Q94 / 180 · AIIMS 2010 · Physics

Which of the following pairs does not have same dimensions ?

  • A
    impulse and momentum
  • Correct answer
    B
    moment of inertia and moment of force
  • C
    angular momentum and Planck's constant
  • D
    work 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'.

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