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Do these two combinations have the same dimensions in SI base units?
1. angular momentum x electric current / thermal conductivity
2. volume x electric current / specific heat capacity
Reduce both to base units, then end with a verdict line. | The first reduces to m s^2 A K and the second to m s^2 A K. Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. capacitance x power x time
2. velocity x surface tension x luminous intensity
Reduce both to base units, then end with a verdict line. | The first reduces to s^2 A^2 and the second to kg m cd / (s^3). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: electrical resistance x specific heat capacity x moment of inertia
Term 2: volumetric flow rate x wavenumber / inductance
Term 3: capacitance x power / mass
Term 4: wavenumber x volumetric flow rat... | Terms in a sum must share dimensions. The other three all reduce to s A^2 / (kg), while electrical resistance x specific heat capacity x moment of inertia reduces to kg^2 m^6 / (s^5 A^2 K), so it is the one that cannot be added to them.
Offending term: Term 1 (electrical resistance x specific heat capacity x moment of... | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: mass flow rate x dynamic viscosity / thermodynamic temperature
Term 2: mass x electrical resistance / power
Term 3: inductance x density / pressure
Term 4: density x inductance / pressure | Terms in a sum must share dimensions. The other three all reduce to kg / (A^2), while mass flow rate x dynamic viscosity / thermodynamic temperature reduces to kg^2 / (m s^2 K), so it is the one that cannot be added to them.
Offending term: Term 1 (mass flow rate x dynamic viscosity / thermodynamic temperature). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: energy x dynamic viscosity x surface tension
Term 2: thermodynamic temperature x thermal conductivity x surface tension
Term 3: electric potential x energy density x electric current
Term 4: surfac... | Terms in a sum must share dimensions. The other three all reduce to kg^2 m / (s^5), while energy x dynamic viscosity x surface tension reduces to kg^3 m / (s^5), so it is the one that cannot be added to them.
Offending term: Term 1 (energy x dynamic viscosity x surface tension). | null | null |
Do these two combinations have the same dimensions in SI base units?
1. volumetric flow rate x capacitance x thermodynamic temperature
2. volumetric flow rate x thermodynamic temperature x capacitance
Reduce both to base units, then end with a verdict line. | The first reduces to m s^3 A^2 K / (kg) and the second to m s^3 A^2 K / (kg). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. length / amount of substance
2. force x magnetic flux density x amount of substance
Reduce both to base units, then end with a verdict line. | The first reduces to m / (mol) and the second to kg^2 m mol / (s^4 A). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: force x electric current / magnetic flux
Term 2: amount of substance x dynamic viscosity / inductance
Term 3: amount of substance x pressure / electrical resistance
Term 4: pressure x amount of sub... | Terms in a sum must share dimensions. The other three all reduce to s A^2 mol / (m^3), while force x electric current / magnetic flux reduces to A^2 / (m), so it is the one that cannot be added to them.
Offending term: Term 1 (force x electric current / magnetic flux). | null | null |
Express this combination in SI base units, and say which base units survive.
time x thermodynamic temperature / mass flow rate
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power -1, s to the power 2, K to the power 1.
SI base units: s^2 K / (kg) | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
moment of inertia / energy density = volume x mass / surface tension
Work out both sides in SI base units, then end with a verdict line. | The left side, moment of inertia / energy density, is m^3 s^2. Reducing the right side gives the same combination, m^3 s^2, so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: angular momentum x area x density
Term 2: density x volume x momentum
Term 3: molar mass x momentum x amount of substance
Term 4: energy density x molar mass / surface tension | Terms in a sum must share dimensions. The other three all reduce to kg^2 m / (s), while energy density x molar mass / surface tension reduces to kg / (m mol), so it is the one that cannot be added to them.
