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JEE Main2015PhysicsCenter of Mass, Momentum and CollisionActual

A uniform thin rod AB of length L has linear mass density μ x = a + b x L , where x is measured from A. If the CM of the rod lies at a distance of 7 12 L from A, then a and b are related as:

Options

  1. A2 a = b
  2. Ba = 2 b
  3. Ca = b
  4. D3 a = 2 b

Correct answer

A. 2 a = b

Step-by-step solution

Consider an infinitesimal small element of length d x and mars density λ x c m = ∫ d m x ∫ d m = ∫ λ   d x x ∫ λ   d x = ∫ 0 L a + b L x x d x ∫ 0 L a + b L x d x = ∫ 0 L a x d x + ∫ 0 L b L x 2 d x ∫ 0 L a d x + ∫ 0 L b L x d x = a x 2 2 0 L + b L   x 3 3 0 L a x 0 L + b L x 2 2 0 L = a 2 L 2 - 0 2 + b 3 L   L 3 - 0 3 a L - 0 + b 2 L L 2 - 0 2 = a 2 L 2 + b L 2 3 a L + b L 2 = a 2 L + b L 3 a + b 2 ∵ x c m = 7 L 12

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