AIPMT PRELIMS 2002 Physics Rolling Motion MCQ Question

To analyze the problem, we consider the dynamics of a solid sphere rolling on a smooth horizontal surface when a force 'F' is applied at a height 'h' from the lowest point. The force causes both translational and rotational motion. The maximum acceleration of the center of mass occurs when the sphere rolls without slipping. The relationship between the applied force, the height, and the radius does not directly correlate in a simple manner. The torque due to the applied force is given by $\tau = F \cdot h$. The moment of inertia of a solid sphere about its center is $I = \frac{2}{5} m R^2$. The angular acceleration $\alpha$ can be expressed as $\alpha = \frac{\tau}{I} = \frac{F \cdot h}{\frac{2}{5} m R^2}$. The linear acceleration of the center of mass $a$ is related to angular acceleration by $a = \alpha R$. Substituting for $\alpha$, we find that the maximum acceleration does not depend on the height 'h' in a straightforward way, leading us to conclude that there is no specific relationship between 'h' and 'R'. Therefore, the correct answer is D.

Options A, B, and C suggest specific relationships between 'h' and 'R' which do not hold true under the conditions of maximum acceleration of the center of mass for a rolling sphere. Thus, they are incorrect.