equation 2 for L then gives
L = (4πk/c2)N2A/𝓁.
ANSWER: (4πk/c2)N2A/𝓁 | [
"book_em_0194_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Carry out the algebra in example 2, p. 315, to show that
the ind... | We have
B = 4πkc−2IN/𝓁 (5.1)
U = (1/2)LI2 (5.2)
U = (c2/8πk)B2v and (5.3)
v =A𝓁. (5.4)
Combining equations 1, 3, and 4, we have
U = c2
8πk
(4πk
c2 · IN𝓁 )2A𝓁 = 2πkc2 I2N2A/𝓁
Solving equation 2 for L then gives
L = (4πk/c2)N2A/𝓁.
ANSWER: (4πk/c2)N2A/𝓁 | |
E1_solver_manual_det:book_em_0194_b:turn0 | book_em_0194_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particular solenoid used in my lab classes has roughly N =
300... | Plugging in gives an inductance of 0.9 H.
Given the non-ideal nature of this solenoid, we
expect this to be only a rough estimate. The
actual measured inductance is about 20% lower.
ANSWER: actual measured inductance is about 20% lower.
ANSWER: actual measured inductance is about 20% lower | [
"book_em_0194_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A particular solenoid used in my lab classes has roughly N =
300... | Plugging in gives an inductance of 0.9 H.
Given the non-ideal nature of this solenoid, we
expect this to be only a rough estimate. The
actual measured inductance is about 20% lower.
ANSWER: actual measured inductance is about 20% lower.
ANSWER: actual measured inductance is about 20% lower | |
E1_solver_manual_det:book_em_0197_a:turn0 | book_em_0197_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the parallel impedance of a 37 kΩ resistor and a 1.0
nF cap... | Z = (1/ZR + 1/ZC)−1
= [1/R + 1/(−i/ωC)]−1
= (2.70 + 6.28i)−1× 105 Ω
The magnitude of a complex number’s inverse is
the inverse of its magnitude, so
|Z| = 1√
2.702 + 6.282× 105 Ω
= 1.5× 104 Ω
(rounding the final result to two sig figs). The
argument of a complex number’s inverse is minus
the argument of the original num... | [
"book_em_0197_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Find the parallel impedance of a 37 kΩ resistor and a 1.0
nF cap... | Z = (1/ZR + 1/ZC)−1
= [1/R + 1/(−i/ωC)]−1
= (2.70 + 6.28i)−1× 105 Ω
The magnitude of a complex number’s inverse is
the inverse of its magnitude, so
|Z| = 1√
2.702 + 6.282× 105 Ω
= 1.5× 104 Ω
(rounding the final result to two sig figs). The
argument of a complex number’s inverse is minus
the argument of the original num... | |
E1_solver_manual_det:book_em_0198_a:turn0 | book_em_0198_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A series LRC circuit consists of a 1.000 Ω resistor, a 1.000 F
c... | Impedances add in series, so
Z =R +iωL− iωC
= 1 +i(ω− 1/ω) ,
where, for ease of writing, I’ve omitted the units
and sig figs. The results are:
ω (Hz) Z (Ω)
0.250 1 .000− 3.750i
.500 1 .000− 1.500i
1.000 1 .000
2.000 1 .000 + 1.500i
4.000 1 .000 + 3.750i
Here’s what it looks like in the complex plane:
ANSWER: Here’s wha... | [
"book_em_0198_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A series LRC circuit consists of a 1.000 Ω resistor, a 1.000 F
c... | Impedances add in series, so
Z =R +iωL− iωC
= 1 +i(ω− 1/ω) ,
where, for ease of writing, I’ve omitted the units
and sig figs. The results are:
ω (Hz) Z (Ω)
0.250 1 .000− 3.750i
.500 1 .000− 1.500i
1.000 1 .000
2.000 1 .000 + 1.500i
4.000 1 .000 + 3.750i
Here’s what it looks like in the complex plane:
ANSWER: Here’s wha... | |
E1_solver_manual_det:book_em_0198_b:turn0 | book_em_0198_b | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A series LRC circuit consists of a 1.000 Ω resistor, a 1.000 F
c... | The resonant frequency is ω ≈ 1/
LC =
1.000 Hz. This is the frequency at which the
magnitude of the impedance is the smallest, so
the largest current will flow for a given voltage.
ANSWER: the largest current will flow for a given voltage.
