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 find the instantaneous center of rotation for a sphere moving with a horizontal velocity \( v_0 \) and an angular velocity \( \omega \) in an anti-clockwise sense, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a sphere moving horizontally to the right with a velocity \( v_0 \).
- The sphere is also rotating anti-clockwise with an angular velocity \( \omega \).
- We are given that \( |v_0| = |\omega R| \), where \( R \) is the radius of the sphere.

2. **Defining Instantaneous Center of Rotation**:
- The instantaneous center of rotation is the point on the object that has zero velocity at that instant. This point is crucial for analyzing the motion of the sphere.

3. **Analyzing Velocities**:
- The point at the bottom of the sphere (point P) will have a velocity due to both the translational motion \( v_0 \) and the rotational motion \( \omega R \).
- The translational velocity \( v_0 \) acts to the right.
- The rotational velocity at point P (which is at the bottom of the sphere) acts to the left and is equal to \( \omega R \).

4. **Setting Up the Equation**:
- Since we know that \( |v_0| = |\omega R| \), we can express the velocities:
- Velocity at point P due to translation = \( v_0 \) (to the right)
- Velocity at point P due to rotation = \( -\omega R \) (to the left)
- Therefore, we can write:
\[ v_P = v_0 - \omega R \]

5. **Finding the Instantaneous Center**:
- For the instantaneous center of rotation, we set the total velocity at point P to zero:
\[ v_0 - \omega R = 0 \]
- Since \( |v_0| = |\omega R| \), we find that the velocity at point P is indeed zero.

6. **Locating the Instantaneous Center**:
- The instantaneous center of rotation lies directly below the center of the sphere at a distance equal to the radius \( R \) from the center of the sphere. This point is on the line of action of the velocity \( v_0 \) and is located on the surface of the sphere.

### Conclusion:
Thus, the instantaneous center of rotation is located at the point on the ground directly below the center of the sphere.