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A uniform rod of length L is rotated in a horizontal plane about a vertical axis through one of its ends. The angular speed of rotation is . Find increase in length of the rod, if and Y are the density and Young's modulus of the rod respectively,

Options

  1. A^3 Y 4 L^2
  2. B4 ^2 L^3 3 Y
  3. C^3 L^3 3 Y
  4. D^3 L^3 8 Y

Correct answer

C. ^3 L^3 3 Y

Step-by-step solution

Given, length of uniform rod =L angular speed = and density of rod = When rod is rotated in a horizontal plane, centrifugal force is responsible for increasing the length of rod, which is also equal to stress. If d l be the change in length of small element d x due to rotation, by application of force d F . Young's modulus, aligned & Y= stress strain = d F / A d l / x & d l= d F x A Y & d F= A Y d l x aligned Where, d F= centrifugal force due to mass of the element of length, d x . aligned d F & =d m x ^2 d F & = m

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