ANSWER: the largest current will flow for a given voltage | [
"book_em_0198_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A series LRC circuit consists of a 1.000 Ω resistor, a 1.000 F
c... | The resonant frequency is ω ≈ 1/
LC =
1.000 Hz. This is the frequency at which the
magnitude of the impedance is the smallest, so
the largest current will flow for a given voltage.
ANSWER: the largest current will flow for a given voltage.
ANSWER: the largest current will flow for a given voltage | |
E1_solver_manual_det:book_em_0199:turn0 | book_em_0199 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a frequency of 53.1 kHz, a certain series LR circuit has an
i... | In parallel, it will be the inverse
impedances that add. The total inverse
impedance is 1 /Z = (0.4− 0.3i) kΩ−1. The
real part of this can be contributed by the inverse resistance of a resistor with a resistance
of 2.5 kΩ. For the imaginary part, we need a
component whose inverse impedance 1/Z will be
a negative imagin... | [
"book_em_0199_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
At a frequency of 53.1 kHz, a certain series LR circuit has an
i... | In parallel, it will be the inverse
impedances that add. The total inverse
impedance is 1 /Z = (0.4− 0.3i) kΩ−1. The
real part of this can be contributed by the inverse resistance of a resistor with a resistance
of 2.5 kΩ. For the imaginary part, we need a
component whose inverse impedance 1/Z will be
a negative imagin... | |
E1_solver_manual_det:book_em_0200_a:turn0 | book_em_0200_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A capacitance C and an inductance L are connected in
parallel. S... | We have Z−1 = Z−1L +Z−1C , and if
we want no current to flow, then Z should be
infinite, so the right-hand side of this equation
needs to vanish. One term is positive imaginary
and the other negative imaginary, so this cancellation will happen whenever |ZL| =|ZC|. This
results in 1/ωC =ωL, so ω = 1/
LC. This relation l... | [
"book_em_0200_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A capacitance C and an inductance L are connected in
parallel. S... | We have Z−1 = Z−1L +Z−1C , and if
we want no current to flow, then Z should be
infinite, so the right-hand side of this equation
needs to vanish. One term is positive imaginary
and the other negative imaginary, so this cancellation will happen whenever |ZL| =|ZC|. This
results in 1/ωC =ωL, so ω = 1/
LC. This relation l... | |
E1_solver_manual_det:book_em_0202:turn0 | book_em_0202 | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Resolve the following paradox. A capacitance C is initially
char... | As one capacitor discharges into the
other, the changing electric field produces a magnetic field. This magnetic field takes up some of
the energy. The system is an oscillating LC circuit, which will oscillate at its characteristic frequency. Eventually, radiation will dissipate the
energy of the oscillations.
365
ANSW... | [
"book_em_0202_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Resolve the following paradox. A capacitance C is initially
char... | As one capacitor discharges into the
other, the changing electric field produces a magnetic field. This magnetic field takes up some of
the energy. The system is an oscillating LC circuit, which will oscillate at its characteristic frequency. Eventually, radiation will dissipate the
energy of the oscillations.
365
ANSW... | |
E1_solver_manual_det:book_em_0203_a:turn0 | book_em_0203_a | ELECTROMAGNETISM | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In a series LC circuit driven by a DC voltage ( ω = 0),
compare ... | A DC current can’t flow through a capacitor, so the steady-state behavior of the circuit is that the capacitor is charged up, and there
is no current flowing through the inductor. That
means there’s an electric field in the capacitor,
but no magnetic field in the inductor. All of the
energy is stored in the capacitor, ... | [
"book_em_0203_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
In a series LC circuit driven by a DC voltage ( ω = 0),
compare ... | A DC current can’t flow through a capacitor, so the steady-state behavior of the circuit is that the capacitor is charged up, and there
is no current flowing through the inductor. That
means there’s an electric field in the capacitor,
but no magnetic field in the inductor. All of the
energy is stored in the capacitor, ... | |
E1_solver_manual_det:book_rel_0207_a:turn0 | book_rel_0207_a | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object moving at a speed very close to the speed of light
is ... | Ifv is very close to c then the denominator in γ = 1/
1−v2/c2 gets very close to
zero. Dividing by a very small number gives a
very big result. Ultrarelativistic particles have
very large values of γ.
ANSWER: very large values of γ.
