problem stringlengths 110 5.09k | answer stringlengths 1 6.51k | source stringclasses 7
values | year int64 1.97k 2.02k | question_number int64 1 995 | sub_question_number int64 1 14 ⌀ | sub_sub_question_number int64 1 16 ⌀ |
|---|---|---|---|---|---|---|
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | $\frac{v_{1}+v_{2}}{2}$ | WoPhO | 2,011 | 1 | 1 | 1 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | $\frac{\left|v_{1}-v_{2}\right|}{L}$ | WoPhO | 2,011 | 1 | 1 | 1 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | The center of mass of the rod moves in a circular orbit. | WoPhO | 2,011 | 1 | 1 | 2 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | The orbit is a circle with radius $R=\frac{v_{1}+v_{2}}{\left|v_{1}-v_{2}\right|} \frac{L}{2}$ and period $T=\frac{2\pi L}{\left|v_{1}-v_{2}\right|}$. | WoPhO | 2,011 | 1 | 1 | 2 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | $\frac{\left|v_{1}^{2}-v_{2}^{2}\right|}{2 g L}$ | WoPhO | 2,011 | 1 | 1 | 3 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | $\frac{g L \sin \alpha}{2\left|v_{1}-v_{2}\right|}$ | WoPhO | 2,011 | 1 | 2 | 1 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | Horizontal, perpendicular to the steepest line of the incline. | WoPhO | 2,011 | 1 | 2 | 1 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | $\mathbf{v}(t)=\left(v_{0}+\frac{g \sin \alpha}{\omega_{0}}\left(\cos \left(\omega_{0} t\right)-1\right)\right)\left(\begin{array}{c} \cos \left(\omega_{0} t\right) \\ \sin \left(\omega_{0} t\right) \end{array}\right)$ | WoPhO | 2,011 | 1 | 3 | 1 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | $\frac{v_{0} \omega_{0}}{2 g \sin \alpha} \leq 1$ | WoPhO | 2,011 | 1 | 3 | 2 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | Two cases: (1) If $\frac{v_{0} \omega_{0}}{2 g \sin \alpha} \leq 1$, then $y_{\max }=\frac{v_{0}^{2}}{2 g \sin \alpha}$. (2) If $\frac{v_{0} \omega_{0}}{2 g \sin \alpha} > 1$, then $y_{\max }=\frac{2\left(v_{0} \omega_{0}-g \sin \alpha\right)}{\omega_{0}^{2}}$. | WoPhO | 2,011 | 1 | 3 | 3 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | The center of mass of the rod moves along a cycloid. | WoPhO | 2,011 | 1 | 4 | 1 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | The orbit is a cycloid generated by a rolling circle of radius $r^{*}=\frac{g \sin \alpha}{4 \omega_{0}^{2}}$. The period of motion is $\pi / \omega_{0}$ and the vertical extent (along the steepest line) is $\Delta y=2r^{*}=\frac{g \sin \alpha}{2 \omega_{0}^{2}}$. | WoPhO | 2,011 | 1 | 4 | 1 |
In this problem, the motion of a uniform rod (stick) with length $L$, ended with caster-wheels at both ends, will be investigated on a flat surface. The casters at each end of the rod can spin freely and independently and have a negligible mass compared to the rod. The friction between the rod and the caster-wheels is ... | $2 \tan \alpha$ | WoPhO | 2,011 | 1 | 4 | 2 |
Consider a spherical water droplet that falls down through the air with a constant speed. The air resistant force at each point on the surface of the water droplet is tangent to the surface. To simplify the analysis, we assume that the air resistant force is homogeneously distributed over the whole surface of the spher... | $\frac{2 \rho g D}{3 \pi}$ | WoPhO | 2,012 | 3 | 1 | 1 |
Consider a spherical water droplet that falls down through the air with a constant speed. The air resistant force at each point on the surface of the water droplet is tangent to the surface. To simplify the analysis, we assume that the air resistant force is homogeneously distributed over the whole surface of the spher... | $\frac{\rho g D^3}{6 \pi}$ | WoPhO | 2,012 | 3 | 1 | 2 |
Consider a spherical water droplet that falls down through the air with a constant speed. The air resistant force at each point on the surface of the water droplet is tangent to the surface. To simplify the analysis, we assume that the air resistant force is homogeneously distributed over the whole surface of the spher... | $\frac{\pi \sigma D}{2}$ | WoPhO | 2,012 | 3 | 1 | 3 |
Consider a spherical water droplet that falls down through the air with a constant speed. The air resistant force at each point on the surface of the water droplet is tangent to the surface. To simplify the analysis, we assume that the air resistant force is homogeneously distributed over the whole surface of the spher... | $1.51 \mathrm{~mm}$ | WoPhO | 2,012 | 3 | 1 | 4 |
Consider a light ray that is refracted into a spherical water droplet, then reflected back once in the water droplet, and finally refracted into air. $\alpha$ is the central angle of the incident point (the angle between the incident ray and the radius at the point of incidence). Find the angle $\theta$ between the ref... | $4 \arcsin (\sin \alpha / n) - 2 \alpha$ | WoPhO | 2,012 | 3 | 2 | 1 |
Consider a light ray that is refracted into a spherical water droplet, then reflected back once in the water droplet, and finally refracted into air. $\alpha$ is the central angle of the incident point. The angle between the reflected ray and the reverse direction of the incident ray is $\theta = 4 \arcsin (\sin \alpha... | $\frac{I_0 D^2 T_1 T_2 R}{8 \pi} \frac{\sin \alpha \cos \alpha \sqrt{n^2-\sin ^2 \alpha}}{2 \cos \alpha-\sqrt{n^2-\sin ^2 \alpha}}$ | WoPhO | 2,012 | 3 | 2 | 2 |
Consider a light ray that is refracted into a spherical water droplet, then reflected back once in the water droplet, and finally refracted into air. $\alpha$ is the central angle of the incident point. The angle between the reflected ray and the reverse direction of the incident ray is $\theta = 4 \arcsin (\sin \alpha... | $41.9^{\circ}$ | WoPhO | 2,012 | 3 | 2 | 3 |
Consider a light ray that is refracted into a spherical water droplet, then reflected back once in the water droplet, and finally refracted into air. $\alpha$ is the central angle of the incident point. The angle between the reflected ray and the reverse direction of the incident ray is $\theta = 4 \arcsin (\sin \alpha... | $J(\theta_M) \rightarrow \infty$ (the intensity runs to infinity at this angle) | WoPhO | 2,012 | 3 | 2 | 3 |
