Question

Which of the following pairs does not have same dimensions ?

• A. impulse and momentum
• B. moment of inertia and moment of force
• C. angular momentum and Planck’s constant
• D. work and torque

Answer 1

Answer: (B)

Impulse $=F\times t$
$=\frac{m\left(v_2-v_1\right)}{t}\times t=m\left(v_2-v_1\right)$
= change in momentum
$\therefore$ [Impulse] = [Momentum]
Angular momentum, $L=mvr$
Planck's constant, $[h]=$ [energy] $\times$ [time]
$\Rightarrow \left[F\times r\times \text{time}\right]=\frac{m\left(v_2-v_1\right)}{t}\times r\times t$
$\Rightarrow m\left(v_2-v_1\right)\times r=$(change of momentum) $\times r$
$\therefore \left[h\right]=\left[L\right].$
Work, W$=\vec{F}.\vec{d}:$ Torque, $\tau =\vec{r}\times \vec{F}$
$\therefore \left[W\right]=\left[\tau \right]$
Moment of inertia, $I=m{r}^2$ = mass $\times$ (distance)${}^2$
Moment of force, $\tau =\vec{r}\times \vec{F}=$ distance $\times$ force
$=$ distance $\times \frac{\text{change of momentum}}{\text{time}}$
$\therefore \left[I\right]\ne \left[\tau \right].$
Therefore, moment of inertia and moment of force have different dimensions.