A body of mass 100 gram is tied to a spring of spring constant 8 ~N / m , while the other end of a spring is fixed. If the body moves in a circular path on smooth horizontal surface with constant angular speed 8 rad / s then the ratio of extension in the spring to its natured length will be

Step-by-step solution

The centripetal force required for circular motion is provided by the spring force. For a mass m moving with angular speed along a circular path of radius r = L₀ + x , where L₀ is the natural length and x the extension, the forces balance as kx = m ^2 (L₀ + x) . Rearranging gives x(k - m ^2) = m ^2 L₀ , so x L₀ = m ^2 k - m ^2 . With m = 0.1 kg , k = 8 N/m , and = 8 rad/s , we compute m ^2 = 6.4 N/m and k - m ^2 = 1.6 N/m . The ratio is then 6.4 1.6 = 4 . The extension is four times the natural length.