The electromagnetic waves with the wavelength between $390 \mathrm{~nm}-780 \mathrm{~nm}$ are visible lights. The water's refraction index is $n_v=1.3439$ at $\lambda=390 \mathrm{~nm}$, and $n_r=1.3316$ at $\lambda=780 \mathrm{~nm}$. The angular diameter of the sun is $\delta=0.5^{\circ}$. The angle at which the maximu... | $41.9^{\circ}$ | WoPhO | 2,012 | 3 | 3 | 1 |
The electromagnetic waves with the wavelength between $390 \mathrm{~nm}-780 \mathrm{~nm}$ are visible lights. The water's refraction index is $n_v=1.3439$ at $\lambda=390 \mathrm{~nm}$, and $n_r=1.3316$ at $\lambda=780 \mathrm{~nm}$. The angular diameter of the sun is $\delta=0.5^{\circ}$. The angle at which the maximu... | $2.3^{\circ}$ | WoPhO | 2,012 | 3 | 3 | 1 |
The reflected sunlight is also diffracted by water droplets. The diffraction pattern resulting from a water droplet is the same as the diffraction pattern resulting from a circular aperture with the same diameter as the droplet. If the angular radius of the diffraction pattern resulting from a water droplet is larger t... | $11.9 \mu \mathrm{m}$ | WoPhO | 2,012 | 3 | 3 | 2 |
If the altitude of the bottom of clouds was $800 \mathrm{~m}$ during the rain, calculate the maximum time $T_M$ after the rain stopped, the water droplets that can contribute to rainbow formation may still be found at altitudes higher than $200 \mathrm{~m}$. The diameter of the smallest water droplet that may contribut... | $1.4 \cdot 10^5 \mathrm{~s}$ | WoPhO | 2,012 | 3 | 3 | 3 |
The radius of a hydrogen atom in its ground state is $a_0=0.0529 \mathrm{~nm}$ (the "Bohr radius"). What is the radius $a^{\prime}$ of a "muonic-hydrogen" atom in which the electron is replaced by an identically charged muon, with mass 207 times that of the electron? Assume the proton mass is much larger than that of t... | Using the Bohr model, the radius is inversely proportional to the mass of the orbiting particle. Since the muon has 207 times the mass of the electron, the radius is $a^{\prime} = \frac{a_0}{207} = 0.256 \mathrm{pm}$. | IPhO | 1,997 | 88 | null | null |
A static container of mass $M$ and cylindrical shape is placed in vacuum. One of its ends is closed. A fixed piston of mass $m$ and negligible width separates the volume of the container into two equal parts. The closed part contains $n$ moles of monoatomic perfect gas with molar mass $M_0$ and temperature $T$. After r... | The final velocity of the container is $v=v_1+v_2$, where $v_1$ is the velocity when the piston leaves and $v_2$ is the additional velocity increase when the gas leaves.\n\nFor $v_1$: Using conservation of momentum and energy up to the moment when the piston leaves the container:\n$\left(M+n M_0\right) v_1-m u=0$\n$\fr... | IPhO | 1,981 | 97 | null | null |
Consider two identical homogeneous balls, A and B, with the same initial temperatures. Ball A is at rest on a horizontal plane, while ball B hangs from a thread. The same quantities of heat have been supplied to both balls. Are the final temperatures of the balls the same or not? Justify your answer. (All kinds of heat... | The final temperatures of the balls are not the same. When the balls are warmed up, their radii increase due to thermal expansion. For ball A resting on the plane, its center of mass moves downward as the radius increases, which decreases its gravitational potential energy. This decrease in potential energy contributes... | IPhO | 1,967 | 49 | null | null |
A triangular prism of mass $M$ is placed with one side on a frictionless horizontal plane. The other two sides are inclined with respect to the plane at angles $\alpha_1$ and $\alpha_2$ respectively. Two blocks of masses $m_1$ and $m_2$, connected by an inextensible thread, can slide without friction on the surface of ... | $a=a_0 \frac{M+m_1+m_2}{m_1 \cos \alpha_1+m_2 \cos \alpha_2}$ | IPhO | 1,971 | 40 | 1 | null |
A triangular prism of mass $M$ is placed with one side on a frictionless horizontal plane. The other two sides are inclined with respect to the plane at angles $\alpha_1$ and $\alpha_2$ respectively. Two blocks of masses $m_1$ and $m_2$, connected by an inextensible thread, can slide without friction on the surface of ... | $a_0=\frac{\left(m_1 \sin \alpha_1-m_2 \sin \alpha_2\right)\left(m_1 \cos \alpha_1+m_2 \cos \alpha_2\right)}{\left(m_1+m_2+M\right)\left(m_1+m_2\right)-\left(m_1 \cos \alpha_1+m_2 \cos \alpha_2\right)^2}$ | IPhO | 1,971 | 40 | 2 | null |
A triangular prism of mass $M$ is placed with one side on a frictionless horizontal plane. The other two sides are inclined with respect to the plane at angles $\alpha_1$ and $\alpha_2$ respectively. Two blocks of masses $m_1$ and $m_2$, connected by an inextensible thread, can slide without friction on the surface of ... | $\frac{m_1}{m_2}=\frac{\sin \alpha_2}{\sin \alpha_1}$ | IPhO | 1,971 | 40 | 3 | null |
An electron gun T emits electrons accelerated by a potential difference U in a vacuum in the direction of line a. The target M is placed at a distance d from the electron gun in such a way that the line segment connecting the points T and M and the line a subtend the angle α. Find the magnetic induction B of the unifor... | If a uniform magnetic field is perpendicular to the initial direction of motion of an electron beam, the electrons will be deflected by a force that is always perpendicular to their velocity and to the magnetic field. Consequently, the beam will be deflected into a circular trajectory. The origin of the centripetal for... | IPhO | 1,977 | 74 | 1 | null |
An electron gun T emits electrons accelerated by a potential difference U in a vacuum in the direction of line a. The target M is placed at a distance d from the electron gun in such a way that the line segment connecting the points T and M and the line a subtend the angle α. Find the magnetic induction B of the unifor... | If a uniform magnetic field is neither perpendicular nor parallel to the initial direction of motion of an electron beam, the electrons will be deflected into a helical trajectory. Namely, the motion of electrons will be composed of an uniform motion on a circle in the plane perpendicular to the magnetic field and of a... | IPhO | 1,977 | 74 | 2 | null |
