JEE Mains Physics · Units and Measurements

Dimensions of Mechanical and Thermal Quantities

Every quantity's dimensions come from the equation that defines it, written in mass M, length L, time T and temperature K.

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Why this matters

Thirty-five PYQs, thirty-four of them multiple choice, and four from 2026. Twenty-three are match-the-list questions that pair four quantities with four dimensional formulas; six ask which pair of quantities has, or does not have, the same dimensions; the other six ask for the dimensions of one or two named quantities, from a unit, a formula or a ratio. A table learned once answers most of them in seconds.

Concept 1 of 3: Dimensions of mechanical quantities

Start from force,

F = ma

, which is

MLT^{-2}

. Almost everything else is force combined with a length, an area or a time: energy is force × length, pressure is force ÷ area, surface tension is force ÷ length. Remember the defining equation and the dimensions follow in one line.

Definition

Force $MLT^{-2}$ ; energy, work and torque $ML^{2}T^{-2}$ ; power $ML^{2}T^{-3}$ .
Momentum and impulse $MLT^{-1}$ ; angular momentum and angular impulse $ML^{2}T^{-1}$ .
Pressure, stress, every elastic modulus and energy density $ML^{-1}T^{-2}$ .
Surface tension and spring constant (force per length) $MT^{-2}$ .
Viscosity, from $F = \eta A\,\dfrac{dv}{dx}$ : $ML^{-1}T^{-1}$ ; its SI unit is the pascal-second.
A ratio of two quantities has the ratio of their dimensions.

Quantity	Defining relation	Dimensions	SI unit
Force	$F = ma$	$MLT^{-2}$	newton (N)
Work, energy, torque	$W = Fs,\ \tau = rF$	$ML^{2}T^{-2}$	J (N m for torque)
Power	$P = W/t$	$ML^{2}T^{-3}$	watt (W)
Momentum, impulse	$p = mv,\ J = Ft$	$MLT^{-1}$	kg m/s or N s
Angular momentum, angular impulse	$L = mvr,\ \tau t$	$ML^{2}T^{-1}$	kg m²/s
Moment of inertia	$I = mr^{2}$	$ML^{2}$	kg m²
Pressure, stress, Young's modulus, bulk modulus	$F/A$	$ML^{-1}T^{-2}$	pascal (Pa)
Pressure gradient	$dP/dx$	$ML^{-2}T^{-2}$	Pa/m
Compressibility	$1/\text{bulk modulus}$	$M^{-1}LT^{2}$	Pa⁻¹
Surface tension, spring constant	$F/l,\ F/x$	$MT^{-2}$	N/m
Coefficient of viscosity	$F = \eta A\,dv/dx$	$ML^{-1}T^{-1}$	pascal-second (Pa s)
Intensity of a wave	power ÷ area	$MT^{-3}$	W/m²
Gravitational constant G	$F = Gm_1m_2/r^{2}$	$M^{-1}L^{3}T^{-2}$	N m²/kg²
Gravitational potential	energy ÷ mass	$L^{2}T^{-2}$	J/kg
Angular speed, frequency, velocity gradient	$\omega = \theta/t,\ dv/dx$	$T^{-1}$	s⁻¹

Each row follows from its defining relation; force, MLT⁻², is the starting point for most of them.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 28 Jan 2026 Shift 2 · Q20Moderate

Example 1 · Units and Measurements · Dimensions of Mechanical and Thermal Quantities

Match List-I with List-II.

List-I	List-II
(A) Coefficient of viscosity	(I) $\left\lbrack ML^{-1}T^{-2} \right\rbrack$
(B) Surface tension	(II) $\left\lbrack ML^{2}T^{-2} \right\rbrack$
(C) Pressure	(III) $\left\lbrack ML^{0}T^{-2} \right\rbrack$
(D) Surface energy	(IV) $\left\lbrack ML^{-1}T^{-1} \right\rbrack$

Choose the correct answer from the options given below:

Surface energy and surface tension are not the same entry

Surface tension is force per length, MT⁻². A match list that names 'surface energy' means an energy, ML²T⁻². Read the exact words before matching.

Viscosity has one power of time, pressure has two

Viscosity is ML⁻¹T⁻¹ and pressure is ML⁻¹T⁻². The two differ only in the power of T, and match lists put them side by side to catch a slip.

Drill 18 more on dimensions of mechanical quantities

Concept 2 of 3: Dimensions of thermal and modern-physics constants

A constant takes whatever dimensions make its equation balance. Boltzmann's constant turns temperature into energy (

E = k_BT

), so it is energy per kelvin. Planck's constant turns frequency into energy (

E = h\nu

), so it is energy × time. Write the equation, isolate the constant, and read off its dimensions.

