JEE PYQ: Units & Measurements - Question ID f88c43e96c13 (JEE Main 2021)

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 for the given quantities are:

• Torque ($\tau$): Torque is the rotational equivalent of force. It is defined as the cross product of the position vector ($\vec{r}$) and the force vector ($\vec{F}$). The magnitude of torque is given by: $\tau = rF \sin \theta$ Since force has dimensions $MLT^{-2}$ and distance has dimensions $L$, the dimensions of torque are: $[\tau] = [r] \cdot [F] = L \cdot MLT^{-2} = ML^2T^{-2}$
• Impulse ($J$): Impulse is the change in momentum of a body. It is the product of force and the time interval over which it acts: $J = F \cdot \Delta t$ The dimensions of impulse are: $[J] = [F] \cdot [t] = MLT^{-2} \cdot T = MLT^{-1}$
• Tension ($T$): Tension is a force that acts along a string, rope, or cable. It has the same dimensions as force: $[T] = MLT^{-2}$
• Surface Tension ($S$): Surface tension is defined as the force per unit length acting perpendicular to an imaginary line drawn on the liquid surface. Its dimensions are: $S = \frac{F}{L}$ $[S] = \frac{MLT^{-2}}{L} = MT^{-2}$

We will derive the dimensions of each quantity in List-I and match them with the corresponding entries in List-II.

1. Torque (a):

   As derived above, torque has dimensions $ML^2T^{-2}$. This matches with entry (iii) in List-II.

2. Impulse (b):

   Impulse has dimensions $MLT^{-1}$. This matches with entry (i) in List-II.

3. Tension (c):

   Tension has dimensions $MLT^{-2}$. This matches with entry (iv) in List-II.

4. Surface Tension (d):

   Surface tension has dimensions $MT^{-2}$. This matches with entry (ii) in List-II.

Thus, the correct matching is:

• (a) → (iii)
• (b) → (i)
• (c) → (iv)
• (d) → (ii)

This corresponds to Option A.

Students often make the following mistakes in such questions:

• Confusing torque and force: Torque is not the same as force. While force has dimensions $MLT^{-2}$, torque has dimensions $ML^2T^{-2}$. Students sometimes incorrectly match torque with force dimensions.
• Misidentifying impulse and momentum: Impulse and momentum have the same dimensions ($MLT^{-1}$), but students may confuse impulse with force ($MLT^{-2}$) due to its definition involving force.
• Overlooking surface tension dimensions: Surface tension is force per unit length, not force itself. Students may incorrectly assign it the dimensions of force ($MLT^{-2}$) instead of $MT^{-2}$.
• Incorrectly matching tension: Tension is a force, but students may confuse it with other quantities like stress or pressure, which have different dimensions.

Exam Tip: Always write down the defining formula for each quantity before deriving its dimensions. This avoids confusion and ensures accuracy. For example, writing $S = F/L$ for surface tension immediately clarifies its dimensions.