Klystrons are devices used for amplifying very high-frequency signals. A klystron basically consists of two identical pairs of parallel plates (cavities) separated by a distance $b$. An electron beam with an initial speed $v_0$ traverses the entire system, passing through small holes in the plates. The high-frequency v... | $2.272 \times 10^{-2} \mathrm{~m}$ | IPhO | 2,001 | 23 | 1 | 1 |
Klystrons are devices used for amplifying very high-frequency signals. A klystron basically consists of two identical pairs of parallel plates (cavities) separated by a distance $b$. An electron beam with an initial speed $v_0$ traverses the entire system, passing through small holes in the plates. The high-frequency v... | $\pm 220^{\circ}$ OR $\pm 140^{\circ}$ | IPhO | 2,001 | 23 | 1 | 2 |
Let $d_L$ and $d_V$ represent the average distances between molecules of water in the liquid phase and in the vapor phase, respectively. Assume that both phases are at $100{ }^{\circ} \mathrm{C}$ and atmospheric pressure, and the vapor behaves like an ideal gas. Using the following data, calculate the ratio $d_V / d_L$... | 12 | IPhO | 2,001 | 23 | 2 | 1 |
A sawtooth voltage waveform $V_0$ can be obtained across the capacitor $C$ in a circuit. The circuit consists of a variable resistor $R$, an ideal battery $V_i$, and a spark gap $SG$ consisting of two electrodes with an adjustable distance between them. When the voltage across the electrodes exceeds the firing voltage ... | $V_i \gg V_f$ | IPhO | 2,001 | 23 | 3 | 2 |
A sawtooth voltage waveform $V_0$ can be obtained across the capacitor $C$ in a circuit. The circuit consists of a variable resistor $R$, an ideal battery $V_i$, and a spark gap $SG$ consisting of two electrodes with an adjustable distance between them. When the voltage across the electrodes exceeds the firing voltage ... | $T=\left(V_f / V_i\right) R C$ | IPhO | 2,001 | 23 | 3 | 3 |
A sawtooth voltage waveform $V_0$ can be obtained across the capacitor $C$ in a circuit. The circuit consists of a variable resistor $R$, an ideal battery $V_i$, and a spark gap $SG$ consisting of two electrodes with an adjustable distance between them. When the voltage across the electrodes exceeds the firing voltage ... | R | IPhO | 2,001 | 23 | 3 | 4 |
A sawtooth voltage waveform $V_0$ can be obtained across the capacitor $C$ in a circuit. The circuit consists of a variable resistor $R$, an ideal battery $V_i$, and a spark gap $SG$ consisting of two electrodes with an adjustable distance between them. When the voltage across the electrodes exceeds the firing voltage ... | SG and R | IPhO | 2,001 | 23 | 3 | 5 |
An atomic beam is prepared by heating a collection of atoms to a temperature $T$ and allowing them to emerge horizontally through a small hole (of atomic dimensions) of diameter $D$ in one side of the oven. Estimate the diameter of the beam after it has traveled a horizontal length $L$ along its path. The mass of an at... | $D+\frac{L \hbar}{D \sqrt{3 M k T}}$ | IPhO | 2,001 | 23 | 4 | 1 |
Dipping the frame in a soap solution, the soap forms a rectangle film of length $b$ and height $h$. White light falls on the film at an angle $\alpha$ (measured with respect to the normal direction). The reflected light displays a green color of wavelength $\lambda_0$. Constants and given data: relative refractive inde... | The thin layer reflects the monochromatic light of the wavelength $\lambda$ in the best way, if the following equation holds true $ 2 n d \cos \beta=(2 k+1) \frac{\lambda}{2}, \quad k=0,1,2, \ldots, $ where $k$ denotes an integer and $\beta$ is the angle of refraction satisfying $ \frac{\sin \alpha}{\sin \beta}=n \text... | IPhO | 1,977 | 73 | 1 | null |
Dipping the frame in a soap solution, the soap forms a rectangle film of length $b$ and height $h$. White light falls on the film at an angle $\alpha$ (measured with respect to the normal direction). The reflected light displays a green color of wavelength $\lambda_0$. Constants and given data: relative refractive inde... | The film appears dark. When viewed from the perpendicular direction, the interference condition that produces the green color at the $30^{\circ}$ illumination angle no longer holds, resulting in destructive interference for visible wavelengths. | IPhO | 1,977 | 73 | 2 | null |
In this problem, a non-self-sustained gas discharge is studied. To maintain permanent operation, an external ionizer is needed, which creates $Z_{\text{ext}}$ pairs of singly ionized ions and free electrons per unit volume and per unit time uniformly in the volume. When the external ionizer is switched on, the number o... | $n_0=0$, $a=\sqrt{\frac{Z_{\text{ext}}}{r}}$, $b=\sqrt{r Z_{\text{ext}}}$ | IPhO | 2,014 | 33 | 1 | null |
In this problem, a non-self-sustained gas discharge is studied. To maintain permanent operation, an external ionizer is needed, which creates $Z_{\text{ext}}$ pairs of singly ionized ions and free electrons per unit volume and per unit time uniformly in the volume. When the external ionizer is switched on, the number o... | $n_e=\sqrt{n_{e1}^2+n_{e2}^2}=20.0 \cdot 10^{10} \mathrm{~cm}^{-3}$ | IPhO | 2,014 | 33 | 2 | null |
In this problem, a non-self-sustained gas discharge is studied. To maintain permanent operation, an external ionizer is needed, which creates $Z_{\text{ext}}$ pairs of singly ionized ions and free electrons per unit volume and per unit time uniformly in the volume. When the external ionizer is switched on, the number o... | $I=\frac{e \beta^2 U^2 S}{r L^3}\left(\sqrt{1+\frac{4 r Z_{\text{ext}} L^4}{\beta^2 U^2}}-1\right)$ | IPhO | 2,014 | 33 | 3 | null |
In this problem, a non-self-sustained gas discharge is studied. To maintain permanent operation, an external ionizer is needed, which creates $Z_{\text{ext}}$ pairs of singly ionized ions and free electrons per unit volume and per unit time uniformly in the volume. When the external ionizer is switched on, the number o... | $\rho=\frac{1}{2 e \beta} \sqrt{\frac{r}{Z_{\text{ext}}}}$ | IPhO | 2,014 | 33 | 4 | null |
