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Q.
Which of the following pairs does not have same dimensions ?
AIIMSAIIMS 2010Physical World, Units and Measurements
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 $ [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.