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(C) To determine which pair does not have the same dimensions,we analyze each option:$(A)$ Angular momentum $(L)$ is given by $L = mvr$. Its dimension

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Moment of inertia and torque

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(C) To determine which pair does not have the same dimensions,we analyze each option:$(A)$ Angular momentum $(L)$ is given by $L = mvr$. Its dimensions are $[ML^2T^{-1}]$. Planck's constant $(h)$ is related to energy by $E = h
u$,so $[h] = [E]/[
u] = [ML^2T^{-2}]/[T^{-1}] = [ML^2T^{-1}]$. Thus,they have the same dimensions.$(B)$ Impulse $(I)$ is defined as the change in momentum,$I = \Delta p$. Thus,impulse and linear momentum have the same dimensions,$[MLT^{-1}]$.$(C)$ Moment of inertia $(I)$ has dimensions $[ML^2]$. Torque $(	au)$ is given by $	au = r 	imes F$,which has dimensions $[L] 	imes [MLT^{-2}] = [ML^2T^{-2}]$. Since $[ML^2] 
eq [ML^2T^{-2}]$,this pair does not have the same dimensions.$(D)$ Work $(W)$ is force $	imes$ displacement,$[ML^2T^{-2}]$. Torque $(	au)$ also has dimensions $[ML^2T^{-2}]$. Thus,they have the same dimensions.Therefore,the pair that does not have the same dimensions is Moment of inertia and torque.

Which of the following pairs does not have the same dimensions?

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