Finding the Center of Mass of Identical Spheres in Contact

To find the center of mass of three identical spheres that are in contact with each other, we can follow a systematic approach. Let's break down the solution into several steps:

Step 1: Understanding the Problem
We are given three identical spheres placed on a table in contact with each other. The task is to determine the location of their center of mass.

Step 2: Identifying Key Information
To solve the problem, we need to consider certain key points:
1. The spheres are identical, which means they have the same mass and radius.
2. The spheres are in contact, forming a triangular arrangement.

Step 3: Visualizing the Situation
Before proceeding with the calculations, it is helpful to visualize the scenario. Imagine three identical spheres in contact with each other, forming an equilateral triangle. We need to find the center of mass within this triangular arrangement.

Step 4: Symmetry Considerations
Due to the symmetry of the equilateral triangle formed by the spheres, we can conclude that the center of mass of the system will also lie at the center of the triangle.

Step 5: Calculating the Center of Mass
The center of mass of an equilateral triangle is located at the intersection point of its medians. In this case, the medians of the triangle formed by the spheres coincide with the line segments joining the centers of the spheres.

To calculate the center of mass, we can find the midpoint of the line segment connecting any two sphere centers. Since all the spheres are identical, the center of mass will be equidistant from each sphere.

Step 6: Determining the Coordinates
Let's assume the centers of the spheres are A, B, and C. We can assign coordinates to these points as follows:
- Sphere A: (0, 0)
- Sphere B: (2r, 0) (where 'r' is the radius of the spheres)
- Sphere C: (r, √3r)

Using these coordinates, we can find the midpoint of AB and AC to locate the center of mass.

The midpoint of AB is ((0 + 2r)/2, (0 + 0)/2) = (r, 0)
The midpoint of AC is ((0 + r)/2, (0 + √3r)/2) = (r/2, √3r/2)

Hence, the center of mass of the three spheres is located at (r/2, √3r/2) within the equilateral triangle formed by the spheres.

In conclusion, the center of mass of three identical spheres placed on a table in contact with each other is located at (r/2, √3r/2), where 'r' represents the radius of the spheres.