Three identical spheres each of radius .R. are placed touching each other so that their centres A, B and C lie on a straight line. The position of their centre of mass from A is

Options

A. `(2R)/(3)`
B. 2R
C. `(5R)/(3)`
D. `(4R)/(3)`

Answer

To find the position of the center of mass of three identical spheres placed touching each other in a straight line, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
- We have three identical spheres, each with a radius \( R \).
- The centers of the spheres are labeled as \( A \), \( B \), and \( C \).
- The spheres are touching each other, so the distance between the centers of adjacent spheres is \( 2R \).

2. **Identifying Distances**:
- The distance from center \( A \) to center \( B \) is \( 2R \).
- The distance from center \( A \) to center \( C \) is \( 4R \) (since \( A \) to \( B \) is \( 2R \) and \( B \) to \( C \) is another \( 2R \)).

3. **Assigning Masses**:
- Let the mass of each sphere be \( m \).
- Therefore, we have:
- Mass at \( A \) = \( m \)
- Mass at \( B \) = \( m \)
- Mass at \( C \) = \( m \)

4. **Using the Center of Mass Formula**:
- The formula for the center of mass \( x_{cm} \) of a system of particles is given by:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} \]
- Here, we take point \( A \) as the reference point (i.e., \( x_A = 0 \)):
- \( x_1 = 0 \) (for mass at \( A \))
- \( x_2 = 2R \) (for mass at \( B \))
- \( x_3 = 4R \) (for mass at \( C \))

5. **Substituting Values into the Formula**:
- Plugging in the values:
\[ x_{cm} = \frac{m \cdot 0 + m \cdot 2R + m \cdot 4R}{m + m + m} \]
- This simplifies to:
\[ x_{cm} = \frac{0 + 2mR + 4mR}{3m} = \frac{6mR}{3m} \]

6. **Final Calculation**:
- The \( m \) cancels out:
\[ x_{cm} = \frac{6R}{3} = 2R \]

### Conclusion:
The position of the center of mass from point \( A \) is \( 2R \).