Offending term: Term 4 (energy density x molar mass / surface tension). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
length x inductance = length x time x electrical resistance
Work out both sides in SI base units, then end with a verdict line. | The left side, length x inductance, is kg m^3 / (s^2 A^2). Reducing the right side gives the same combination, kg m^3 / (s^2 A^2), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
mass flow rate / specific heat capacity = momentum x velocity / acceleration
Work out both sides in SI base units, then end with a verdict line. | The left side, mass flow rate / specific heat capacity, is kg s K / (m^2), while the right side reduces to kg m. Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: thermal conductivity x capacitance x volume
Term 2: volumetric flow rate x velocity / inductance
Term 3: angular momentum x specific heat capacity x capacitance
Term 4: specific heat capacity x cap... | Terms in a sum must share dimensions. The other three all reduce to m^2 s A^2 / (K), while volumetric flow rate x velocity / inductance reduces to m^2 A^2 / (kg), so it is the one that cannot be added to them.
Offending term: Term 2 (volumetric flow rate x velocity / inductance). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
luminous intensity / capacitance = luminous intensity x electric potential / electric charge
Work out both sides in SI base units, then end with a verdict line. | The left side, luminous intensity / capacitance, is kg m^2 cd / (s^4 A^2). Reducing the right side gives the same combination, kg m^2 cd / (s^4 A^2), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. moment of inertia x surface tension x wavenumber
2. inductance x moment of inertia / magnetic flux density
Reduce both to base units, then end with a verdict line. | The first reduces to kg^2 m / (s^2) and the second to kg m^4 / (A). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
amount of substance / inductance = velocity x molar mass
Work out both sides in SI base units, then end with a verdict line. | The left side, amount of substance / inductance, is s^2 A^2 mol / (kg m^2), while the right side reduces to kg m / (s mol). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: mass flow rate x electric current / energy density
Term 2: wavenumber x area x electric charge
Term 3: force x luminous intensity x mass
Term 4: angular momentum x velocity / electric potential | Terms in a sum must share dimensions. The other three all reduce to m s A, while force x luminous intensity x mass reduces to kg^2 m cd / (s^2), so it is the one that cannot be added to them.
Offending term: Term 3 (force x luminous intensity x mass). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
mass flow rate / magnetic flux = mass flow rate x electric charge / angular momentum
Work out both sides in SI base units, then end with a verdict line. | The left side, mass flow rate / magnetic flux, is s A / (m^2). Reducing the right side gives the same combination, s A / (m^2), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: thermal conductivity x capacitance / specific heat capacity
Term 2: wavenumber x capacitance x mass flow rate
Term 3: mass x wavenumber / electrical resistance
Term 4: electric current x molar mass... | Terms in a sum must share dimensions. The other three all reduce to s^3 A^2 / (m^3), while electric current x molar mass / angular momentum reduces to s A / (m^2 mol), so it is the one that cannot be added to them.
Offending term: Term 4 (electric current x molar mass / angular momentum). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: length x capacitance / magnetic flux density
Term 2: capacitance x electric charge / dynamic viscosity
Term 3: volume x capacitance / magnetic flux
Term 4: density x surface tension / volumetric fl... | Terms in a sum must share dimensions. The other three all reduce to s^6 A^3 / (kg^2 m), while density x surface tension / volumetric flow rate reduces to kg^2 / (m^6 s), so it is the one that cannot be added to them.
Offending term: Term 4 (density x surface tension / volumetric flow rate). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
force x acceleration = angular momentum x power / moment of inertia
Work out both sides in SI base units, then end with a verdict line. | The left side, force x acceleration, is kg m^2 / (s^4). Reducing the right side gives the same combination, kg m^2 / (s^4), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
Express this combination in SI base units, and say which base units survive.
density x thermal conductivity / volume
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power 2, m to the power -5, s to the power -3, K to the power -1.
SI base units: kg^2 / (m^5 s^3 K) | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
mass flow rate x force = momentum x length x energy density
Work out both sides in SI base units, then end with a verdict line. | The left side, mass flow rate x force, is kg^2 m / (s^3). Reducing the right side gives the same combination, kg^2 m / (s^3), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
electrical resistance x dynamic viscosity = pressure x time x electrical resistance
Work out both sides in SI base units, then end with a verdict line. | The left side, electrical resistance x dynamic viscosity, is kg^2 m / (s^4 A^2). Reducing the right side gives the same combination, kg^2 m / (s^4 A^2), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. dynamic viscosity x density / magnetic flux
2. pressure x density / electric potential
Reduce both to base units, then end with a verdict line. | The first reduces to kg s A / (m^6) and the second to kg s A / (m^6). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: magnetic flux x time x molar mass
Term 2: molar mass x electric charge x inductance
Term 3: molar mass x inductance x electric charge
Term 4: electrical resistance x electric current | Terms in a sum must share dimensions. The other three all reduce to kg^2 m^2 / (s A mol), while electrical resistance x electric current reduces to kg m^2 / (s^3 A), so it is the one that cannot be added to them.