Definition

$k_B = E/T$ : $ML^{2}T^{-2}K^{-1}$ . Gas constant $R = PV/(nT)$ : $ML^{2}T^{-2}K^{-1}\text{mol}^{-1}$ .
Specific heat $c = Q/(m\Delta T)$ : $L^{2}T^{-2}K^{-1}$ . Latent heat $L = Q/m$ : $L^{2}T^{-2}$ (no K).
Thermal conductivity, from $\dfrac{Q}{t} = kA\dfrac{\Delta T}{l}$ : $MLT^{-3}K^{-1}$ .
Stefan's constant, from $\dfrac{P}{A} = \sigma T^{4}$ : $MT^{-3}K^{-4}$ .
Planck's constant $h = E/\nu$ : $ML^{2}T^{-1}$ , the same as angular momentum.
Stopping potential is a voltage, $ML^{2}T^{-3}A^{-1}$ ; work function is an energy, $ML^{2}T^{-2}$ .
Rydberg constant (it gives $1/\lambda$ ): $L^{-1}$ . Decay constant ( $N = N_0e^{-\lambda t}$ ): $T^{-1}$ .

Constant or quantity	Equation it comes from	Dimensions
Boltzmann constant $k_B$	$E = \tfrac{3}{2}k_BT$	$ML^{2}T^{-2}K^{-1}$
Gas constant R	$PV = nRT$	$ML^{2}T^{-2}K^{-1}\text{mol}^{-1}$
Specific heat capacity	$Q = mc\Delta T$	$L^{2}T^{-2}K^{-1}$
Latent heat	$Q = mL$	$L^{2}T^{-2}$ No temperature in latent heat, so it differs from specific heat.
Thermal conductivity	$Q/t = kA\,\Delta T/l$	$MLT^{-3}K^{-1}$
Stefan's constant $\sigma$	$P/A = \sigma T^{4}$	$MT^{-3}K^{-4}$
Planck's constant h	$E = h\nu$	$ML^{2}T^{-1}$
Work function	$h\nu = \phi + K_{\max}$	$ML^{2}T^{-2}$
Stopping potential	$eV_0 = K_{\max}$	$ML^{2}T^{-3}A^{-1}$
Rydberg constant	$1/\lambda = R(1/n_1^{2} - 1/n_2^{2})$	$L^{-1}$
Decay constant	$N = N_0e^{-\lambda t}$	$T^{-1}$

Each constant has the dimensions that balance its equation.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 2 · Q2Moderate

Example 2 · Units and Measurements · Dimensions of Mechanical and Thermal Quantities

Match List-I with List-II.

List-I	List-II
(A) Boltzmann constant	(I) $[M^{-1}L^{3}T^{-2}]$
(B) Stefan's constant	(II) $[ML^{2}T^{-1}]$
(C) Planck's constant	(III) $[ML^{2}T^{-2}K^{-1}]$
(D) Gravitational constant	(IV) $[ML^{0}T^{-3}K^{-4}]$

Choose the correct answer from the options given below.

Stopping potential is a voltage, not an energy

The stopping potential V₀ satisfies eV₀ = K_max, so V₀ itself is energy per charge, ML²T⁻³A⁻¹. Only eV₀ is an energy. The work function, by contrast, is an energy, ML²T⁻².

Specific heat carries K⁻¹, latent heat does not

Specific heat is energy per mass per kelvin, L²T⁻²K⁻¹. Latent heat is energy per mass, L²T⁻². They are a standard 'different dimensions' pair.

Drill 9 more on dimensions of thermal and modern-physics constants

Concept 3 of 3: Pairs of quantities with the same dimensions

Different physical ideas can share dimensions. Torque and energy are both force × length; Planck's constant and angular momentum are both energy × time. To test a pair, write both dimensional formulas and compare every power. One power out of place makes the pair different.

Definition

Same: torque and energy; Planck's constant and angular momentum; pressure, stress, Young's modulus and energy density; velocity gradient, decay constant, angular speed and frequency; impulse and momentum.
Different: specific heat and latent heat; linear momentum and torque; surface tension and impulse.
A product such as pressure × time is viscosity: $ML^{-1}T^{-2} \times T = ML^{-1}T^{-1}$ .
$\sqrt{uG}$ , with u an energy density, is $\sqrt{ML^{-1}T^{-2} \cdot M^{-1}L^{3}T^{-2}} = LT^{-2}$ : force per unit mass.

Pair	Dimensions of the first	Dimensions of the second	Same?
Torque and energy	$ML^{2}T^{-2}$	$ML^{2}T^{-2}$	Yes
Planck's constant and angular momentum	$ML^{2}T^{-1}$	$ML^{2}T^{-1}$	Yes
Stress and energy density	$ML^{-1}T^{-2}$	$ML^{-1}T^{-2}$	Yes
Velocity gradient and decay constant	$T^{-1}$	$T^{-1}$	Yes
Pressure × time and viscosity	$ML^{-1}T^{-1}$	$ML^{-1}T^{-1}$	Yes
Specific heat and latent heat	$L^{2}T^{-2}K^{-1}$	$L^{2}T^{-2}$	No The only difference is K⁻¹.
Linear momentum and torque	$MLT^{-1}$	$ML^{2}T^{-2}$	No
Surface tension and impulse	$MT^{-2}$	$MLT^{-1}$	No

Compare every power of M, L, T and K; one mismatch makes the pair different.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 29 Jan 2025 · Q4Moderate

Example 3 · Units and Measurements · Dimensions of Mechanical and Thermal Quantities

The pair of physical quantities not having same dimensions is :

Same dimensions do not mean the same quantity

Torque and work are both ML²T⁻², but torque is not an energy: it is a turning effect, and it is not measured in joules. A statement that they 'have the same dimensions' is true; one that they 'are the same quantity' is not.

