A sphere is moving at some instant with horizontal velocity `v_(0)` in right and angular velocity `omega` in anti clockwise sense. If `|v_(0)| = |omega R|`, the instantaneous centre of rotation is

Options

A. at the bottom of the sphere
B. at the top of the sphere
C. at the centre of the sphere
D. anywhere inside the sphere

Answer

To solve the problem, we need to find the instantaneous center of rotation of a sphere that is rolling with a horizontal velocity \( v_0 \) to the right and an angular velocity \( \omega \) in the anticlockwise direction. We are given that \( |v_0| = |\omega R| \), where \( R \) is the radius of the sphere.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The sphere is rolling on a smooth surface. The horizontal velocity \( v_0 \) is directed to the right, and the angular velocity \( \omega \) is directed in the anticlockwise direction.

2. **Relating Linear and Angular Velocities**:
- The linear velocity at the point of contact with the ground due to rotation is given by \( \omega R \). Since \( |v_0| = |\omega R| \), the linear velocity of the center of mass \( v_0 \) is equal to the linear velocity due to rotation at the bottom point of the sphere.

3. **Finding the Instantaneous Center of Rotation**:
- The instantaneous center of rotation is the point on the sphere where the velocity is zero at that instant. For rolling without slipping, this point is located at the bottom of the sphere when the sphere rolls to the right.
- However, since \( v_0 = \omega R \), the sphere is not just rolling; it is also translating. We need to find a point where the translational velocity \( v_0 \) cancels out the rotational velocity \( \omega R \).

4. **Analyzing the Velocity Components**:
- The point at the bottom of the sphere (point of contact) has a velocity of \( \omega R \) directed to the left (due to rotation).
- The center of mass of the sphere is moving to the right with velocity \( v_0 \).
- To find the instantaneous center of rotation, we can consider a point above the center of the sphere where the leftward velocity component due to rotation cancels out the rightward velocity of the center of mass.

5. **Determining the Position of the Instantaneous Center**:
- Let’s denote the instantaneous center of rotation as point \( P \). The velocity at point \( P \) can be expressed as:
\[ v_P = v_0 - \omega R \]
- Setting \( v_P = 0 \) gives us:
\[ v_0 - \omega R = 0 \implies v_0 = \omega R \]
- This means that point \( P \) is located directly above the center of the sphere, at a distance \( R \) from the center.

6. **Conclusion**:
- Therefore, the instantaneous center of rotation is at the top of the sphere.

### Final Answer:
The instantaneous center of rotation is at the **top of the sphere**.