JEE PYQ: Units & Measurements - Question ID bcf1d4819817 (JEE Main 2024)

Core Formula & Concept:

The principle of dimensional homogeneity states that in any physically meaningful equation, the dimensions (units) of every term on both sides must be identical. This principle is a powerful tool to check the correctness of derived formulas or to deduce the form of an unknown relationship.

In this problem, we are given the time period $T$ of an orbit, the gravitational constant $G$, mass $M$, and radius $r$. We must determine which of the four proposed expressions for $T^2$ is dimensionally consistent.

Key dimensional formulas:

• Time period $T$: dimension $[T]$ (time).
• Gravitational constant $G$: dimension $[M^{-1} L^3 T^{-2}]$.
• Mass $M$: dimension $[M]$.
• Radius $r$: dimension $[L]$.

Step-by-Step Derivation:

Step 1: Write the dimensional formula for $T^2$.

$T^2$ has dimension $[T^2]$.

Step 2: Analyze each option dimensionally.

Option A: $T^2 = \frac{4 \pi^2 r}{G M^2}$

Dimensional formula of RHS: $\frac{[L]}{[M^{-1} L^3 T^{-2}] \cdot [M^2]} = \frac{[L]}{[M^{-1} L^3 T^{-2} M^2]} = \frac{[L]}{[M L^3 T^{-2}]} = [M^{-1} L^{-2} T^2]$.

This does not match $[T^2]$. Hence, Option A is incorrect.

Option B: $T^2 = 4 \pi^2 r^3$

Dimensional formula of RHS: $[L^3]$.

This does not match $[T^2]$. Hence, Option B is incorrect.

Option C: $T^2 = \frac{4 \pi^2 r^3}{G M}$

Dimensional formula of RHS: $\frac{[L^3]}{[M^{-1} L^3 T^{-2}] \cdot [M]} = \frac{[L^3]}{[M^{-1} L^3 T^{-2} M]} = \frac{[L^3]}{[L^3 T^{-2}]} = [T^2]$.

This matches the dimension of $T^2$. Hence, Option C is correct.

Option D: $T^2 = \frac{4 \pi^2 r^2}{G M}$

Dimensional formula of RHS: $\frac{[L^2]}{[M^{-1} L^3 T^{-2}] \cdot [M]} = \frac{[L^2]}{[L^3 T^{-2}]} = [L^{-1} T^2]$.

This does not match $[T^2]$. Hence, Option D is incorrect.

Common Traps & Exam Tip:

Students often confuse the powers of $r$ and $M$ in the denominator. A common mistake is to assume $T^2 \propto r^2$ or $T^2 \propto M^{-2}$, leading to incorrect options like A or D. Always verify dimensions carefully—especially the powers of $G$, which carries $[M^{-1} L^3 T^{-2}]$. The correct expression must balance all dimensions to yield pure $[T^2]$.