Four identical spheres each of radius 10 cm and mass 1 kg are placed on a horizontal surface touching one another so that their centres are located at the corners of square of side 20 cm. What is the distance of their centre of mass from centre of either sphere ?

Options

A. 5 cm
B. 10 cm
C. 20 cm
D. None of these

Answer

To find the distance of the center of mass of four identical spheres from the center of either sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have four identical spheres, each with a radius of 10 cm and a mass of 1 kg.
- The spheres are placed such that their centers form the corners of a square with a side length of 20 cm.

2. **Identifying the Centers of the Spheres**:
- The centers of the spheres are at the corners of the square. Let's denote the corners as A, B, C, and D.
- The coordinates of the centers can be assigned as follows:
- A (0, 0)
- B (20, 0)
- C (20, 20)
- D (0, 20)

3. **Calculating the Center of Mass**:
- The formula for the center of mass (CM) of a system of particles is given by:
\[ \text{CM} = \frac{\sum m_i \cdot r_i}{\sum m_i} \]
- Here, \( m_i \) is the mass of each sphere (1 kg) and \( r_i \) are the position vectors of the centers.

4. **Calculating the x-coordinate of CM**:
- The x-coordinates of the centers are 0, 20, 20, and 0.
- Therefore, the x-coordinate of the center of mass is:
\[ x_{CM} = \frac{1 \cdot 0 + 1 \cdot 20 + 1 \cdot 20 + 1 \cdot 0}{1 + 1 + 1 + 1} = \frac{40}{4} = 10 \text{ cm} \]

5. **Calculating the y-coordinate of CM**:
- The y-coordinates of the centers are 0, 0, 20, and 20.
- Therefore, the y-coordinate of the center of mass is:
\[ y_{CM} = \frac{1 \cdot 0 + 1 \cdot 0 + 1 \cdot 20 + 1 \cdot 20}{1 + 1 + 1 + 1} = \frac{40}{4} = 10 \text{ cm} \]

6. **Finding the Distance from the Center of Either Sphere**:
- The center of any sphere (for example, sphere A) is at (0, 0).
- The center of mass is at (10, 10).
- The distance \( d \) from the center of sphere A to the center of mass is calculated using the distance formula:
\[ d = \sqrt{(x_{CM} - x_A)^2 + (y_{CM} - y_A)^2} = \sqrt{(10 - 0)^2 + (10 - 0)^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \text{ cm} \]

### Final Answer:
The distance of the center of mass from the center of either sphere is \( 10\sqrt{2} \) cm.