Question: Which pair do not have equal dimensions? [PMT/NEET-2000]
a. Energy and torque
b. Force and impulse
c. Angular momentum and Planck’s constant
d. Elastic modulus and pressure

Answer: (b) Force and impulse do not have equal dimensions.

Dimensional check for each pair

• Energy and torque
  • $ [E] = [\text{work}] = [F \cdot s] = ML^{2}T^{-2} $
  • $ [\tau] = [r \times F] = ML^{2}T^{-2} $
  • Equal (though different physical nature: scalar vs axial vector).
• Force and impulse
  • $ [F] = MLT^{-2} $
  • $ [\text{impulse}] = [F \cdot \Delta t] = MLT^{-1} $
  • Not equal.
• Angular momentum and Planck’s constant
  • $ [L] = [r \times p] = ML^{2}T^{-1} $
  • $ [h] = [E \cdot t] = ML^{2}T^{-1} $
  • Equal.
• Elastic modulus and pressure
  • $ [Y] = [\text{stress}] = [F/A] = ML^{-1}T^{-2} $
  • $ [P] = [F/A] = ML^{-1}T^{-2} $
  • Equal.