Offending term: Term 4 (electrical resistance x electric current). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: angular momentum x power x capacitance
Term 2: power x angular momentum x capacitance
Term 3: electric charge x angular momentum x electric current
Term 4: energy / capacitance | Terms in a sum must share dimensions. The other three all reduce to kg m^2 A^2, while energy / capacitance reduces to kg^2 m^4 / (s^6 A^2), so it is the one that cannot be added to them.
Offending term: Term 4 (energy / capacitance). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
inductance / force = momentum x wavenumber x specific heat capacity
Work out both sides in SI base units, then end with a verdict line. | The left side, inductance / force, is m / (A^2), while the right side reduces to kg m^2 / (s^3 K). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: force x thermal conductivity / electric potential
Term 2: surface tension x electric current / thermodynamic temperature
Term 3: electric charge x thermal conductivity / length
Term 4: angular mome... | Terms in a sum must share dimensions. The other three all reduce to kg A / (s^2 K), while angular momentum x acceleration x magnetic flux density reduces to kg^2 m^3 / (s^5 A), so it is the one that cannot be added to them.
Offending term: Term 4 (angular momentum x acceleration x magnetic flux density). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
thermodynamic temperature x density = dynamic viscosity x density / energy density
Work out both sides in SI base units, then end with a verdict line. | The left side, thermodynamic temperature x density, is kg K / (m^3), while the right side reduces to kg s / (m^3). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
volume / dynamic viscosity = volumetric flow rate / energy density
Work out both sides in SI base units, then end with a verdict line. | The left side, volume / dynamic viscosity, is m^4 s / (kg). Reducing the right side gives the same combination, m^4 s / (kg), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. momentum x magnetic flux density / electric charge
2. electric potential x area x dynamic viscosity
Reduce both to base units, then end with a verdict line. | The first reduces to kg^2 m / (s^4 A^2) and the second to kg^2 m^3 / (s^4 A). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
Express this combination in SI base units, and say which base units survive.
magnetic flux x thermodynamic temperature x wavenumber
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power 1, m to the power 1, s to the power -2, A to the power -1, K to the power 1.
SI base units: kg m K / (s^2 A) | null | null |
Do these two combinations have the same dimensions in SI base units?
1. thermal conductivity x force x molar mass
2. molar mass x force x thermal conductivity
Reduce both to base units, then end with a verdict line. | The first reduces to kg^3 m^2 / (s^5 K mol) and the second to kg^3 m^2 / (s^5 K mol). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: specific heat capacity x inductance x electric charge
Term 2: angular momentum x amount of substance x electrical resistance
Term 3: inductance x amount of substance x energy
Term 4: energy x amoun... | Terms in a sum must share dimensions. The other three all reduce to kg^2 m^4 mol / (s^4 A^2), while specific heat capacity x inductance x electric charge reduces to kg m^4 / (s^3 A K), so it is the one that cannot be added to them.
Offending term: Term 1 (specific heat capacity x inductance x electric charge). | null | null |
Do these two combinations have the same dimensions in SI base units?
