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?? constant
• D. work and torque

Answer 1

Answer: (B)

moment of inertia and moment of force

Answer 2

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, $I = mr^{2}$ = 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.