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A uniform plank of mass m, free to move in horizontal direction, is placed on the top of a solid cylinder of mass 2 m and radius R. The plank is attached with one end of a mass less spring of force constant k and the other end of spring is fixed with a wall as shown in fig. The plank is shiftily displaced towards spring and released. There is no slipping anv where then

Options

  1. ATotal energy of the system, when compression is spring is x and velocity of plank is v, is 7 8 mv ^2+ 1 2 kx ^
  2. BAngular frequency of oscillation is 4 k 7 ~m
  3. CTime period of oscillation is 7 ~m 4 k
  4. DAngular frequency oscillation is 2 k 7 m

Correct answer

B. Angular frequency of oscillation is 4 k 7 ~m

Step-by-step solution

At the instant when velocity of plank is v velocity of C.M. of cylinder is V 2 and angular velocity of cylinder is aligned & E = 1 2 mv ^2+ 1 2 kx ^2+ 1 2 2 ~m ( v 2 )^2+ 1 2 I ( ~V / 2 R )^2 & I = 2 mR ^2 R = mR ^2 & = 7 8 mv ^2+ 1 2 kx ^2 aligned For free oscillation d E d t =0 a=- ( 4 k 7 m ) x ^2= 4 k 7 ~m and t = 2 =2 7 ~m 4 k

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