JEE PYQ: Units & Measurements - Question ID d3a9a9baa08f (JEE Main 2023)

Core Formula & Concept:

In dimensional analysis, every physical quantity is expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), and time ($T$). The key formulas and concepts used here are:

• Spring constant ($k$): From Hooke’s law, $F = kx$, where $F$ is force and $x$ is displacement. The dimensions of force are $[MLT^{-2}]$, and displacement is $[L]$. Thus, $k = \frac{F}{x}$ has dimensions $[MT^{-2}]$.
• Angular speed ($\omega$): Angular speed is the rate of change of angular displacement, $\omega = \frac{\Delta \theta}{\Delta t}$. Since angular displacement is dimensionless, $\omega$ has dimensions $[T^{-1}]$.
• Angular momentum ($L$): Angular momentum is given by $L = I\omega$, where $I$ is moment of inertia and $\omega$ is angular speed. The dimensions of $I$ are $[ML^2]$, and $\omega$ is $[T^{-1}]$, so $L$ has dimensions $[ML^2T^{-1}]$.
• Moment of inertia ($I$): Moment of inertia for a point mass is $I = mr^2$, where $m$ is mass and $r$ is distance. Thus, $I$ has dimensions $[ML^2]$.

Step-by-Step Derivation:

We derive the dimensions for each quantity in List I and match them with List II.

1. Spring constant (A):

   From Hooke’s law: $F = kx$.
   Dimensions of force: $[F] = [MLT^{-2}]$.
   Dimensions of displacement: $[x] = [L]$.
   Thus, dimensions of $k$: $[k] = \frac{[F]}{[x]} = \frac{[MLT^{-2}]}{[L]} = [MT^{-2}]$.
   This matches with II in List II.

2. Angular speed (B):

   Angular speed is $\omega = \frac{\Delta \theta}{\Delta t}$.
   Angular displacement $\theta$ is dimensionless, so $[\omega] = \frac{1}{[T]} = [T^{-1}]$.
   This matches with I in List II.

3. Angular momentum (C):

   Angular momentum is $L = I\omega$.
   Dimensions of moment of inertia: $[I] = [ML^2]$.
   Dimensions of angular speed: $[\omega] = [T^{-1}]$.
   Thus, $[L] = [I][\omega] = [ML^2][T^{-1}] = [ML^2T^{-1}]$.
   This matches with IV in List II.

4. Moment of inertia (D):

   Moment of inertia is $I = mr^2$.
   Dimensions of mass: $[m] = [M]$.
   Dimensions of distance: $[r] = [L]$.
   Thus, $[I] = [M][L^2] = [ML^2]$.
   This matches with III in List II.

Thus, the correct matching is:

• A → II
• B → I
• C → IV
• D → III

This corresponds to Option C.

Common Traps & Exam Tip:

Students often confuse the dimensions of angular momentum and moment of inertia. A common mistake is swapping their dimensions, as both involve $ML^2$ but differ in the time dimension. Another frequent error is misidentifying the spring constant’s dimensions, often confusing it with force ($[MLT^{-2}]$) instead of recognizing it as force per unit length ($[MT^{-2}]$). Always derive dimensions from first principles to avoid such pitfalls.