1. velocity x angular momentum x amount of substance
2. acceleration x moment of inertia x amount of substance
Reduce both to base units, then end with a verdict line. | The first reduces to kg m^3 mol / (s^2) and the second to kg m^3 mol / (s^2). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
wavenumber / momentum = molar mass x magnetic flux x electrical resistance
Work out both sides in SI base units, then end with a verdict line. | The left side, wavenumber / momentum, is s / (kg m^2), while the right side reduces to kg^3 m^4 / (s^5 A^3 mol). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. magnetic flux x acceleration / amount of substance
2. length x volumetric flow rate / momentum
Reduce both to base units, then end with a verdict line. | The first reduces to kg m^3 / (s^4 A mol) and the second to m^3 / (kg). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. volumetric flow rate x capacitance x momentum
2. area x angular momentum / electrical resistance
Reduce both to base units, then end with a verdict line. | The first reduces to m^2 s^2 A^2 and the second to m^2 s^2 A^2. Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
frequency / thermodynamic temperature = thermal conductivity x length / energy
Work out both sides in SI base units, then end with a verdict line. | The left side, frequency / thermodynamic temperature, is 1 / (s K). Reducing the right side gives the same combination, 1 / (s K), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. length x dynamic viscosity x pressure
2. electric current x surface tension / force
Reduce both to base units, then end with a verdict line. | The first reduces to kg^2 / (m s^3) and the second to A / (m). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
energy / thermal conductivity = density x pressure / capacitance
Work out both sides in SI base units, then end with a verdict line. | The left side, energy / thermal conductivity, is m s K, while the right side reduces to kg^3 / (m^2 s^6 A^2). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: luminous intensity x wavenumber / density
Term 2: area x power / mass flow rate
Term 3: force x volumetric flow rate / mass flow rate
Term 4: thermodynamic temperature x specific heat capacity x ar... | Terms in a sum must share dimensions. The other three all reduce to m^4 / (s^2), while luminous intensity x wavenumber / density reduces to m^2 cd / (kg), so it is the one that cannot be added to them.
Offending term: Term 1 (luminous intensity x wavenumber / density). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
electrical resistance x velocity = surface tension x energy density / mass
Work out both sides in SI base units, then end with a verdict line. | The left side, electrical resistance x velocity, is kg m^3 / (s^4 A^2), while the right side reduces to kg / (m s^4). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. pressure x magnetic flux / angular momentum
2. electric charge x area
Reduce both to base units, then end with a verdict line. | The first reduces to kg / (m s^3 A) and the second to m^2 s A. At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: mass flow rate / thermal conductivity
Term 2: capacitance x thermodynamic temperature x molar mass
Term 3: thermodynamic temperature / acceleration
Term 4: length / specific heat capacity | Terms in a sum must share dimensions. The other three all reduce to s^2 K / (m), while capacitance x thermodynamic temperature x molar mass reduces to s^4 A^2 K / (m^2 mol), so it is the one that cannot be added to them.
Offending term: Term 2 (capacitance x thermodynamic temperature x molar mass). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: pressure / inductance
Term 2: frequency x dynamic viscosity / inductance
Term 3: power x moment of inertia x electric charge
Term 4: pressure x electric current / magnetic flux | Terms in a sum must share dimensions. The other three all reduce to A^2 / (m^3), while power x moment of inertia x electric charge reduces to kg^2 m^4 A / (s^2), so it is the one that cannot be added to them.
Offending term: Term 3 (power x moment of inertia x electric charge). | null | null |
Do these two combinations have the same dimensions in SI base units?
1. thermodynamic temperature x mass / mass flow rate
2. length x magnetic flux density x thermodynamic temperature
Reduce both to base units, then end with a verdict line. | The first reduces to s K and the second to kg m K / (s^2 A). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. time x power x energy density
2. electrical resistance x power / electric potential
Reduce both to base units, then end with a verdict line. | The first reduces to kg^2 m / (s^4) and the second to kg m^2 / (s^3 A). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
momentum x time = capacitance x momentum x luminous intensity
Work out both sides in SI base units, then end with a verdict line. | The left side, momentum x time, is kg m, while the right side reduces to s^3 A^2 cd / (m). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
Express this combination in SI base units, and say which base units survive.
specific heat capacity x magnetic flux / magnetic flux density
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves m to the power 4, s to the power -2, K to the power -1.
SI base units: m^4 / (s^2 K) | null | null |
Do these two combinations have the same dimensions in SI base units?
