A solid sphere (m kg, R meter) placed on a horizontal rough plane, is attached to a spring S 1 , fixed to a light rod OA of length L meter ,as shown in the figure. The other end O of the rod is hing

Question

A solid sphere (m kg, R meter) placed on a horizontal rough plane, is attached to a spring S 1 ​ , fixed to a light rod OA of length L meter ,as shown in the figure. The other end O of the rod is hinged to rotate in the plane about vertical axis passing through it. A spring S 2 ​ is fixed to the midpoint B of the rod. If the spring constants of both springs are kNm − 1 each, then the angular frequency of oscillations of the mass would be (in rad s − 1 )

TestHub · Accepted Answer

Torque balance on the rod about O: K ( 2 L θ ​ ) 2 L ​ = K ( x − L θ ) L , giving x = 4 5 L θ ​ . The sphere rolls without slipping; torque about its contact point: K ( x − L θ ) R = 5 7 ​ m R 2 α . With x − L θ = 5 x ​ and a = R α : 5 K x ​ = 5 7 ​ ma ⇒ a = 7 m K ​ x . ω = 7 m K ​ ​ - (A)

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