JEE PYQ: Units & Measurements - Question ID cd7494f7bba8 (JEE Main 2025)

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), and time ($T$). The key formulas and concepts for the given quantities are:

• Young’s Modulus ($Y$): Defined as the ratio of stress to strain. Stress is force per unit area ($F/A$), and strain is dimensionless. $Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\text{Strain}}$ Since strain is dimensionless, the dimensions of Young’s Modulus are the same as those of stress: $[Y] = \frac{[F]}{[A]} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2}$
• Torque ($\tau$): Torque is the cross product of force and the perpendicular distance from the axis of rotation. $\tau = \vec{r} \times \vec{F}$ The dimensions of torque are: $[\tau] = [r] \cdot [F] = L \cdot (M L T^{-2}) = M L^2 T^{-2}$
• Coefficient of Viscosity ($\eta$): From Newton’s law of viscosity, the viscous force is proportional to the velocity gradient: $F = \eta A \frac{dv}{dx}$ Rearranging for $\eta$: $\eta = \frac{F}{A \cdot (dv/dx)}$ The dimensions of $\eta$ are: $[\eta] = \frac{M L T^{-2}}{L^2 \cdot (L T^{-1}/L)} = \frac{M L T^{-2}}{L^2 \cdot T^{-1}} = M L^{-1} T^{-1}$
• Gravitational Constant ($G$): From Newton’s law of gravitation: $F = G \frac{m_1 m_2}{r^2}$ Rearranging for $G$: $G = \frac{F r^2}{m_1 m_2}$ The dimensions of $G$ are: $[G] = \frac{(M L T^{-2}) \cdot L^2}{M \cdot M} = M^{-1} L^3 T^{-2}$

We now match the dimensions derived above with the options in List-II:

Comparing with the given options:

• Option A: Incorrect (e.g., Young’s Modulus is not $M L^{-1} T^{-1}$).
• Option B: Incorrect (e.g., Torque is not $M L^{-1} T^{-2}$).
• Option C: Incorrect (e.g., Coefficient of Viscosity is not $M L^3 T^{-2}$).
• Option D: Correct (matches all derived dimensions).

Students often make the following mistakes:

1. Confusing Torque and Work: Both torque and work have the same dimensions ($M L^2 T^{-2}$), but they are distinct physical quantities. Torque is a vector, while work is a scalar. In this question, torque is correctly matched with (IV).
2. Misidentifying Viscosity Dimensions: Some students confuse the coefficient of viscosity with pressure or stress, leading to incorrect matches like (II) instead of (I).
3. Gravitational Constant Confusion: The dimensions of $G$ are often mistakenly associated with force or energy. Remember that $G$ has dimensions of $M^{-1} L^3 T^{-2}$, which is unique.
4. Overlooking Negative Exponents: Students sometimes misplace the exponents, especially for $G$ (e.g., writing $M L^3 T^{-2}$ instead of $M^{-1} L^3 T^{-2}$).

Exam Tip: Always derive the dimensions from first principles rather than relying on memory. This avoids confusion between similar-looking dimensions (e.g., $M L^{-1} T^{-1}$ vs. $M L^{-1} T^{-2}$).