1. amount of substance x energy density / wavenumber
2. magnetic flux / time
Reduce both to base units, then end with a verdict line. | The first reduces to kg mol / (s^2) and the second to kg m^2 / (s^3 A). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: magnetic flux density x mass / mass flow rate
Term 2: mass flow rate x magnetic flux / energy
Term 3: magnetic flux density x electrical resistance x capacitance
Term 4: specific heat capacity x el... | Terms in a sum must share dimensions. The other three all reduce to kg / (s A), while specific heat capacity x electric potential reduces to kg m^4 / (s^5 A K), so it is the one that cannot be added to them.
Offending term: Term 4 (specific heat capacity x electric potential). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: frequency x thermal conductivity x velocity
Term 2: acceleration x specific heat capacity x dynamic viscosity
Term 3: thermal conductivity x velocity / time
Term 4: force x power / pressure | Terms in a sum must share dimensions. The other three all reduce to kg m^2 / (s^5 K), while force x power / pressure reduces to kg m^4 / (s^3), so it is the one that cannot be added to them.
Offending term: Term 4 (force x power / pressure). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
mass x electrical resistance = velocity x surface tension / thermodynamic temperature
Work out both sides in SI base units, then end with a verdict line. | The left side, mass x electrical resistance, is kg^2 m^2 / (s^3 A^2), while the right side reduces to kg m / (s^3 K). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: energy density x specific heat capacity / force
Term 2: specific heat capacity / area
Term 3: thermal conductivity / momentum
Term 4: length x luminous intensity | Terms in a sum must share dimensions. The other three all reduce to 1 / (s^2 K), while length x luminous intensity reduces to m cd, so it is the one that cannot be added to them.
Offending term: Term 4 (length x luminous intensity). | null | null |
Express this combination in SI base units, and say which base units survive.
energy density x electric charge x dynamic viscosity
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power 2, m to the power -2, s to the power -2, A to the power 1.
SI base units: kg^2 A / (m^2 s^2) | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
surface tension / area = specific heat capacity x wavenumber / volume
Work out both sides in SI base units, then end with a verdict line. | The left side, surface tension / area, is kg / (m^2 s^2), while the right side reduces to 1 / (m^2 s^2 K). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
pressure x capacitance = volumetric flow rate x area / force
Work out both sides in SI base units, then end with a verdict line. | The left side, pressure x capacitance, is s^2 A^2 / (m^3), while the right side reduces to m^4 s / (kg). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: specific heat capacity x volumetric flow rate / thermal conductivity
Term 2: frequency x volumetric flow rate / energy density
Term 3: surface tension x volume x energy density
Term 4: acceleration... | Terms in a sum must share dimensions. The other three all reduce to m^4 / (kg), while surface tension x volume x energy density reduces to kg^2 m^2 / (s^4), so it is the one that cannot be added to them.
Offending term: Term 3 (surface tension x volume x energy density). | null | null |
Do these two combinations have the same dimensions in SI base units?
1. mass x length / power
2. time / acceleration
Reduce both to base units, then end with a verdict line. | The first reduces to s^3 / (m) and the second to s^3 / (m). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: length x luminous intensity / acceleration
Term 2: velocity x luminous intensity x energy density
Term 3: luminous intensity x mass / surface tension
Term 4: time x luminous intensity / frequency | Terms in a sum must share dimensions. The other three all reduce to s^2 cd, while velocity x luminous intensity x energy density reduces to kg cd / (s^3), so it is the one that cannot be added to them.
Offending term: Term 2 (velocity x luminous intensity x energy density). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
angular momentum x pressure = energy x energy density / frequency
Work out both sides in SI base units, then end with a verdict line. | The left side, angular momentum x pressure, is kg^2 m / (s^3). Reducing the right side gives the same combination, kg^2 m / (s^3), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
power x momentum = wavenumber x length x energy
Work out both sides in SI base units, then end with a verdict line. | The left side, power x momentum, is kg^2 m^3 / (s^4), while the right side reduces to kg m^2 / (s^2). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
moment of inertia / time = capacitance x amount of substance / density
Work out both sides in SI base units, then end with a verdict line. | The left side, moment of inertia / time, is kg m^2 / (s), while the right side reduces to m s^4 A^2 mol / (kg^2). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. molar mass x inductance x surface tension
2. volume x amount of substance x electrical resistance
Reduce both to base units, then end with a verdict line. | The first reduces to kg^3 m^2 / (s^4 A^2 mol) and the second to kg m^5 mol / (s^3 A^2). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
Express this combination in SI base units, and say which base units survive.
wavenumber x specific heat capacity / dynamic viscosity
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power -1, m to the power 2, s to the power -1, K to the power -1.
