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Units and Measurements question

2005 · Shift 0 · Q168
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Units and Measurements question

2005 · Shift 0 · Q168

JEE MainPhysicsUnits and MeasurementsMCQ+4 / −1
Out of the following pair, which one does NOT have identical dimensions is
  1. A
    angular momentum and Planck's constant
  2. B
    impulse and momentum
  3. C
    moment of inertia and moment of a force
  4. D
    work and torque
View written solutionFree

Correct answer: C

  1. We compare the dimensions of each pair.

Useful dimensional formulas:

  • Momentum: [p]=[MLT−1][p] = [M L T^{-1}][p]=[MLT−1]
  • Impulse == = force ×\times× time =[MLT−2] [T]=[MLT−1]= [M L T^{-2}]\,[T] = [M L T^{-1}]=[MLT−2][T]=[MLT−1]
  • Angular momentum: [L]=[r×p]=[L] [MLT−1]=[ML2T−1][L] = [r \times p] = [L]\,[M L T^{-1}] = [M L^2 T^{-1}][L]=[r×p]=[L][MLT−1]=[ML2T−1]
  • Planck's constant: from E=hνE = h\nuE=hν [h]=[E][ν]=[ML2T−2][T−1]=[ML2T−1][h] = \frac{[E]}{[\nu]} = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}][h]=[ν][E]​=[T−1][ML2T−2]​=[ML2T−1]
  • Moment of inertia: I=∑mr2I = \sum m r^2I=∑mr2 [I]=[ML2][I] = [M L^2][I]=[ML2]
  • Moment of force (torque): force ×\times× distance [τ]=[MLT−2] [L]=[ML2T−2][\tau] = [M L T^{-2}]\,[L] = [M L^2 T^{-2}][τ]=[MLT−2][L]=[ML2T−2]
  • Work: force ×\times× displacement [W]=[MLT−2] [L]=[ML2T−2][W] = [M L T^{-2}]\,[L] = [M L^2 T^{-2}][W]=[MLT−2][L]=[ML2T−2]
  1. Check each option.

A: angular momentum and Planck's constant

  • Angular momentum: [ML2T−1][M L^2 T^{-1}][ML2T−1]
  • Planck's constant: [ML2T−1][M L^2 T^{-1}][ML2T−1]

They are identical.

B: impulse and momentum

  • Impulse: [MLT−1][M L T^{-1}][MLT−1]
  • Momentum: [MLT−1][M L T^{-1}][MLT−1]

They are identical.

C: moment of inertia and moment of a force

  • Moment of inertia: [ML2][M L^2][ML2]
  • Moment of force: [ML2T−2][M L^2 T^{-2}][ML2T−2]

These are not identical.

D: work and torque

  • Work: [ML2T−2][M L^2 T^{-2}][ML2T−2]
  • Torque: [ML2T−2][M L^2 T^{-2}][ML2T−2]

They are identical.

  1. Therefore, the pair that does not have identical dimensions is: C\boxed{C}C​
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