Consider a case of fixed smooth sphere of radius R . A block of mass m is placed at the most poit of the sphere . A sharp impulse is applied on the block to impart it a speed v .
What is the normal reaction between block and the sphere just after the impulse is applied ?

Options

A. `(mv^(2))/R`
B. `mg-(mv^(2))/R`
C. mg
D. 0

Explanation

To solve the problem, we need to analyze the forces acting on the block just after the impulse is applied. Here's a step-by-step solution:

### Step 1: Identify the forces acting on the block
When the block is at the topmost point of the sphere, two main forces act on it:
1. The weight of the block, \( W = mg \), acting downwards.
2. The normal reaction force \( N \) exerted by the sphere on the block, acting perpendicular to the surface of the sphere (also downwards at this point).

### Step 2: Understand the motion of the block
After the impulse is applied, the block starts moving with a speed \( v \) horizontally. As it moves, it will also experience a centripetal acceleration because it is moving in a circular path around the center of the sphere.

### Step 3: Apply the centripetal force concept
For the block to maintain circular motion at the topmost point, the net force acting towards the center of the sphere must provide the necessary centripetal force. The centripetal force \( F_c \) required for the block moving with speed \( v \) at radius \( R \) is given by:
\[ F_c = \frac{mv^2}{R} \]

### Step 4: Set up the equation for forces
At the topmost point, the net force towards the center of the sphere is the difference between the weight of the block and the normal reaction force:
\[ F_c = mg - N \]
Substituting the expression for centripetal force:
\[ \frac{mv^2}{R} = mg - N \]

### Step 5: Solve for the normal reaction force \( N \)
Rearranging the equation gives us:
\[ N = mg - \frac{mv^2}{R} \]

### Conclusion
Thus, the normal reaction between the block and the sphere just after the impulse is applied is:
\[ N = mg - \frac{mv^2}{R} \]