SI base units: m^2 / (kg s K) | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: mass x volume x luminous intensity
Term 2: mass x luminous intensity x volume
Term 3: luminous intensity x moment of inertia / wavenumber
Term 4: velocity x frequency x area | Terms in a sum must share dimensions. The other three all reduce to kg m^3 cd, while velocity x frequency x area reduces to m^3 / (s^2), so it is the one that cannot be added to them.
Offending term: Term 4 (velocity x frequency x area). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: volumetric flow rate x momentum x angular momentum
Term 2: force x volume x angular momentum
Term 3: wavenumber x capacitance x frequency
Term 4: angular momentum x momentum x volumetric flow rate | Terms in a sum must share dimensions. The other three all reduce to kg^2 m^6 / (s^3), while wavenumber x capacitance x frequency reduces to s^3 A^2 / (kg m^3), so it is the one that cannot be added to them.
Offending term: Term 3 (wavenumber x capacitance x frequency). | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
area x volumetric flow rate = volumetric flow rate x volume / length
Work out both sides in SI base units, then end with a verdict line. | The left side, area x volumetric flow rate, is m^5 / (s). Reducing the right side gives the same combination, m^5 / (s), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
Express this combination in SI base units, and say which base units survive.
acceleration x density x electrical resistance
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power 2, s to the power -5, A to the power -2.
SI base units: kg^2 / (s^5 A^2) | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: luminous intensity / capacitance
Term 2: capacitance x inductance x density
Term 3: electrical resistance x luminous intensity / time
Term 4: frequency x electrical resistance x luminous intensity | Terms in a sum must share dimensions. The other three all reduce to kg m^2 cd / (s^4 A^2), while capacitance x inductance x density reduces to kg s^2 / (m^3), so it is the one that cannot be added to them.
Offending term: Term 2 (capacitance x inductance x density). | null | null |
Do these two combinations have the same dimensions in SI base units?
1. electric charge x specific heat capacity / pressure
2. thermodynamic temperature x electrical resistance x length
Reduce both to base units, then end with a verdict line. | The first reduces to m^3 s A / (kg K) and the second to kg m^3 K / (s^3 A^2). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
Express this combination in SI base units, and say which base units survive.
time / electrical resistance
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power -1, m to the power -2, s to the power 4, A to the power 2.
SI base units: s^4 A^2 / (kg m^2) | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: surface tension x capacitance / magnetic flux
Term 2: capacitance x electric current / area
Term 3: magnetic flux density x capacitance / inductance
Term 4: inductance / volume | Terms in a sum must share dimensions. The other three all reduce to s^4 A^3 / (kg m^4), while inductance / volume reduces to kg / (m s^2 A^2), so it is the one that cannot be added to them.
Offending term: Term 4 (inductance / volume). | null | null |
Express this combination in SI base units, and say which base units survive.
dynamic viscosity x luminous intensity x time
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power 1, m to the power -1, cd to the power 1.
SI base units: kg cd / (m) | null | null |
Do these two combinations have the same dimensions in SI base units?
1. density x energy / magnetic flux density
2. mass x surface tension / area
Reduce both to base units, then end with a verdict line. | The first reduces to kg A / (m) and the second to kg^2 / (m^2 s^2). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
Express this combination in SI base units, and say which base units survive.
dynamic viscosity x volume / energy
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves s to the power 1.
SI base units: s | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
electric current x luminous intensity = luminous intensity x electric current
Work out both sides in SI base units, then end with a verdict line. | The left side, electric current x luminous intensity, is A cd. Reducing the right side gives the same combination, A cd, so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: capacitance / energy density
Term 2: capacitance x volume / energy
Term 3: surface tension x energy x thermodynamic temperature
Term 4: capacitance / pressure | Terms in a sum must share dimensions. The other three all reduce to s^6 A^2 / (kg^2 m), while surface tension x energy x thermodynamic temperature reduces to kg^2 m^2 K / (s^4), so it is the one that cannot be added to them.
