A non-uniform ball of radius R and radius of gyration about geometric centre `=R//2` is kept on a frictionless surface. The geometric centre coincides with the centre of mass. The ball is struck horizontally with a sharp impulse `=J` the point of application of the impulse is at a height `h` above the surface. then.

Options

A. (a)The ball will slip on surface for all cases
B. (b)the ball will roll purely if `h=5R//4`
C. (c)the ball will roll purely if `h=3R//2`
D. (d)there will be no rotation if `h=R`

Explanation

To solve the problem step by step, we will analyze the situation described and apply the principles of linear and angular momentum.

### Step 1: Understand the System
We have a non-uniform ball of radius \( R \) with a radius of gyration \( k = \frac{R}{2} \). The ball is on a frictionless surface, and the center of mass coincides with the geometric center. An impulse \( J \) is applied horizontally at a height \( h \) above the surface.

### Step 2: Define the Parameters
- Radius of the ball: \( R \)
- Radius of gyration: \( k = \frac{R}{2} \)
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