Drill 5 more on pairs of quantities with the same dimensions

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Reference tables (3)

Dimensions of mechanical quantities15 rows

Quantity	Defining relation	Dimensions	SI unit
Force	$F = ma$	$MLT^{-2}$	newton (N)
Work, energy, torque	$W = Fs,\ \tau = rF$	$ML^{2}T^{-2}$	J (N m for torque)
Power	$P = W/t$	$ML^{2}T^{-3}$	watt (W)
Momentum, impulse	$p = mv,\ J = Ft$	$MLT^{-1}$	kg m/s or N s
Angular momentum, angular impulse	$L = mvr,\ \tau t$	$ML^{2}T^{-1}$	kg m²/s
Moment of inertia	$I = mr^{2}$	$ML^{2}$	kg m²
Pressure, stress, Young's modulus, bulk modulus	$F/A$	$ML^{-1}T^{-2}$	pascal (Pa)
Pressure gradient	$dP/dx$	$ML^{-2}T^{-2}$	Pa/m
Compressibility	$1/\text{bulk modulus}$	$M^{-1}LT^{2}$	Pa⁻¹
Surface tension, spring constant	$F/l,\ F/x$	$MT^{-2}$	N/m
Coefficient of viscosity	$F = \eta A\,dv/dx$	$ML^{-1}T^{-1}$	pascal-second (Pa s)
Intensity of a wave	power ÷ area	$MT^{-3}$	W/m²
Gravitational constant G	$F = Gm_1m_2/r^{2}$	$M^{-1}L^{3}T^{-2}$	N m²/kg²
Gravitational potential	energy ÷ mass	$L^{2}T^{-2}$	J/kg
Angular speed, frequency, velocity gradient	$\omega = \theta/t,\ dv/dx$	$T^{-1}$	s⁻¹

Each row follows from its defining relation; force, MLT⁻², is the starting point for most of them.

Dimensions of thermal and modern-physics constants11 rows

Constant or quantity	Equation it comes from	Dimensions
Boltzmann constant $k_B$	$E = \tfrac{3}{2}k_BT$	$ML^{2}T^{-2}K^{-1}$
Gas constant R	$PV = nRT$	$ML^{2}T^{-2}K^{-1}\text{mol}^{-1}$
Specific heat capacity	$Q = mc\Delta T$	$L^{2}T^{-2}K^{-1}$
Latent heat	$Q = mL$	$L^{2}T^{-2}$ No temperature in latent heat, so it differs from specific heat.
Thermal conductivity	$Q/t = kA\,\Delta T/l$	$MLT^{-3}K^{-1}$
Stefan's constant $\sigma$	$P/A = \sigma T^{4}$	$MT^{-3}K^{-4}$
Planck's constant h	$E = h\nu$	$ML^{2}T^{-1}$
Work function	$h\nu = \phi + K_{\max}$	$ML^{2}T^{-2}$
Stopping potential	$eV_0 = K_{\max}$	$ML^{2}T^{-3}A^{-1}$
Rydberg constant	$1/\lambda = R(1/n_1^{2} - 1/n_2^{2})$	$L^{-1}$
Decay constant	$N = N_0e^{-\lambda t}$	$T^{-1}$

Each constant has the dimensions that balance its equation.

Pairs of quantities with the same dimensions8 rows

Pair	Dimensions of the first	Dimensions of the second	Same?
Torque and energy	$ML^{2}T^{-2}$	$ML^{2}T^{-2}$	Yes
Planck's constant and angular momentum	$ML^{2}T^{-1}$	$ML^{2}T^{-1}$	Yes
Stress and energy density	$ML^{-1}T^{-2}$	$ML^{-1}T^{-2}$	Yes
Velocity gradient and decay constant	$T^{-1}$	$T^{-1}$	Yes
Pressure × time and viscosity	$ML^{-1}T^{-1}$	$ML^{-1}T^{-1}$	Yes
Specific heat and latent heat	$L^{2}T^{-2}K^{-1}$	$L^{2}T^{-2}$	No The only difference is K⁻¹.
Linear momentum and torque	$MLT^{-1}$	$ML^{2}T^{-2}$	No
Surface tension and impulse	$MT^{-2}$	$MLT^{-1}$	No

Compare every power of M, L, T and K; one mismatch makes the pair different.

Watch out for (5)

Surface energy and surface tension are not the same entry
→ Dimensions of mechanical quantities
Viscosity has one power of time, pressure has two
→ Dimensions of mechanical quantities
Stopping potential is a voltage, not an energy
→ Dimensions of thermal and modern-physics constants
Specific heat carries K⁻¹, latent heat does not
→ Dimensions of thermal and modern-physics constants
Same dimensions do not mean the same quantity
→ Pairs of quantities with the same dimensions

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