JEE PYQ: Units & Measurements - Question ID 2d7d33a72196 (JEE Main 2002)

In dimensional analysis, 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 ($J$). The dimensions of derived quantities are obtained by combining these fundamental dimensions using the defining equations of the quantities.

Key formulas for the quantities involved in the question:

• Force ($F$): Defined by Newton’s second law: $F = ma$. Dimensions: $[F] = M L T^{-2}$.
• Work ($W$): Work is force times displacement: $W = F \cdot s$. Dimensions: $[W] = M L^2 T^{-2}$.
• Torque ($\tau$): Torque is force times the perpendicular distance from the pivot: $\tau = F \cdot r$. Dimensions: $[\tau] = M L^2 T^{-2}$.
• Stress ($\sigma$): Stress is force per unit area: $\sigma = \frac{F}{A}$. Dimensions: $[\sigma] = M L^{-1} T^{-2}$.
• Energy ($E$): Energy is the capacity to do work, and its dimensions are the same as work. Dimensions: $[E] = M L^2 T^{-2}$.

We compare the dimensions of each pair given in the options:

1. Option A: Torque and Work
   $[\tau] = M L^2 T^{-2}$
   $[W] = M L^2 T^{-2}$
   Both have identical dimensions.
2. Option B: Stress and Energy
   $[\sigma] = M L^{-1} T^{-2}$
   $[E] = M L^2 T^{-2}$
   Dimensions differ.
3. Option C: Force and Stress
   $[F] = M L T^{-2}$
   $[\sigma] = M L^{-1} T^{-2}$
   Dimensions differ.
4. Option D: Force and Work
   $[F] = M L T^{-2}$
   $[W] = M L^2 T^{-2}$
   Dimensions differ.

Only Option A has a pair with equal dimensions.

Students often confuse torque with force or stress with energy because they involve force in their definitions. However, torque and work both involve force multiplied by a distance (displacement or lever arm), giving them identical dimensions. Stress, being force per unit area, has dimensions of $M L^{-1} T^{-2}$, which is different from energy. Always write down the dimensional formula explicitly to avoid such mistakes.