JEE PYQ: Units & Measurements - Question ID fad6c77bf33a (JEE Main 2024)

Angular impulse is the rotational analogue of linear impulse. Just as linear impulse ($J$) is the integral of force over time, angular impulse ($H$) is the integral of torque over time.

• Linear impulse: $J = \int F \, dt \quad \Rightarrow \quad [J] = [F][T] = \mathrm{M\,L\,T^{-2}} \times \mathrm{T} = \mathrm{M\,L\,T^{-1}}$
• Torque ($\tau$): Torque is the cross product of position vector ($r$) and force ($F$), so its dimensional formula is $[\tau] = [r][F] = \mathrm{L} \times \mathrm{M\,L\,T^{-2}} = \mathrm{M\,L^2\,T^{-2}}$
• Angular impulse ($H$): $H = \int \tau \, dt \quad \Rightarrow \quad [H] = [\tau][T] = \mathrm{M\,L^2\,T^{-2}} \times \mathrm{T} = \mathrm{M\,L^2\,T^{-1}}$

1. Write the definition of angular impulse: $H = \int \tau \, dt$
2. Express torque in terms of force and distance: $\tau = r \times F$
3. Determine the dimensions of torque: $[\tau] = [r][F] = \mathrm{L} \times \mathrm{M\,L\,T^{-2}} = \mathrm{M\,L^2\,T^{-2}}$
4. Multiply by the dimension of time to obtain the dimension of angular impulse: $[H] = [\tau] \times [T] = \mathrm{M\,L^2\,T^{-2}} \times \mathrm{T} = \mathrm{M\,L^2\,T^{-1}}$
5. Compare with the given options. Option C matches $\mathrm{M\,L^2\,T^{-1}}$.

Many students confuse angular impulse with linear impulse or with torque itself.

• Mistake 1: Forgetting to multiply torque’s dimension by time, leading to option A ($\mathrm{M\,L^2\,T^{-2}}$), which is the dimension of torque, not angular impulse.
• Mistake 2: Incorrectly treating angular impulse as force times time (option D, $\mathrm{M\,L\,T^{-1}}$), which is linear impulse.
• Exam Tip: Always start from the definition $H = \int \tau \, dt$ and carry out the dimensional multiplication step by step.