A superconducting magnet with inductance L = 10 H is connected in parallel with a superconducting switch. The switch has resistance r that can be controlled: r = 0 in the superconducting state, or r = r_n = 5 Ω in the normal state. The parallel combination is connected to a power source with resistance R. The total cur... | For t_1 to t_3: Since r = 0, the voltage across the magnet V_M = L(dI_1/dt) = 0, so I_1 remains constant at I_1 = I_0/2. The switch current is I_2 = I - I_0/2. For t_3 to t_4: Since I_2 = 0 at t_3 and I is held at I_0/2, V_M = I_2 r_n = 0, so I_1 and I_2 do not change. Therefore, I_1 = I_0/2 and I_2 = 0. | IPhO | 1,994 | 82 | 1 | null |
A superconducting magnet with inductance L = 10 H is connected in parallel with a superconducting switch. The switch has resistance r that can be controlled: r = 0 in the superconducting state, or r = r_n = 5 Ω in the normal state. The parallel combination is connected to a power source with electromotive force E = 3.7... | For 0 ≤ t < 1 min: Since r = 0, V_M = L(dI_1/dt) = 0, so I_1 = I_1(0) = 0 and I_2 = I - I_1 = 0.5 A. At t = 1 min, r jumps from 0 to r_n. Since I_1 cannot change abruptly due to inductance, I drops from E/R = 0.5 A to E/(R + r_n) = 3.75/(7.5 + 5) = 0.3 A instantaneously. For 1 min ≤ t < 2 min: I, I_1, and I_2 gradually... | IPhO | 1,994 | 82 | 2 | null |
A superconducting magnet with inductance L = 10 H is connected in parallel with a superconducting switch. The switch has resistance r that can be controlled: r = 0 in the superconducting state, or r = r_n = 5 Ω in the normal state. The parallel combination is connected to a power source with resistance R. The total cur... | Step 1: Turn on the power switch K and increase the total current I to 20 A (equal to I_1). Since r = 0, V_M = L(dI_1/dt) = 0, so I_1 cannot change and I_2 increases by 20 A (from -20 A to 0 A). Step 2: Switch r from 0 to r_n. Step 3: Gradually reduce I to zero while keeping |I_2| < 0.5 A. Since I_2 = V_M/r_n and V_M =... | IPhO | 1,994 | 82 | 3 | null |
A superconducting magnet with inductance L = 10 H is connected in parallel with a superconducting switch. The switch has resistance r that can be controlled: r = 0 in the superconducting state, or r = r_n = 5 Ω in the normal state. The parallel combination is connected to a power source with resistance R. The total cur... | Step 1: Turn on the power switch K and increase the total current I to 20 A (equal to I_1). Since r = 0, V_M = L(dI_1/dt) = 0, so I_1 cannot change and I_2 increases by 20 A (from -20 A to 0 A). Step 2: Switch r from 0 to r_n. Step 3: Increase I by 10 A to 30 A with a rate subject to the requirement |I_2| < 0.5 A (corr... | IPhO | 1,994 | 82 | 4 | null |
Two dispersive prisms having apex angles $\hat{A}_1=60^{\circ}$ and $\hat{A}_2=30^{\circ}$ are glued together at their common face AC, forming a composite prism system where the angle at vertex C is $\hat{C}=90^{\circ}$. The refractive indices of the prisms depend on wavelength according to the relations $n_1(\lambda)=... | $\lambda_0=500 \mathrm{~nm}$ | IPhO | 1,983 | 53 | 1 | null |
Two dispersive prisms having apex angles $\hat{A}_1=60^{\circ}$ and $\hat{A}_2=30^{\circ}$ are glued together at their common face AC, forming a composite prism system where the angle at vertex C is $\hat{C}=90^{\circ}$. The refractive indices of the prisms depend on wavelength according to the relations $n_1(\lambda)=... | $n_1(\lambda_0)=n_2(\lambda_0)=1.5$ | IPhO | 1,983 | 53 | 1 | null |
Two dispersive prisms having apex angles $\hat{A}_1=60^{\circ}$ and $\hat{A}_2=30^{\circ}$ are glued together at their common face AC, forming a composite prism system where the angle at vertex C is $\hat{C}=90^{\circ}$. The refractive indices of the prisms depend on wavelength according to the relations $n_1(\lambda)=... | $\delta_{\text{min}} \cong 30.7^{\circ}$ | IPhO | 1,983 | 53 | 3 | null |
Two dispersive prisms having apex angles $\hat{A}_1=60^{\circ}$ and $\hat{A}_2=30^{\circ}$ are glued together at their common face AC, forming a composite prism system where the angle at vertex C is $\hat{C}=90^{\circ}$. The refractive indices of the prisms depend on wavelength according to the relations $n_1(\lambda)=... | $\lambda \cong 1.2 \mu \mathrm{m}$ | IPhO | 1,983 | 53 | 4 | null |
Consider an infinite electrical network consisting of identical resistors, each with resistance $r$. The network is arranged in a ladder configuration. Starting from terminal A, there is a resistor $r$ connected in series. After this resistor, the circuit reaches a junction. From this junction, one path goes through a ... | It is easy to observe that if we remove the first section of the network (the series resistor and the first shunt resistor), the remaining network is identical to the initial infinite network. Thus, the equivalent resistance of the remaining part is also $R_{AB}$. This allows us to set up the following equation for the... | IPhO | 1,967 | 48 | null | null |
In the theory of special relativity, the relation between energy E and momentum p for a free particle with rest mass m₀ is E = √(p²c² + m₀²c⁴) = mc². When such a particle is subject to a conservative force, the total energy of the particle, which is the sum of √(p²c² + m₀²c⁴) and the potential energy, is conserved. If ... | The turning points occur at (x, p) = (±p₀c/f, 0). | IPhO | 1,994 | 81 | 1 | null |
In the theory of special relativity, the relation between energy E and momentum p for a free particle with rest mass m₀ is E = √(p²c² + m₀²c⁴) = mc². When such a particle is subject to a conservative force, the total energy of the particle, which is the sum of √(p²c² + m₀²c⁴) and the potential energy, is conserved. If ... | The particle moves clockwise in the (p, x) diagram: starting from (0, p₀), it moves to (L, 0) with decreasing p, then to (0, -p₀) with negative p, then to (-L, 0) with increasing p, and back to (0, p₀). | IPhO | 1,994 | 81 | 1 | null |
