A spherical shell of mass 3 kg is rolling without slipping on a horizontal surface with a speed of 4 m s ⁻¹ . It collides with a horizontal ideal spring and compresses it. If the maximum compression of the spring before the shell momentarily stops is 0.2 m , the spring constant of the spring is ______ N m ⁻¹ .

Step-by-step solution

By the conservation of mechanical energy, the total initial kinetic energy of the rolling spherical shell is entirely converted into the elastic potential energy of the spring at maximum compression. The total kinetic energy of a rolling spherical shell (hollow sphere) is: K = 1 2 mv^2 + 1 2 I ^2 For a spherical shell, I = 2 3 mr^2 . With = v r , we get: K = 1 2 mv^2 + 1 2 ( 2 3 mr^2 ) ( v r )^2 K = 1 2 mv^2 + 1 3 mv^2 = 5 6 mv^2 Substituting the given values: K = 5 6 3 (4)^2 = 5 2 16 = 40 J The elastic potential e