A solid sphere of mass M and radius R is attached to a spring of negligible mass kept on a horizontal plane such that it can roll without slipping. The sphere is made to execute SHM by stretching through a distance and released, then the time period of such oscillation is ( K = spring constant )

Step-by-step solution

Total energy = E = 1 2 mV V^2+ 1 2 I ^2+ 1 2 Kx ^2 aligned & = 1 2 mV ^2+ 1 2 ( 2 5 mR ^2 ) V ^2 R ^2 + 1 2 Kx ^2 & = 7 10 mV ^2+ 1 2 Kx ^2 aligned As E = constant aligned & dE dt =0 & 7 10 ~m (2 ~V ) dV dt + 1 2 ~K (2 x ) dx dt =0 & 7 5 mVa + KVx =0 & 7 5 ma + Kx =0 & a = -5 7 K m x aligned In SHM, a=- ^2 x So, = 5 ~K 7 ~m Hence, T =2 7 ~m 5 ~K