Question

A solid sphere of radius R is placed on smooth horizontal surface. A horizontal force F is applied at height h from the lowest point. For the maximum acceleration of centre of mass, which is correct?

• A. h = R
• B. h = 2R
• C. h = 0
• D. no relation between h and R

Answer 1

The Correct Option is D

The problem involves understanding the dynamics of a solid sphere subjected to a horizontal force applied at a certain height. To solve it, we need to analyze the motion considering both linear and rotational dynamics.

Given a solid sphere of radius R, with a force F applied at height h from the lowest point, the aim is to find the condition for maximum acceleration of its center of mass.

Let's delve into the physics:

1. The force F applied creates both a translational motion in the sphere and a torque about the center of mass.
2. The linear acceleration a of the center of mass is given by Newton's second law: F = ma where m is the mass of the sphere.
3. The torque \tau caused by the force about the center of mass is: \tau = F \cdot h
4. This torque leads to an angular acceleration \alpha given by: \tau = I \alpha where I is the moment of inertia of the solid sphere about its center, which is \frac{2}{5}mR^2. Thus: Fh = \left( \frac{2}{5}mR^2 \right) \alpha.
5. The linear acceleration a and angular acceleration \alpha are related by: a = \alpha R. Using this relation in the torque equation, we have: Fh = \frac{2}{5}mR^2 \cdot \frac{a}{R}, Fh = \frac{2}{5}maR \Rightarrow h = \frac{2}{5}\frac{aR}{F}.
6. The aim is to maximize a, and the result shows that h depends on F and R in such a way that there is no fixed or direct relationship that guarantees maximum acceleration for particular values of h relative to R.

From the analysis, it is evident there is no simple relationship between h and R that would always result in maximum acceleration of the center of mass. Therefore, the correct answer is no relation between h and R.