A meson is a particle made up of two quarks. The rest mass M of the meson is equal to the total energy of the two-quark system divided by c². Consider a one-dimensional model for a meson at rest, in which the two quarks are assumed to move along the x-axis and attract each other with a force of constant magnitude f. It... | For quark 1, the turning points are at (x₁, p₁) = (±Mc²/2f, 0). For quark 2, the turning points are at (x₂, p₂) = (∓Mc²/2f, 0). | IPhO | 1,994 | 81 | 2 | null |
A meson is a particle made up of two quarks. The rest mass M of the meson is equal to the total energy of the two-quark system divided by c². Consider a one-dimensional model for a meson at rest, in which the two quarks are assumed to move along the x-axis and attract each other with a force of constant magnitude f. It... | The maximum distance between the two quarks is d = Mc²/f. | IPhO | 1,994 | 81 | 2 | null |
A meson is a particle made up of two quarks. The rest mass M of the meson is equal to the total energy of the two-quark system divided by c². Consider a one-dimensional model for a meson at rest in frame S, in which the two quarks are assumed to move along the x-axis and attract each other with a force of constant magn... | For quark 1, the turning points in S' are: (x₁', t₁') = (0, 0), (2L, 2τ), (3L/2, 5τ/2), (L, 3τ), (3L, 5τ) where L = Mc²/2f and τ = Mc/2f. For quark 2, the turning points in S' are: (x₂', t₂') = (0, 0), (-L/2, τ/2), (3L/2, 5τ/2), (7L/2, 9τ/2), (3L, 5τ). | IPhO | 1,994 | 81 | 3 | null |
A meson is a particle made up of two quarks. The rest mass M of the meson is equal to the total energy of the two-quark system divided by c². Consider a one-dimensional model for a meson at rest in frame S, in which the two quarks are assumed to move along the x-axis and attract each other with a force of constant magn... | The maximum distance between the two quarks in Lab frame S' is d' = Mc²/2f. | IPhO | 1,994 | 81 | 3 | null |
A meson is a particle made up of two quarks. The rest mass M of the meson is equal to the total energy of the two-quark system divided by c². For a meson with rest energy Mc² = 140 MeV and velocity 0.60c relative to the Lab frame S', determine its energy E' in the Lab Frame S'. | E' = Mc²/√(1-β²) = 140 MeV/0.8 = 175 MeV. | IPhO | 1,994 | 81 | 4 | null |
In a long bar having the shape of a rectangular parallelepiped with sides a, b, and c (a ≫ b ≫ c), made from the semiconductor InSb, flows a current I parallel to the edge a. The bar is in an external magnetic field B which is parallel to the edge c. The magnetic field produced by the current I can be neglected. The cu... | 4.06 V/m | IPhO | 1,967 | 21 | 1 | null |
In a long bar having the shape of a rectangular parallelepiped with sides a, b, and c (a ≫ b ≫ c), made from the semiconductor InSb, flows a current I parallel to the edge a. The bar is in an external magnetic field B which is parallel to the edge c. The magnetic field produced by the current I can be neglected. The cu... | The electric field has a component parallel to edge a (in the direction of the current) and a perpendicular component along edge b (perpendicular to both the current and magnetic field directions). The perpendicular component points in the direction of edge b. | IPhO | 1,967 | 21 | 1 | null |
In a long bar having the shape of a rectangular parallelepiped with sides a, b, and c (a ≫ b ≫ c), made from the semiconductor InSb, flows a current I parallel to the edge a. The bar is in an external magnetic field B which is parallel to the edge c. The magnetic field produced by the current I can be neglected. The cu... | 25 mV | IPhO | 1,967 | 21 | 2 | null |
In a long bar having the shape of a rectangular parallelepiped with sides a, b, and c (a ≫ b ≫ c), made from the semiconductor InSb, flows a current I parallel to the edge a. The bar is in an external magnetic field B which is parallel to the edge c. The magnetic field produced by the current I can be neglected. The cu... | \frac{I_0 B_0}{2 n e_0 c} \cos \delta | IPhO | 1,967 | 21 | 3 | null |
Consider a hot-air balloon with fixed volume $V_B = 1.1 \text{ m}^3$. The mass of the balloon envelope, whose volume is to be neglected in comparison to $V_B$, is $m_H = 0.187 \text{ kg}$. The balloon is at ground level where the external air temperature is $\vartheta_1 = 20^\circ \text{C}$ and the normal external air ... | The floating condition requires that the total mass of the balloon (envelope mass plus internal air mass) equals the mass of displaced external air: $V_B \cdot \rho_2 + m_H = V_B \cdot \rho_1$. This gives $\rho_2 = \rho_1 - m_H/V_B = 1.2 - 0.187/1.1 = 1.031 \text{ kg/m}^3$. Using the ideal gas relation $\rho_1/\rho_2 =... | IPhO | 1,982 | 86 | 1 | null |
Consider a hot-air balloon with fixed volume $V_B = 1.1 \text{ m}^3$. The mass of the balloon envelope, whose volume is to be neglected in comparison to $V_B$, is $m_H = 0.187 \text{ kg}$. The balloon is at ground level where the external air temperature is $\vartheta_1 = 20^\circ \text{C}$ and the normal external air ... | The force $F_B$ acting on the rope is the difference between the buoyant force $F_A$ and the weight force $F_G$: $F_B = V_B \cdot \rho_1 \cdot g - (V_B \cdot \rho_3 + m_H) \cdot g$. Using the ideal gas relation $\rho_3 \cdot T_3 = \rho_1 \cdot T_1$ with $T_1 = 293.15 \text{ K}$ and $T_3 = 383.15 \text{ K}$, we find $\r... | IPhO | 1,982 | 86 | 2 | null |
Consider a hot-air balloon with fixed volume $V_B = 1.1 \text{ m}^3$. The mass of the balloon envelope, whose volume is to be neglected in comparison to $V_B$, is $m_H = 0.187 \text{ kg}$. The balloon is at ground level where the external air temperature is $\vartheta_1 = 20^\circ \text{C}$ and the normal external air ... | The balloon rises to the height $h$ where the external air density $\rho_h$ equals the effective density $\rho_{\text{eff}}$ of the balloon system. The effective density is $\rho_{\text{eff}} = (\rho_3 \cdot V_B + m_H)/V_B = \rho_3 + m_H/V_B$. Using $\rho_3 = \rho_1 \cdot T_1/T_3 = 1.2 \cdot 293.15/383.15 = 0.921 \text... | IPhO | 1,982 | 86 | 3 | null |