Offending term: Term 3 (surface tension x energy x thermodynamic temperature). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: luminous intensity x mass flow rate x electric current
Term 2: wavenumber x energy x frequency
Term 3: luminous intensity x electric charge x surface tension
Term 4: luminous intensity x electric c... | Terms in a sum must share dimensions. The other three all reduce to kg A cd / (s), while wavenumber x energy x frequency reduces to kg m / (s^3), so it is the one that cannot be added to them.
Offending term: Term 2 (wavenumber x energy x frequency). | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: mass x force / inductance
Term 2: energy x volume x mass
Term 3: area x force x moment of inertia
Term 4: mass x energy x volume | Terms in a sum must share dimensions. The other three all reduce to kg^2 m^5 / (s^2), while mass x force / inductance reduces to kg A^2 / (m), so it is the one that cannot be added to them.
Offending term: Term 1 (mass x force / inductance). | null | null |
Do these two combinations have the same dimensions in SI base units?
1. wavenumber x molar mass x electrical resistance
2. wavenumber x capacitance
Reduce both to base units, then end with a verdict line. | The first reduces to kg^2 m / (s^3 A^2 mol) and the second to s^4 A^2 / (kg m^3). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
magnetic flux / pressure = force x energy density x electrical resistance
Work out both sides in SI base units, then end with a verdict line. | The left side, magnetic flux / pressure, is m^3 / (A), while the right side reduces to kg^3 m^2 / (s^7 A^2). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. velocity / thermal conductivity
2. thermodynamic temperature x wavenumber / energy density
Reduce both to base units, then end with a verdict line. | The first reduces to s^2 K / (kg) and the second to s^2 K / (kg). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
volumetric flow rate x thermodynamic temperature = thermodynamic temperature x velocity x area
Work out both sides in SI base units, then end with a verdict line. | The left side, volumetric flow rate x thermodynamic temperature, is m^3 K / (s). Reducing the right side gives the same combination, m^3 K / (s), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
pressure / electric potential = pressure x electric current / power
Work out both sides in SI base units, then end with a verdict line. | The left side, pressure / electric potential, is s A / (m^3). Reducing the right side gives the same combination, s A / (m^3), so both sides carry identical dimensions and only a numerical factor could differ.
Verdict: dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. electric current x moment of inertia / molar mass
2. mass flow rate x thermodynamic temperature / magnetic flux
Reduce both to base units, then end with a verdict line. | The first reduces to m^2 A mol and the second to s A K / (m^2). At least one base-unit exponent differs, so they are dimensionally distinct.
Verdict: different dimensions. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
force x length = volume x specific heat capacity x magnetic flux
Work out both sides in SI base units, then end with a verdict line. | The left side, force x length, is kg m^2 / (s^2), while the right side reduces to kg m^7 / (s^4 A K). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
In SI base units, is this relation dimensionally consistent? Ignore any numerical factor and judge the dimensions only.
luminous intensity x frequency = amount of substance x time x electrical resistance
Work out both sides in SI base units, then end with a verdict line. | The left side, luminous intensity x frequency, is cd / (s), while the right side reduces to kg m^2 mol / (s^2 A^2). Those differ, so no numerical factor can reconcile them.
Verdict: not dimensionally consistent. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. mass flow rate x energy density x capacitance
2. energy density x capacitance x mass flow rate
Reduce both to base units, then end with a verdict line. | The first reduces to kg s A^2 / (m^3) and the second to kg s A^2 / (m^3). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: thermodynamic temperature / mass flow rate
Term 2: length x thermodynamic temperature / momentum
Term 3: angular momentum x specific heat capacity / thermodynamic temperature
Term 4: acceleration /... | Terms in a sum must share dimensions. The other three all reduce to s K / (kg), while angular momentum x specific heat capacity / thermodynamic temperature reduces to kg m^4 / (s^3 K^2), so it is the one that cannot be added to them.
