Three identical metal balls each of radius `r` are placed touching each other on a horizontal surface such that an equilateral triangle is formed, when the center of three balls are joined. The center of mass of system is located at the

Options

A. horizontal surface
B. centre of one of the balls
C. line joining the centres of any two balls
D. point of intersection of the medians

Answer

To find the center of mass of the system consisting of three identical metal balls placed in a triangular formation, we can follow these steps:

### Step 1: Understand the Configuration
We have three identical metal balls, each with a radius \( r \). When they touch each other, they form an equilateral triangle with the distance between the centers of any two balls equal to \( 2r \).

### Step 2: Identify the Centers of the Balls
Let’s denote the centers of the three balls as \( C_1 \), \( C_2 \), and \( C_3 \). The coordinates of these centers can be assigned as follows:
- \( C_1 = (0, 0) \)
- \( C_2 = (2r, 0) \)
- \( C_3 = \left( r, r\sqrt{3} \right) \)

### Step 3: Calculate the Center of Mass
The center of mass \( (x_{cm}, y_{cm}) \) of the system can be calculated using the formula:
\[ x_{cm} = \frac{x_1 + x_2 + x_3}{n} \]
\[ y_{cm} = \frac{y_1 + y_2 + y_3}{n} \]
where \( n \) is the number of balls (which is 3 in this case).

Substituting the coordinates:
\[ x_{cm} = \frac{0 + 2r + r}{3} = \frac{3r}{3} = r \]
\[ y_{cm} = \frac{0 + 0 + r\sqrt{3}}{3} = \frac{r\sqrt{3}}{3} \]

### Step 4: Conclusion
Thus, the center of mass of the system of three identical metal balls is located at:
\[ \left( r, \frac{r\sqrt{3}}{3} \right) \]