Consider a hot-air balloon with fixed volume $V_B = 1.1 \text{ m}^3$. The mass of the balloon envelope, whose volume is to be neglected in comparison to $V_B$, is $m_H = 0.187 \text{ kg}$. The balloon is at ground level where the external air temperature is $\vartheta_1 = 20^\circ \text{C}$ and the normal external air ... | For small height differences (10 m in comparison to 843 m), the exponential pressure drop (or density drop respectively) with height can be approximated by a linear function of height. Therefore the driving force (buoyant force minus weight) is proportional to the displacement from the equilibrium position. This is the... | IPhO | 1,982 | 86 | 4 | null |
A photon of frequency $f$ possesses an effective inertial mass $m$ determined by its energy. Assume that it has a gravitational mass equal to this inertial mass. Accordingly, a photon emitted at the surface of a star will lose energy when it escapes from the star's gravitational field. Show that the frequency shift $\D... | If a photon has an effective inertial mass $m$ determined by its energy then $m c^2 = h f$ or $m = \frac{h f}{c^2}$. Now, assume that gravitational mass equals inertial mass, and consider a photon of energy $h f$ (mass $m = \frac{h f}{c^2}$) emitted upwards at a distance $r$ from the centre of the star. It will lose en... | IPhO | 1,995 | 57 | 1 | null |
An unmanned spacecraft is launched in an experiment to measure both the mass $M$ and radius $R$ of a star in our galaxy. Photons are emitted from $\mathrm{He}^{+}$ ions on the surface of the star. These photons can be monitored through resonant absorption by $\mathrm{He}^{+}$ ions contained in a test chamber in the spa... | The change in photon energy in ascending from $r_i$ to $r_f$ is given by $h f_i - h f_f = -\frac{G M m_f}{r_f} + \frac{G M m_i}{r_i} \simeq \frac{G M h f_i}{c^2}\left[\frac{1}{r_i} - \frac{1}{r_f}\right]$. Therefore $\frac{f_f}{f_i} = 1 - \frac{G M}{c^2}\left[\frac{1}{r_i} - \frac{1}{r_f}\right]$. In the experiment, $R... | IPhO | 1,995 | 57 | 2 | null |
In order to determine $R$ and $M$ in a gravitational redshift experiment, it is usual to consider the frequency correction due to the recoil of the emitting atom. [Thermal motion causes emission lines to be broadened without displacing emission maxima, and we may therefore assume that all thermal effects have been take... | For the photon, photon momentum is $p = \frac{h f}{c}$ and photon energy is $E = h f$. Use the mass-energy equivalence, $E = m c^2$, to relate the internal energy change of the atom to the rest-mass change. Thus: $\Delta E = (m_0 - m_0') c^2$. In the laboratory frame of reference the energy before emission is $E = m_0 ... | IPhO | 1,995 | 57 | 3 | 1 |
In order to determine $R$ and $M$ in a gravitational redshift experiment, it is usual to consider the frequency correction due to the recoil of the emitting atom. [Thermal motion causes emission lines to be broadened without displacing emission maxima, and we may therefore assume that all thermal effects have been take... | For the emitted photon, $h f = \Delta E \left[1 - \frac{\Delta E}{2 m_0 c^2}\right]$. If relativistic effects are ignored, then $h f_0 = \Delta E$. Hence the relativistic frequency shift $\frac{\Delta f}{f_0}$ is given by $\frac{\Delta f}{f_0} = \frac{\Delta E}{2 m_0 c^2}$. For $\mathrm{He}^{+}$ transition $(n = 2 \rig... | IPhO | 1,995 | 57 | 3 | 2 |
In a well-known model of an ideal gas, whose equation of state obeys the Clapeyron-Mendeleev law, the following important physical effects are neglected. First, molecules of a real gas have a finite size and, secondly, they interact with one another. In this problem one mole of water is considered. Taking into account ... | $b \approx N_A d^3$, where $N_A$ is Avogadro's number. | IPhO | 2,014 | 30 | 1 | null |
With account of intermolecular attraction forces, van der Waals proposed the following equation of state that neatly describes both the gaseous and liquid states of matter: $\left(P+\frac{a}{V^2}\right)(V-b)=R T$, where $a$ is a specific constant. At temperatures $T$ below a certain critical value $T_c$, the isotherm o... | $a=\frac{27 R^2 T_c^2}{64 P_c}$ | IPhO | 2,014 | 30 | 2 | null |
With account of intermolecular attraction forces, van der Waals proposed the following equation of state that neatly describes both the gaseous and liquid states of matter: $\left(P+\frac{a}{V^2}\right)(V-b)=R T$, where $a$ is a specific constant. At temperatures $T$ below a certain critical value $T_c$, the isotherm o... | $b=\frac{R T_c}{8 P_c}$ | IPhO | 2,014 | 30 | 2 | null |
For water, the critical temperature is $T_c=647 \mathrm{~K}$ and the critical pressure is $P_c=2.2 \cdot 10^7 \mathrm{Pa}$. The van der Waals constant $a$ is given by $a=\frac{27 R^2 T_c^2}{64 P_c}$, where $R$ is the universal gas constant. Calculate the numerical value of $a_w$ for water. | $a_w=0.56 \frac{\mathrm{m}^6 \cdot \mathrm{Pa}}{\mathrm{mole}^2}$ | IPhO | 2,014 | 30 | 3 | null |
For water, the critical temperature is $T_c=647 \mathrm{~K}$ and the critical pressure is $P_c=2.2 \cdot 10^7 \mathrm{Pa}$. The van der Waals constant $b$ is given by $b=\frac{R T_c}{8 P_c}$, where $R$ is the universal gas constant. Calculate the numerical value of $b_w$ for water. | $b_w=3.1 \cdot 10^{-5} \frac{\mathrm{m}^3}{\mathrm{~mole}}$ | IPhO | 2,014 | 30 | 3 | null |
For water, the van der Waals constant $b_w=3.1 \cdot 10^{-5} \frac{\mathrm{m}^3}{\mathrm{~mole}}$. The parameter $b$ is approximately equal to the volume of all molecules in one mole, i.e., $b \approx N_A d^3$, where $N_A$ is Avogadro's number and $d$ is the diameter of the molecules. Estimate the diameter of water mol... | $d_w=\sqrt[3]{\frac{b_w}{N_A}}=3.7 \cdot 10^{-10} \mathrm{~m} \approx 4 \cdot 10^{-10} \mathrm{~m}$ | IPhO | 2,014 | 30 | 4 | null |
A unit cell of a crystal of natrium chloride (common salt, $\mathrm{NaCl}$) is a cube with edge length $a = 5.6 \times 10^{-10} \mathrm{~m}$. The unit cell structure is as follows: Natrium atoms are located at the body center of the cube (1 atom) and at the midpoints of the 12 edges of the cube. Chlorine atoms are loca... | First, calculate the number of natrium atoms ($n_1$) and chlorine atoms ($n_2$) in a single $\mathrm{NaCl}$ unit cell. One natrium atom is at the body center and belongs entirely to the cell. 12 natrium atoms are on the edges, and each belongs to 4 adjacent cells, so $1/4$ of each belongs to this cell. Thus, $n_1 = 1 +... | IPhO | 1,970 | 17 | null | null |
