Question

In which of the following pairs, the two physical quantities have different dimensions?

• A. Planck's constant and angular momentum
• B. Impulse and linear momentum
• C. Moment of inertia and moment of a force
• D. Energy and torque

Answer 1

The correct option is C Moment of inertia and moment of a force

(a): Using $L=mvr=\frac{nh}{2\pi }$
Thus the angular momentum $\left(L\right)$ has the same dimensions as that of Planck's constant $\left(h\right)$.
(b): Impulse $I=\Delta P$
Thus impulse $\left(I\right)$ has the same dimensions as that of linear momentum $\left(P\right)$
(c): Using $\tau =I\alpha \Rightarrow I=\frac{\tau }{\alpha }$
where angular acceleration is not dimensionless.
Thus the dimensions of moment of inertia are different from those of moment of force (OR torque)
(d): Dimension of energy $E=\left[M{L}^2{T}^{-2}\right]$
Dimension of torque $\tau =Fr=M{L}^2{T}^{-2}$
Hence, torque and energy have same dimensions.