JEE Mains Physics · Units and Measurements

Deriving Relations and Changing Base Quantities

Writing a quantity as a product of powers of other quantities and equating the powers of M, L and T finds the unknown powers; the same algebra rewrites any quantity in a new set of base quantities.

All Units and Measurements notes JEE Mains Physics strategy

Why this matters

Fourteen PYQs, thirteen of them multiple choice, and four from 2026. Six find the powers in a relation such as v = λᵃgᵇρᶜ by equating the powers of M, L and T; eight rewrite a quantity in new base quantities, such as force, velocity and time, or h, c and G. Both are three linear equations in three unknowns.

Concept 1 of 2: Finding unknown powers by equating dimensions

Suppose a quantity depends on three others as a product of powers. Both sides must have the same dimensions, so the power of M on the left equals the total power of M on the right, and the same for L and T. Three equations fix three unknown powers.

Definition

Write $Q = k\,A^{a}B^{b}C^{c}$ , with k a pure number.
Put in the dimensions of A, B, C and Q.
Equate the powers of M, of L and of T (and of A or K if they appear), then solve.
A base quantity carried by only one factor, and absent from Q, forces that factor's power to zero.
Limits: the method cannot find k, cannot handle a sum of terms, and needs no more unknowns than equations.

Exponent method

Q = k\,A^{a}B^{b}C^{c} \Rightarrow \text{equate the powers of } M,\ L,\ T

Worked example

The period of a simple pendulum is taken as

T = k\,m^{a}l^{b}g^{c}

, with m the mass of the bob, l the length and g the acceleration due to gravity. Find a, b and c.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 15 Apr 2023 · Q17Moderate

Example 1 · Units and Measurements · Deriving Relations and Changing Base Quantities

The speed of a wave produced in water is given by

v =\lambda^{a}g^{b}\rho^{c}

. Where

\lambda,g

and

\rho

are wavelength of wave, acceleration due to gravity and density of water respectively. The values of

a,b

and

c

respectively, are:

A base quantity on one side only forces its power to zero

If only one quantity on the right carries mass, and the left side has no mass, that quantity's power must be zero. Dropping the M equation because 'nothing is heavy' loses this result.

Drill 5 more on finding unknown powers by equating dimensions

Concept 2 of 2: Writing dimensions in new base quantities

Any three independent quantities can serve as the base instead of M, L and T. First express M, L and T in the new base. Then substitute them into the quantity's usual dimensions. The same idea converts a number from one system of units to another.

Definition

Step 1: write each new base quantity in M, L, T; solve for M, L, T in terms of the new base.
Step 2: substitute into the quantity's usual formula.
Or directly: set $Q = X^{a}Y^{b}Z^{c}$ and equate powers of M, L, T.
Planck units from h, c, G: mass $\sqrt{hc/G}$ , length $\sqrt{Gh/c^{3}}$ , time $\sqrt{Gh/c^{5}}$ .
Converting a value: $n_2 = n_1\left(\dfrac{M_1}{M_2}\right)^{a}\left(\dfrac{L_1}{L_2}\right)^{b}\left(\dfrac{T_1}{T_2}\right)^{c}$ for a quantity $M^{a}L^{b}T^{c}$ .

Change of units

n_2 = n_1\left(\frac{M_1}{M_2}\right)^{a}\left(\frac{L_1}{L_2}\right)^{b}\left(\frac{T_1}{T_2}\right)^{c}

Worked example

If energy E, velocity V and time T are taken as the base quantities, find the dimensions of mass, of length and of force.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 2 Apr 2026 Shift 2 · Q1Moderate

Example 2 · Units and Measurements · Deriving Relations and Changing Base Quantities

Dimensions of universal gravitational constant (G) in terms of Planck's constant (h), distance (

L

), mass (

M

) and time (

T

) are

\_\_\_\_

Solve for the old base first

Writing the new quantities in M, L and T is the easy half. The answer needs the reverse: M, L and T in the new quantities. Substituting before inverting gives the reciprocal powers.

Drill 7 more on writing dimensions in new base quantities

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.

Formulas (2)

Finding unknown powers by equating dimensions
Exponent method
$Q = k\,A^{a}B^{b}C^{c} \Rightarrow \text{equate the powers of } M,\ L,\ T$
Writing dimensions in new base quantities
Change of units
$n_2 = n_1\left(\frac{M_1}{M_2}\right)^{a}\left(\frac{L_1}{L_2}\right)^{b}\left(\frac{T_1}{T_2}\right)^{c}$

Watch out for (2)

A base quantity on one side only forces its power to zero
→ Finding unknown powers by equating dimensions
Solve for the old base first
→ Writing dimensions in new base quantities

Test yourself on Units and Measurements

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Units and Measurements chapter test Drill all 14 questions