A glass sphere of radius $R$ contains a spherical air bubble of radius $r$. The center of the bubble is located at a distance $d$ from the center of the sphere. Describe methods to determine the diameter of the bubble ($2r$) without damaging the sphere. | We cannot rely on any value about the density of the glass as it is quite uncertain. The index of refraction can be determined using a light beam which does not touch the bubble. Another method consists of immersing the sphere into a liquid of the same index of refraction: its surface becomes invisible. A great number ... | IPhO | 1,976 | 71 | null | null |
The mean temperature of the earth is $T=287 \mathrm{~K}$. What would the new mean temperature $T^{\prime}$ be if the mean distance between the earth and the sun was reduced by $1 \%$? Assume the Earth is in radiative equilibrium with the Sun. | If the solar power output is $P$ and the radius of the earth's orbit is $R$, then $T$ is given by equating incoming and outgoing radiation: $ (1-r) \frac{P}{4 \pi R^2} \cdot \pi R_E^2 = 4 \pi R_E^2 \varepsilon \sigma T^4 $ Where $r$ is the reflectance of the earth with respect to solar radiation (albedo), $R_E$ is the ... | IPhO | 1,997 | 89 | null | null |
A ball, thrown with an initial speed $v_0$, moves in a homogeneous gravitational field in the $x$-$z$ plane, where the $x$-axis is horizontal, and the $z$-axis is vertical and antiparallel to the free fall acceleration $g$. Neglect the effect of air drag. By adjusting the launching angle for a ball thrown with a fixed ... | $\frac{v_0^2}{2g}$ | IPhO | 2,012 | 63 | 1 | null |
A ball, thrown with an initial speed $v_0$, moves in a homogeneous gravitational field in the $x$-$z$ plane, where the $x$-axis is horizontal, and the $z$-axis is vertical and antiparallel to the free fall acceleration $g$. Neglect the effect of air drag. By adjusting the launching angle for a ball thrown with a fixed ... | $\frac{g}{2v_0^2}$ | IPhO | 2,012 | 63 | 1 | null |
A ball is thrown from the ground level $z=0$ toward a spherical building of radius $R$ centered at $(0, R)$ (so the building sits on the ground with its topmost point at $(0, 2R)$). The launching point can be freely selected on the ground level $z=0$, and the launching angle can be adjusted as needed. The aim is to hit... | $3\sqrt{\frac{gR}{2}}$ | IPhO | 2,012 | 63 | 3 | null |
Consider a parallel, transparent plate of thickness $d$. Its refractive index varies as $n=\frac{n_0}{1-\frac{x}{R}}$. A light beam enters from the air perpendicularly to the plate at the point A ($x_A=0$) and emerges from it at the point B at an angle $\alpha$. Data: $n_0=1.2$, $R=13$ cm, $\alpha=30^\circ$. Find the r... | $n_B=\sqrt{n_0^2+\sin^2\alpha}=1.3$ | IPhO | 1,972 | 101 | 1 | null |
Consider a parallel, transparent plate of thickness $d$. Its refractive index varies as $n=\frac{n_0}{1-\frac{x}{R}}$. A light beam enters from the air perpendicularly to the plate at the point A ($x_A=0$) and emerges from it at the point B at an angle $\alpha$. Data: $n_0=1.2$, $R=13$ cm, $\alpha=30^\circ$. Find $x_B$... | $x_B=R\left(1-\frac{n_0}{\sqrt{n_0^2+\sin^2\alpha}}\right)=1$ cm | IPhO | 1,972 | 101 | 2 | null |
Consider a parallel, transparent plate of thickness $d$. Its refractive index varies as $n=\frac{n_0}{1-\frac{x}{R}}$. A light beam enters from the air perpendicularly to the plate at the point A ($x_A=0$) and emerges from it at the point B at an angle $\alpha$. Data: $n_0=1.2$, $R=13$ cm, $\alpha=30^\circ$. Find the t... | From Snell's law applied to the varying refractive index, we have $n(x)\sin\beta(x)=n_0$, where $\beta(x)$ is the angle between the ray and the vertical. This gives $\sin\beta(x)=\frac{n_0}{n(x)}=\frac{R-x}{R}$, which is exactly the sine of the angle for a circular arc with center at $(R, 0)$ and radius $R$. From the g... | IPhO | 1,972 | 101 | 3 | null |
This problem is concerned with the difficulties of detecting gravitational waves generated by astronomical events. It should be realised that the explosion of a distant supernova may produce fluctuations in the gravitational field strength at the surface of the Earth of about $10^{-19} \mathrm{~N} \mathrm{~kg}^{-1}$. A... | $\mu=4.5 \times 10^{-3} \mathrm{~s}^{-1}$ | IPhO | 2,000 | 37 | 1 | 1 |
This problem is concerned with the difficulties of detecting gravitational waves generated by astronomical events. It should be realised that the explosion of a distant supernova may produce fluctuations in the gravitational field strength at the surface of the Earth of about $10^{-19} \mathrm{~N} \mathrm{~kg}^{-1}$. A... | $\omega=8.1 \times 10^3 \mathrm{rad} \cdot \mathrm{s}^{-1}$ | IPhO | 2,000 | 37 | 1 | 2 |
This problem is concerned with the difficulties of detecting gravitational waves generated by astronomical events. It should be realised that the explosion of a distant supernova may produce fluctuations in the gravitational field strength at the surface of the Earth of about $10^{-19} \mathrm{~N} \mathrm{~kg}^{-1}$. A... | $\delta l=3.8 \times 10^{-6} \mathrm{~m}$ | IPhO | 2,000 | 37 | 1 | 3 |
This problem is concerned with the difficulties of detecting gravitational waves generated by astronomical events. It should be realised that the explosion of a distant supernova may produce fluctuations in the gravitational field strength at the surface of the Earth of about $10^{-19} \mathrm{~N} \mathrm{~kg}^{-1}$. A... | $\Delta l=(\rho \Delta g / 2 E) l^2$ | IPhO | 2,000 | 37 | 1 | 4 |
This problem is concerned with the difficulties of detecting gravitational waves generated by astronomical events. It should be realised that the explosion of a distant supernova may produce fluctuations in the gravitational field strength at the surface of the Earth of about $10^{-19} \mathrm{~N} \mathrm{~kg}^{-1}$. A... | $l=9.2 \times 10^7 \mathrm{~m}$ | IPhO | 2,000 | 37 | 1 | 5 |
