Question

Show that the center of mass of three identical particles situated at the points $z_1,z_2,z_3$ is $(z_1+z_2+z_3)/3$.

Short answer

The center of mass of three identical particles is $(z_1+z_2+z_3)/3$.

Step-by-step solution

- Understand the Concept of Center of Mass

The center of mass for a system of particles is the average position of all the mass in the system, weighted according to their masses. For identical particles, it simplifies to the average of their positions.

- List Given Data

The particles are situated at points represented by complex numbers or coordinates: $z_1,z_2,z_3$. Each particle has the same mass, so the masses do not affect their relative positions.

- Write the Formula

For three identical particles, the formula for the center of mass is:$$Z_{cm}=\frac{z_1+z_2+z_3}{3}.$$

- Substitute and Simplify

Since the particles are identical, substitute the positions into the formula:$(z_1+z_2+z_3)/3$.This shows that the center of mass is the average of the positions of the three particles.

Key concepts

identical particles

When dealing with identical particles, it means that each particle has the same mass and, thus, contributes equally to the overall system. In our exercise, we consider three identical particles located at positions represented by complex numbers or coordinates: $z_1$, $z_2$, and $z_3$. Since their masses are the same, we can simplify calculations. This is because the mass of each particle will cancel out during the calculation of the center of mass. In simpler words, when particles are identical, their relative positions become the focus rather than their individual masses.

This equal treatment further simplifies our work because each particle 'pulls' on the center of mass with equal strength, making the problem only about their locations.

average position

The average position in a system like ours is straightforward to find. When working with identical particles, we can see the center of mass as a type of average of their positions. The center of mass is simply the point where you could evenly balance all particles if they were connected.

In mathematical terms, we achieve this by taking the sum of all particle positions and dividing by the number of particles. For our three particles located at positions $z_1$, $z_2$, and $z_3$, the formula to find their average position is:

$$\text{Average Position}=\frac{z_1+z_2+z_3}{3}$$

This formula provides the middle point (or balance point) among all these positions.

weighted average

The concept of a weighted average becomes simpler with identical particles. Typically, calculating a weighted average involves multiplying each data point by a weight that represents its significance (e.g., masses in physics problems). However, because our particles are identical, each mass is the same, and therefore, each weight is the same.

Thus, our problem does not need special weights and the center of mass calculation is simply the arithmetic mean of the positions. This is mathematically expressed as:

$$Z_{cm}=\frac{z_1+z_2+z_3}{3}$$

Here, $Z_{cm}$ represents the center of mass, calculated as the sum of all positions divided by the number of particles.

Since all particles have the same weight, this simplifies to the average position formula from earlier.