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. `10sqrt(2) cm`

Answer

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

### Step 1: Understand the Configuration
We have four identical spheres, each with a radius of 10 cm and mass of 1 kg. They are arranged such that their centers form the corners of a square with a side length of 20 cm.

### Step 2: Identify the Coordinates of the Centers
Let's assign coordinates to the centers of the spheres:
- Sphere 1 (bottom left corner): \( (0, 0) \)
- Sphere 2 (bottom right corner): \( (20, 0) \)
- Sphere 3 (top left corner): \( (0, 20) \)
- Sphere 4 (top right corner): \( (20, 20) \)

### Step 3: Calculate the Center of Mass
The center of mass (CM) for a system of particles can be calculated using the formula:
\[ \text{CM} = \frac{1}{M} \sum m_i \vec{r_i} \]
where \( M \) is the total mass and \( \vec{r_i} \) are the position vectors of the masses.

In our case, since all spheres have the same mass \( m = 1 \, \text{kg} \), the total mass \( M = 4 \, \text{kg} \).

The coordinates of the center of mass can be calculated as:
\[ x_{CM} = \frac{1}{M} \sum m_i x_i = \frac{1}{4} (0 + 20 + 0 + 20) = \frac{40}{4} = 10 \, \text{cm} \]
\[ y_{CM} = \frac{1}{M} \sum m_i y_i = \frac{1}{4} (0 + 0 + 20 + 20) = \frac{40}{4} = 10 \, \text{cm} \]

Thus, the center of mass is located at \( (10, 10) \).

### Step 4: Calculate the Distance from the Center of Either Sphere
Now, we need to find the distance from the center of one of the spheres (for example, Sphere 1 at \( (0, 0) \)) to the center of mass \( (10, 10) \).

Using the distance formula:
\[ d = \sqrt{(x_{CM} - x_{sphere})^2 + (y_{CM} - y_{sphere})^2} \]
Substituting the values:
\[ d = \sqrt{(10 - 0)^2 + (10 - 0)^2} = \sqrt{10^2 + 10^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \, \text{cm} \]

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

---