A non-directional form of gravitational wave detector consists of a sphere of copper alloy of mass $1168 \mathrm{~kg}$, suspended in a vacuum from a vibration-reducing assembly. Transducers, containing tuned circuits, are attached to the sphere to detect changes in its dimensions. The transducers will, however, pick up... | The amplitude of atomic vibrations is proportional to $\sqrt{T}$, so the reduction factor is $\sqrt{100 \mathrm{mK} / 300 \mathrm{K}} = \sqrt{0.1 / 300} = \sqrt{1/3000} \approx 0.0182$ or approximately $1/55$. | IPhO | 2,000 | 37 | 1 | 6 |
A non-directional form of gravitational wave detector consists of a sphere of copper alloy of mass $1168 \mathrm{~kg}$, suspended in a vacuum from a vibration-reducing assembly. Transducers, containing tuned circuits, are attached to the sphere to detect changes in its dimensions. The sphere is initially cooled to $4.2... | The heat capacity is $s = s_0(T/T_0)^3$ where $s_0 = 0.072 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$ at $T_0 = 4.2 \mathrm{~K}$. The energy removed is $Q = m\int_{T_2}^{T_1} s \, dT = m(s_0/T_0^3)\int_{T_2}^{T_1} T^3 \, dT = m(s_0/T_0^3)(T_1^4 - T_2^4)/4$. With $m = 1168 \mathrm{~kg}$, $T_1 = 4.2 \mathrm{~K}$, $T... | IPhO | 2,000 | 37 | 1 | 7 |
This problem is concerned with the effect of a gravitational field on the propagation of light in space. A photon emitted from the surface of the Sun (mass $M$, radius $R$) is red-shifted. By assuming a rest-mass equivalent for the photon energy, apply Newtonian gravitational theory to show that the effective (or measu... | Using $m c^2=h f \Rightarrow m=h f / c^2$, and energy conservation $h f^{\prime}=h f-G M m / R$, we get $h f^{\prime}=h f\left(1-G M / R c^2\right)$, therefore $f^{\prime}=f\left(1-G M / R c^2\right)$. | IPhO | 2,000 | 37 | 2 | 1 |
This problem is concerned with the effect of a gravitational field on the propagation of light in space. A photon emitted from the surface of the Sun (mass $M$, radius $R$) is red-shifted. By assuming a rest-mass equivalent for the photon energy, Newtonian gravitational theory shows that the effective frequency of the ... | From the redshift factor, $c_r^{\prime} = c(1-GM/rc^2)$. Since time dilation and length contraction both occur by the same factor, $n_r = c/c_r^{\prime} = 1/(1-GM/rc^2) \approx 1 + GM/rc^2$ for small $GM/rc^2$. However, considering both time and length effects, $n_r = c/[c(1-GM/rc^2)^2] = 1 + 2GM/rc^2$ for small $GM/rc... | IPhO | 2,000 | 37 | 2 | 2 |
This problem is concerned with the effect of a gravitational field on the propagation of light in space. A photon emitted from the surface of the Sun (mass $M$, radius $R$) is red-shifted. By assuming a rest-mass equivalent for the photon energy, Newtonian gravitational theory shows that the effective frequency of the ... | By Snell's law: $n(r+\delta r) \sin \theta=n(r) \sin (\theta-\delta \xi)$, which gives $(d n / d r) \delta r \sin \theta=-n(r) \cos \theta \delta \xi$. With $n(r)=1+2 G M / r c^2$, we have $(d n / d r)=-2 G M / c^2 r^2$, so $\delta \xi=\left(2 G M \tan \theta / c^2 r^2\right) \delta r$. With $r^2=x^2+R^2$ and $r d r=x ... | IPhO | 2,000 | 37 | 2 | 3 |
End of preview. Expand in Data Studio
This is the part of the physics dataset that we sanitized in the paper From Weeks to Hours: Fast and Principled SFT Curation for LLM.
It contains 3,084 text-only physics olympiad question–answer pairs covering seven competitions, with recorded years spanning 1967–2018. We also recorded source metadata for each question.
Data Sanitizing
The source material is collected from PhoPile (See citation below). Their original collection contains 2,662 entries, including problem statements, source solutions, competition and year metadata, question numbering, and references to question or solution images.
We sanitized the dataset with Qwen3.5-27B.
- We split multipart questions and subquestions to make sure that each question is independent and self-contained, containing all the information and conditions need to solve it.
- When later subquestions refer to the formulas and conditions in previous subquestions, such information is incuded in the later subquestions as well when we split it.
- If previous subquestions only provide hint or scaffolding, when formulating the latter subquestions that does not refer to the previous subquestions, we drop the previous subquestions.
- We further handled the images in the problems. Since the original dataset did not provide the images (only their local paths), we instructed the model to analyze based on the question and answer text, whether the image referred to contain additional information that cannot be recovered from text or not.
- We explicitly request the model to try to reverse-engineer what information is presented in the image, and ask the model to provide a text-only problem statement with the image information incorporated.
- Eventually, even though 33% of the original questions contained images which we have no access to, the information in 89% of these images can be recovered and incorporated into the text-only problem statements.
Citation
This dataset was built upon the following work
@inproceedings{zheng2025phopile,
title = "Benchmarking Foundation Models with Retrieval-Augmented Generation in Olympic-Level Physics Problem Solving",
author = "Zheng, Shunfeng and Zhang, Yudi and Fang, Meng and Zhang, Zihan and Wu, Zhitan and Pechenizkiy, Mykola and Chen, Ling",
booktitle = "Findings of the Association for Computational Linguistics: EMNLP 2025",
year = "2025",
}
If you find the dataset useful, please consider citing our work
@inproceedings{
jin2026from,
title={From Weeks to Hours: Fast and Principled {SFT} Curation for {LLM}},
author={Hongyi Henry Jin and Wenhan Yang and Meysam Ghaffari and Carlos Morato and Baharan Mirzasoleiman},
booktitle={The Fortieth Annual Conference on Neural Information Processing Systems},
year={2026},
url={https://openreview.net/forum?id=4jOHUJJVCg}
}
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