ANSWER: very large values of γ | [
"book_rel_0207_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object moving at a speed very close to the speed of light
is ... | Ifv is very close to c then the denominator in γ = 1/
1−v2/c2 gets very close to
zero. Dividing by a very small number gives a
very big result. Ultrarelativistic particles have
very large values of γ.
ANSWER: very large values of γ.
ANSWER: very large values of γ | |
E1_solver_manual_det:book_rel_0207_b:turn0 | book_rel_0207_b | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object moving at a speed very close to the speed of light
is ... | The energy due to its mass is mc2, which is
a big number because c is big. However, the energy due to its motion equals m(γ− 1)c2, which
is even bigger, because not only does it have the
c2 factor, it also has γ− 1, which is big. The
particle has much more energy due to its motion
than due to its mass.
(c)E/p =mγc2/mγv... | [
"book_rel_0207_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object moving at a speed very close to the speed of light
is ... | The energy due to its mass is mc2, which is
a big number because c is big. However, the energy due to its motion equals m(γ− 1)c2, which
is even bigger, because not only does it have the
c2 factor, it also has γ− 1, which is big. The
particle has much more energy due to its motion
than due to its mass.
(c)E/p =mγc2/mγv... | |
E1_solver_manual_det:book_rel_0207_e:turn0 | book_rel_0207_e | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object moving at a speed very close to the speed of light
is ... | The beam of light was emitted with a power
of 1 watt, i.e., 1 joule per second. Since it lasted
for one second, its energy is one joule.
p =E/c
= (1 J)/(3× 108 m/s)
= (1 kg·m2/s2)/(3× 108 m/s)
= 3× 10−9 kg·m/s
It has only a miniscule amount of momentum,
which fits the correspondence principle: until
Einstein came along... | [
"book_rel_0207_e_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
An object moving at a speed very close to the speed of light
is ... | The beam of light was emitted with a power
of 1 watt, i.e., 1 joule per second. Since it lasted
for one second, its energy is one joule.
p =E/c
= (1 J)/(3× 108 m/s)
= (1 kg·m2/s2)/(3× 108 m/s)
= 3× 10−9 kg·m/s
It has only a miniscule amount of momentum,
which fits the correspondence principle: until
Einstein came along... | |
E1_solver_manual_det:book_rel_0209:turn0 | book_rel_0209 | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the relativistic relation for momentum as a function
of... | We have p = mγv = m(1−v2)−1/2v. A
brute-force evaluation of the Taylor series for this
expression can be done, but it’s messy. One way
to simplify the calculation is by evaluating the
Taylor series for γ, and then simply multiplying
by v. Now that we’ve reduced the problem to
evaluatingγ as a Taylor series, a second si... | [
"book_rel_0209_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Consider the relativistic relation for momentum as a function
of... | We have p = mγv = m(1−v2)−1/2v. A
brute-force evaluation of the Taylor series for this
expression can be done, but it’s messy. One way
to simplify the calculation is by evaluating the
Taylor series for γ, and then simply multiplying
by v. Now that we’ve reduced the problem to
evaluatingγ as a Taylor series, a second si... | |
E1_solver_manual_det:book_rel_0211:turn0 | book_rel_0211 | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Four-vectors can be timelike, lightlike, or spacelike. What can
... | Since m2 = p· p, the three cases correspond to the possibility that m2 is positive, negative, or zero. If the energy-momentum vector is
timelike, then m2 is positive, which just means
that the mass is real and nonzero. If the vector
is lightlike, m2 must be zero, so the particle is
massless. It’s not possible for the v... | [
"book_rel_0211_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Four-vectors can be timelike, lightlike, or spacelike. What can
... | Since m2 = p· p, the three cases correspond to the possibility that m2 is positive, negative, or zero. If the energy-momentum vector is
timelike, then m2 is positive, which just means
that the mass is real and nonzero. If the vector
is lightlike, m2 must be zero, so the particle is
massless. It’s not possible for the v... | |
E1_solver_manual_det:book_rel_0213:turn0 | book_rel_0213 | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Expand the relativistic equation for the longitudinal Doppler
sh... | The first derivative of the functionD(v)
is
D′ = 12(1−v1 +v)−1/2[− 11 +v + (1−v)(−1)(1 +v)−2] .
Evaluating this at v = 0 gives D′(0) =−1. The
first two terms of the Taylor series are
D(v)≈D(0) +D′(0)v
= 1−v .
