A solid sphere of mass $2 ext{ kg}$ is purely rolling on a horizontal surface with a translational

Question

A solid sphere of mass $2 	ext{ kg}$ is purely rolling on a horizontal surface with a translational velocity of $5 	ext{ m/s}$. It collides head-on with a relaxed horizontal spring of force constant $70 	ext{ N/m}$. The maximum compression produced in the spring before the sphere momentarily stops will be

Professor PC Thomas Classes · Answer

(c) : At maximum compression, all the initial kinetic energy of the rolling sphere is completely stored as elastic potential energy in the spring. The total kinetic energy of a rolling solid sphere is $KE = \frac{1}{2}mv^2(1 + K^2/R^2) = \frac{1}{2}mv^2(1 + 2/5) = \frac{7}{10}mv^2$. Substituting the values: $KE = \frac{7}{10} 	imes 2 	imes (5)^2 = 35 	ext{ J}$. Equating this to spring potential energy $\frac{1}{2}kx^2$: $\frac{1}{2} 	imes 70 	imes x^2 = 35 \Rightarrow 35x^2 = 35 \Rightarrow x^2 = 1$. Therefore, the maximum compression is $x = 1.0 	ext{ m}$.