In the figure shown, mass of the plank is m and that of the solid cylinder is 8 m . Springs are light. The plank is slightly displaced from equilibrium and then released. Find the period of small oscillations (in seconds) of the plank. There is no slipping at any contact point. The ratio of the mass of the plank and stiffness of the spring i.e. m K = 2 ^2

Step-by-step solution

Kx - f ₁= m a ₂ ....(i) f₁-f₂=8 m a₁ ....(ii) _ cm = I _ cm or (f₁+f₂ ) R= (8 m) R^2 2 ( f ₁+ f ₂ )= (8 ~m ) a ₁ 2 ....(iii) From (ii) and (iii), 2 f₁=(8 m) 3 a₁ 2 or f₁=(8 ~m ) 3 a₁ 4 and f ₂=2 ma ₁ . Hence, f ₂ will be in forward direction. Also, for no slipping, at the point of contact between plank and cylinder. 2 a₁=a₂ ....(iv) f ₁=(8 ~m ) ( 3 a ₂ 8 )=(3 ~m ) a ₂ ....(v) from (i) and (v), 2 K x-(3 m) a₂=m a₂ a ₂= 2 ~K 4 ~m x or ( -d^2 x d t^2 )= ( 2 K 4 m ) x or d ^2 x dt ^2 =- ( 2 4 ^2 2 ) x or ^2= ^2 4 or =