JEE PYQ: Units & Measurements - Question ID 2405dd75f86a (JEE Main 2025)

Core Formula & Concept:

In the chapter Units & Measurements, the cornerstone is the principle of dimensional homogeneity. Every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($I$), thermodynamic temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$). Two physical quantities have the same dimensions if and only if their dimensional formulae are identical.

Key formulae used in this question:

• Angular momentum: $L = mvr \implies [L] = M L^2 T^{-1}$
• Planck’s constant: $h = \frac{E}{\nu} \implies [h] = M L^2 T^{-1}$
• Torque: $\tau = r \times F \implies [\tau] = M L^2 T^{-2}$
• Energy: $E = \frac{1}{2}mv^2 \implies [E] = M L^2 T^{-2}$
• Surface tension: $\sigma = \frac{F}{l} \implies [\sigma] = M T^{-2}$
• Impulse: $J = F \Delta t \implies [J] = M L T^{-1}$
• Pressure: $P = \frac{F}{A} \implies [P] = M L^{-1} T^{-2}$
• Young’s modulus: $Y = \frac{\text{Stress}}{\text{Strain}} \implies [Y] = M L^{-1} T^{-2}$

Step-by-Step Derivation:

Step 1: Dimensional Formulae of Each Quantity

We write the dimensional formula for each physical quantity listed in the options.

Step 2: Compare Dimensions Pairwise

We compare the dimensions of each pair in the options:

• Option A: Angular momentum vs Planck’s constant
  Both have dimensions $M L^2 T^{-1}$. Same dimensions.
• Option B: Torque vs Energy
  Both have dimensions $M L^2 T^{-2}$. Same dimensions.
• Option C: Surface tension vs Impulse
  Surface tension: $M T^{-2}$
  Impulse: $M L T^{-1}$
  Different dimensions.
• Option D: Pressure vs Young’s modulus
  Both have dimensions $M L^{-1} T^{-2}$. Same dimensions.

Step 3: Identify the Pair with Different Dimensions

Only Option C (Surface tension and Impulse) has different dimensions. Therefore, the pair not having the same dimensions is Option C.

Common Traps & Exam Tip:

Students often confuse surface tension with force per unit area, mistakenly assigning it dimensions $M L^{-1} T^{-2}$ (like pressure). However, surface tension is force per unit length, giving it dimensions $M T^{-2}$. Similarly, impulse is force multiplied by time, yielding $M L T^{-1}$. These two are dimensionally distinct, and this is the trap the question exploits.

Exam Tip: Always write down the dimensional formula explicitly. Never rely on memory alone—derive it from the definition of the quantity.