Units & Dimensions

JEE Main level, single correct, medium

67 of 149

Question 67 of 149

Given below are two statements:
Statement (I) : Planck's constant and angular momentum have the same dimensions.
Statement (II) : Linear momentum and moment of force have the same dimensions.
In light of the above statements, choose the correct answer from the options given below :

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Answer: (A) Statement I is true but Statement II is false

The dimensions of the required quantities can be written as
E=hν⇒[h]=[E][ν]⇒[h]=[ML2T−2][T−1]=[ML2T−1]E=h\nu \Rightarrow \left[h\right]=\dfrac{\left[E\right]}{\left[\nu \right]} \Rightarrow [h]=\dfrac{\left[{ML}^{2}{T}^{-2}\right]}{\left[{T}^{-1}\right]} =\left[{ML}^{2}{T}^{-1}\right]
[L]=[mvr]=[M][LT−1][L]=[ML2T−1]\begin{matrix}[L] & = & \left[mvr\right] \\ & = & \left[M\right]\left[{LT}^{-1}\right]\left[L\right] \\ & = & \left[{ML}^{2}{T}^{-1}\right]\end{matrix}
[p]=[mv]=[M][LT−1]=[ML1T−1]\begin{matrix}[p] & = & \left[mv\right] \\ & = & \left[M\right]\left[{LT}^{-1}\right] \\ & = & \left[{ML}^{1}{T}^{-1}\right]\end{matrix}
[τ]=[Fr]=[MLT−2][L]=[ML2T−2]\begin{matrix}[\tau ] & = & \left[Fr\right] \\ & = & \left[{MLT}^{-2}\right]\left[L\right] \\ & = & \left[{ML}^{2}{T}^{-2}\right]\end{matrix}
where, hh is Planck's constant, LL is angular momentum, pp is linear momentum and τ\tau is moment of force.
From the above analysis, it can be concluded that Statement I is true, but statement II is false.

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