JEE PYQ: Units & Measurements - Question ID 346581fe9c27 (JEE Main 2023)

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:

• Angular momentum ($L$): Defined as the cross product of position vector ($r$) and linear momentum ($p$), i.e., $L = r \times p$. Since $r$ has dimensions $[L]$ and $p$ has dimensions $[MLT^{-1}]$, the dimensions of angular momentum are $[ML^2T^{-1}]$.
• Torque ($\tau$): Defined as the cross product of position vector ($r$) and force ($F$), i.e., $\tau = r \times F$. Force has dimensions $[MLT^{-2}]$, so torque has dimensions $[ML^2T^{-2}]$.
• Stress ($\sigma$): Defined as force per unit area, i.e., $\sigma = F/A$. Force has dimensions $[MLT^{-2}]$ and area has dimensions $[L^2]$, so stress has dimensions $[ML^{-1}T^{-2}]$.
• Pressure gradient ($\nabla P$): Defined as the rate of change of pressure with respect to distance, i.e., $\nabla P = \Delta P / \Delta x$. Pressure has dimensions $[ML^{-1}T^{-2}]$ and distance has dimensions $[L]$, so the pressure gradient has dimensions $[ML^{-2}T^{-2}]$.

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

1. Angular Momentum (A):

   Angular momentum is given by $L = r \times p$, where $r$ is the position vector and $p$ is the linear momentum.
   Dimensions of $r$: $[L]$
   Dimensions of $p$: $[MLT^{-1}]$
   Thus, dimensions of $L$: $[L] \times [MLT^{-1}] = [ML^2T^{-1}]$.
   This matches with III in List II.

2. Torque (B):

   Torque is given by $\tau = r \times F$, where $r$ is the position vector and $F$ is the force.
   Dimensions of $r$: $[L]$
   Dimensions of $F$: $[MLT^{-2}]$
   Thus, dimensions of $\tau$: $[L] \times [MLT^{-2}] = [ML^2T^{-2}]$.
   This matches with I in List II.

3. Stress (C):

   Stress is given by $\sigma = F/A$, where $F$ is the force and $A$ is the area.
   Dimensions of $F$: $[MLT^{-2}]$
   Dimensions of $A$: $[L^2]$
   Thus, dimensions of $\sigma$: $[MLT^{-2}] / [L^2] = [ML^{-1}T^{-2}]$.
   This matches with IV in List II.

4. Pressure Gradient (D):

   Pressure gradient is given by $\nabla P = \Delta P / \Delta x$, where $\Delta P$ is the change in pressure and $\Delta x$ is the change in distance.
   Dimensions of $P$: $[ML^{-1}T^{-2}]$
   Dimensions of $x$: $[L]$
   Thus, dimensions of $\nabla P$: $[ML^{-1}T^{-2}] / [L] = [ML^{-2}T^{-2}]$.
   This matches with II in List II.

Thus, the correct matching is:
A - III, B - I, C - IV, D - II.
This corresponds to Option B.

Students often make the following mistakes in such questions:

• Confusing torque and angular momentum: Both involve $r \times F$ or $r \times p$, but torque has dimensions of work ($[ML^2T^{-2}]$), while angular momentum has dimensions of $[ML^2T^{-1}]$. Students sometimes mix up the exponents of time.
• Misidentifying stress and pressure: Stress and pressure have the same dimensions ($[ML^{-1}T^{-2}]$), but the pressure gradient has an additional $L^{-1}$ factor because it is pressure per unit distance. Students may overlook this distinction.
• Incorrectly calculating pressure gradient: Some students forget to divide by the distance term, leading to incorrect dimensions for the pressure gradient.

Exam Tip: Always write down the defining formula for each quantity before deriving its dimensions. This avoids confusion and ensures accuracy.