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. center one of the balls
C. line joining centers of any two balls
D. point of intersection of medians

Explanation

To solve the problem of finding the center of mass of three identical metal balls arranged in an equilateral triangle, we will follow these steps:

### Step 1: Understand the Arrangement
We have three identical metal balls, each with a radius `r`. They are arranged in such a way that they touch each other, forming the vertices of an equilateral triangle. The centers of the balls will also form the vertices of this triangle.

### Step 2: Determine the Coordinates of the Centers
Let’s assign coordinates to the centers of the balls:
- Ball 1 (C1) at (0, 0)
- Ball 2 (C2) at (2r, 0) (since the distance between the centers of two touching balls is 2r)
- Ball 3 (C3) at (r, r√3) (the third vertex of the equilateral triangle)

### Step 3: Calculate the Center of Mass
The center of mass (CM) of a system of particles can be calculated using the formula:
\[ CM_x = \frac{x_1 + x_2 + x_3}{3} \]
\[ CM_y = \frac{y_1 + y_2 + y_3}{3} \]

Substituting the coordinates:
- For x-coordinates:
\[ CM_x = \frac{0 + 2r + r}{3} = \frac{3r}{3} = r \]

- For y-coordinates:
\[ CM_y = \frac{0 + 0 + r\sqrt{3}}{3} = \frac{r\sqrt{3}}{3} \]

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