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JEE Main 202427 JanuaryMorning Shift
Exam
JEE Main
Year
2024
Date
27 January
Shift
Morning Shift
Subject
Physics
Chapter
Units and Dimensions

JEE Main 2024 · 27 January · Morning Shift · Physics · Units and Dimensions

Given below are two statements: Statement (I) : Planck's constant...

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 :

Options

  1. AStatement I is true but Statement II is false
  2. BBoth Statement I and Statement II are false
  3. CBoth Statement I and Statement II are true
  4. DStatement I is false but Statement II is true

Correct answer

A. Statement I is true but Statement II is false

Step-by-step solution

The dimensions of the required quantities can be written as

E=hν⇒[h]=[E][ν]⇒[h]=[ML2T−2][T−1]=[ML2T−1]E = hν ⇒ \left[h\right] = \dfrac{\left[E\right]}{\left[ν\right]} ⇒ [ 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{cases}[ L ] & = & \left[mvr\right] \\ & = & \left[M\right]\left[{LT}^{- 1}\right]\left[L\right] \\ & = & \left[{ML}^{2}{T}^{- 1}\right]\end{cases}

{[p]=[mv]=[M][LT−1]=[ML1T−1]\begin{cases}[ p ] & = & \left[mv\right] \\ & = & \left[M\right]\left[{LT}^{- 1}\right] \\ & = & \left[{ML}^{1}{T}^{- 1}\right]\end{cases}

{[τ]=[Fr]=[MLT−2][L]=[ML2T−2]\begin{cases}[ τ ] & = & \left[Fr\right] \\ & = & \left[{MLT}^{- 2}\right]\left[L\right] \\ & = & \left[{ML}^{2}{T}^{- 2}\right]\end{cases}

where, hh  is Planck's constant, LL is angular momentum, pp is linear momentum and ττ 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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