Q94 / 180 · AIIMS 2010 · Physics
Which of the following pairs does not have same dimensions ?
- Aimpulse and momentum
- Correct answerBmoment of inertia and moment of force
- Cangular momentum and Planck's constant
- Dwork and torque.
Solution
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
$\therefore \quad$ [Impulse $]=[$ Momentum $]$
Angular momentum, $\mathrm{L}=m v r$
Planck's constant, $[h]=[$ energy $] \times[$ time $]$
$$
\Rightarrow[F \times r \times \text { time })=\frac{m\left(v_2-v_1\right)}{t} \times r \times t
$$
$\Rightarrow m\left(v_2-v_1\right) \times r=($ change of momentum $) \times r$
$\therefore \quad[h]=[L]$.
Work, $W=\vec{F} \cdot \vec{d}:$ Torque, $\tau=\vec{r} \times \vec{F}$
$$
\therefore \quad[W]=[\tau]
$$
Moment of inertia, $I=m r^2=\operatorname{mass} \times(\text { distance })^2$
Moment of force, $\tau=\vec{r} \times \vec{F}=$ distance $\times$ force
$$
\begin{aligned}
& \quad=\text { distance } \times \frac{\text { change of momentum }}{\text { time }} \\
& \therefore \quad[\mathrm{I}] \neq[\tau] .
\end{aligned}
$$
Therefore, moment of inertia and moment of force have different dimensions.
Hence, the correct answer is option: 'B'.