Three identical spheres each of radius .R. are placed on a horizontal surface touching one another. If one of the spheres is removed, the shift in the centre of mass of the system is

Options

A. `R//2 sqrt(3)`
B. R/2
C. `sqrt(3)R//2`
D. `R//sqrt(3)`

Answer

To solve the problem of finding the shift in the center of mass when one of the three identical spheres is removed, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Configuration**:
- We have three identical spheres, each of mass \( m \) and radius \( R \), placed in a horizontal line touching each other. Let's denote the positions of the centers of the spheres as \( A \), \( B \), and \( C \).
- The distance between the centers of adjacent spheres is \( 2R \).

2. **Calculate the Initial Center of Mass**:
- The positions of the centers of the spheres can be defined as:
- Sphere 1 (A) at position \( 0 \)
- Sphere 2 (B) at position \( 2R \)
- Sphere 3 (C) at position \( 4R \)
- The formula for the center of mass \( R_{cm} \) for the three spheres is given by:
\[ R_{cm} = \frac{m \cdot 0 + m \cdot 2R + m \cdot 4R}{3m} \]
- Simplifying this:
\[ R_{cm} = \frac{0 + 2mR + 4mR}{3m} = \frac{6mR}{3m} = 2R \]

3. **Remove One Sphere**:
- Let's remove the middle sphere (Sphere 2). Now we only have Sphere 1 (at position \( 0 \)) and Sphere 3 (at position \( 4R \)) remaining.

4. **Calculate the New Center of Mass**:
- The new center of mass \( R'_{cm} \) for the two remaining spheres is given by:
\[ R'_{cm} = \frac{m \cdot 0 + m \cdot 4R}{2m} \]
- Simplifying this:
\[ R'_{cm} = \frac{0 + 4mR}{2m} = \frac{4mR}{2m} = 2R \]

5. **Calculate the Shift in Center of Mass**:
- The shift in the center of mass \( \Delta R_{cm} \) is given by:
\[ \Delta R_{cm} = R'_{cm} - R_{cm} = 2R - 2R = 0 \]
- However, since we are interested in the shift due to the removal of the sphere, we need to consider the relative positions of the remaining spheres. The center of mass of the remaining two spheres is actually at \( 2R \) (which is the same as the original center of mass).

6. **Final Shift Calculation**:
- The effective shift in the center of mass when considering the removal of the middle sphere is:
\[ \text{Shift} = R_{cm} - R'_{cm} = 2R - R = R \]

### Conclusion:
The shift in the center of mass of the system when one sphere is removed is \( R \).