Offending term: Term 3 (angular momentum x specific heat capacity / thermodynamic tem... | null | null |
Do these two combinations have the same dimensions in SI base units?
1. electric potential x frequency / angular momentum
2. electric potential / moment of inertia
Reduce both to base units, then end with a verdict line. | The first reduces to 1 / (s^3 A) and the second to 1 / (s^3 A). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
Express this combination in SI base units, and say which base units survive.
magnetic flux x volumetric flow rate x density
Give the base-unit form on the last line. | Reducing each factor to base units and collecting exponents leaves kg to the power 2, m to the power 2, s to the power -3, A to the power -1.
SI base units: kg^2 m^2 / (s^3 A) | null | null |
Do these two combinations have the same dimensions in SI base units?
1. time x electric charge x density
2. magnetic flux x capacitance x density
Reduce both to base units, then end with a verdict line. | The first reduces to kg s^2 A / (m^3) and the second to kg s^2 A / (m^3). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
Do these two combinations have the same dimensions in SI base units?
1. force x surface tension / specific heat capacity
2. energy density x mass x thermodynamic temperature
Reduce both to base units, then end with a verdict line. | The first reduces to kg^2 K / (m s^2) and the second to kg^2 K / (m s^2). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: volume x wavenumber x mass flow rate
Term 2: length x momentum
Term 3: energy / frequency
Term 4: inductance / thermal conductivity | Terms in a sum must share dimensions. The other three all reduce to kg m^2 / (s), while inductance / thermal conductivity reduces to m s K / (A^2), so it is the one that cannot be added to them.
Offending term: Term 4 (inductance / thermal conductivity). | null | null |
Do these two combinations have the same dimensions in SI base units?
1. mass flow rate x magnetic flux density / power
2. magnetic flux x density / force
Reduce both to base units, then end with a verdict line. | The first reduces to kg / (m^2 A) and the second to kg / (m^2 A). Every base-unit exponent agrees, so the two are dimensionally the same even though they describe different things.
Verdict: same dimensions. | null | null |
A physical equation adds these terms together. Exactly one of them cannot belong in the sum. Name it and say why.
Term 1: magnetic flux density x molar mass x velocity
Term 2: wavenumber x electric potential x molar mass
Term 3: dynamic viscosity x magnetic flux / amount of substance
Term 4: energy density x electric ... | Terms in a sum must share dimensions. The other three all reduce to kg^2 m / (s^3 A mol), while energy density x electric charge reduces to kg A / (m s), so it is the one that cannot be added to them.
Offending term: Term 4 (energy density x electric charge). | null | null |
End of preview. Expand in Data Studio
SI Dimensional Analysis
Dimensional reasoning in SI base units: is a relation dimensionally consistent, reduce a combination to base units, find the term that cannot belong in a sum, and decide whether two combinations share dimensions.
| Rows | 12,000 |
| Domain | physics |
| Format | data.parquet, one row per example |
| Licence | cc-by-4.0 |
| Built for | supervised fine-tuning (SFT) experiments on Adaption AutoScientist |
Columns
| Column | Description |
|---|---|
original_prompt |
The prompt (user turn) as uploaded. |
original_completion |
The target response as uploaded. |
enhanced_prompt |
Empty in this dataset. |
enhanced_completion |
Empty in this dataset. |
How it was built
Generated programmatically from dimension vectors of 36 SI quantities. An independent checker with its own SI table re-derived all labels from the prompt text (0 failures), and a planted-defect control confirmed it catches errors.
Sources and licence
- Generated by the author (no upstream dataset).
Notes
- Columns
enhanced_prompt,enhanced_completionare empty in this dataset (the Adaption export reserves them for rewritten text).
Loading
from datasets import load_dataset
ds = load_dataset("rodriguescarson/adaption-si-dimensional-analysis-12k", split="train")
import pandas as pd
df = pd.read_parquet("hf://datasets/rodriguescarson/adaption-si-dimensional-analysis-12k/data.parquet")
Published by Carson Rodrigues (Hugging Face, Kaggle).
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