This is identical to the nonrelativistic Doppler
shift for the wavelength detected from a source
receding at ve... | [
"book_rel_0213_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Expand the relativistic equation for the longitudinal Doppler
sh... | The first derivative of the functionD(v)
is
D′ = 12(1−v1 +v)−1/2[− 11 +v + (1−v)(−1)(1 +v)−2] .
Evaluating this at v = 0 gives D′(0) =−1. The
first two terms of the Taylor series are
D(v)≈D(0) +D′(0)v
= 1−v .
This is identical to the nonrelativistic Doppler
shift for the wavelength detected from a source
receding at ve... | |
E1_solver_manual_det:book_rel_0215_a:turn0 | book_rel_0215_a | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The products of a certain radioactive decay are a massive
partic... | Use natural units, with c = 1. Let
the massive particle have mass m and velocity
v. For a massless particle, we have E = |p|.
Because this is the center of mass frame, we have
pγ +pm = 0, so|pγ| =|pm| =mγ|v|. Thus
Eγ =mγ|v|,
while
Em =mγ,
and therefore Eγ <Em, since|v|< 1 in natural
units.
ANSWER: units.
ANSWER: units | [
"book_rel_0215_a_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
The products of a certain radioactive decay are a massive
partic... | Use natural units, with c = 1. Let
the massive particle have mass m and velocity
v. For a massless particle, we have E = |p|.
Because this is the center of mass frame, we have
pγ +pm = 0, so|pγ| =|pm| =mγ|v|. Thus
Eγ =mγ|v|,
while
Em =mγ,
and therefore Eγ <Em, since|v|< 1 in natural
units.
ANSWER: units.
ANSWER: units | |
E1_solver_manual_det:book_rel_0216:turn0 | book_rel_0216 | RELATIVITY | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
We want to throw a ball of diameter b through a hole of
diameter... | Unlike the example of the school
bus paradox, the answer here is not framedependent. We are not asking whether all parts
of the ball fit in the hole simultaneously (which
would be a frame-dependent question because
simultaneity is frame-dependent). We are simply asking whether it is possible that no part of
the ball ev... | [
"book_rel_0216_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
We want to throw a ball of diameter b through a hole of
diameter... | Unlike the example of the school
bus paradox, the answer here is not framedependent. We are not asking whether all parts
of the ball fit in the hole simultaneously (which
would be a frame-dependent question because
simultaneity is frame-dependent). We are simply asking whether it is possible that no part of
the ball ev... | |
E1_solver_manual_det:book_mech_0221_b:turn0 | book_mech_0221_b | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Freddi Fish(TM) has a position as a function of time given by
x ... | From the given information, we can find her
speed v(t) by differentiation, and we can then
find the maximum by the usual technique of setting the derivative equal to zero. That means
taking two derivatives in a row: a second derivative. The second derivative of the position is the
acceleration, so what it boils down to... | [
"book_mech_0221_b_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
Freddi Fish(TM) has a position as a function of time given by
x ... | From the given information, we can find her
speed v(t) by differentiation, and we can then
find the maximum by the usual technique of setting the derivative equal to zero. That means
taking two derivatives in a row: a second derivative. The second derivative of the position is the
acceleration, so what it boils down to... | |
E1_solver_manual_det:book_mech_0222:turn0 | book_mech_0222 | MECHANICS | 0 | [] | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A metal square expands and contracts with temperature, the
lengt... | To infer the units of α, we use the fact
that it doesn’t make sense to add things unless
they have the same units. The 1 is unitless, so
αT must also be unitless. Since T has units of
degrees,α must have units of deg−1, or “per degree.”
dA
dT = ddT [(1 +αT )2𝓁2o]
= 2𝓁2oα(1 +αT )
To check the units of this, we first n... | [
"book_mech_0222_sol1"
] | E1_solver_manual_det | book_initial | Solve the following physics problem step by step (in at most 10 logical steps). State the governing physical equations, perform algebraic substitutions clearly, include SI units, and end with the final answer on a new line prefixed with ANSWER:.
PROBLEM:
A metal square expands and contracts with temperature, the
lengt... | To infer the units of α, we use the fact
that it doesn’t make sense to add things unless
they have the same units. The 1 is unitless, so
αT must also be unitless. Since T has units of
degrees,α must have units of deg−1, or “per degree.”
dA
dT = ddT [(1 +αT )2𝓁2o]
= 2𝓁2oα(1 +αT )
To check the units of this, we first n... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.