Three identical spheres each of mass `m` and 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 centre of `A` is

Options

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

Explanation

To find the position of the center of mass of three identical spheres, each with mass \( m \) and radius \( R \), placed touching each other in a straight line, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Positions of the Centers:**
- Let the center of the first sphere (Sphere A) be at the origin, \( A(0, 0) \).
- The center of the second sphere (Sphere B) will be at a distance of \( 2R \) from Sphere A, so its position is \( B(2R, 0) \).
- The center of the third sphere (Sphere C) will be at a distance of \( 2R \) from Sphere B, making its position \( C(4R, 0) \).

2. **Use 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, \( m_1 = m_2 = m_3 = m \), and the positions are \( x_1 = 0 \), \( x_2 = 2R \), and \( x_3 = 4R \).

3. **Substitute the Values:**
\[ x_{cm} = \frac{m \cdot 0 + m \cdot 2R + m \cdot 4R}{m + m + m} \]
Simplifying this:
\[ x_{cm} = \frac{0 + 2mR + 4mR}{3m} = \frac{6mR}{3m} = 2R \]

4. **Determine the Position Relative to Sphere A:**
The center of mass is located at \( 2R \) from the origin (center of Sphere A). Since the center of Sphere A is at \( 0 \), the center of mass is \( 2R \) away from Sphere A.

### Final Answer:
The position of the center of mass from the center of Sphere A is \( 2R \).

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