Answer:
Explanation:
I'd be glad to help you determine the distance of the center of mass (COM) of the system from the origin.
**Given:**
- n particles with masses m, 2m, 3m, ..., nm (arithmetic progression)
- Distances from the origin: x₁ = 1, x₂ = 2, x₃ = 3, ..., xₙ = n (arithmetic progression)
**Assumptions:**
- The particles are located on the x-axis.
**Finding the Center of Mass (COM):**
The center of mass (COM) of a system of particles is a weighted average of their positions, where the weights are the masses of the particles. It can be calculated using the following formula:
```
x_COM = (Σ (m_i * x_i)) / Σ m_i
```
where:
- x_COM is the x-coordinate of the center of mass
- m_i is the mass of the i-th particle
- x_i is the x-coordinate of the i-th particle
- Σ represents the summation over all particles (i = 1 to n)
**Solution:**
1. **Recognize the Arithmetic Progressions:**
Both the masses (m, 2m, 3m, ..., nm) and the distances (1, 2, 3, ..., n) form arithmetic progressions. This simplifies the calculations.
2. **Express the Summations:**
- Sum of masses (Σ m_i): We can use the formula for the sum of an arithmetic series:
```
Σ m_i = n * (m + nm) / 2
```
- Sum of distances (Σ x_i): Similar to the sum of masses, we can use the formula for an arithmetic series:
```
Σ x_i = n * (1 + n) / 2
```
3. **Calculate the Center of Mass (COM):**
Substitute the expressions for Σ m_i and Σ x_i into the formula for x_COM:
```
x_COM = (n * (m + nm) / 2 * n * (1 + n) / 2) / (n * (m + nm) / 2)
```
Cancel out common factors:
```
x_COM = n * (1 + n) / 2
```
**Simplifying further for an even distribution of masses (m = 2m = 3m = ... = nm):**
In this special case, all masses are equal, so m = 2m = 3m = ... = nm = M (let's say). This simplifies the calculation to:
```
x_COM = n * (1 + n) / (2n)
= (1 + n) / 2
```
**Interpretation:**
The center of mass (COM) is always located somewhere between the minimum distance (x₁ = 1) and the maximum distance (xₙ = n) from the origin. This makes sense intuitively, as the COM takes into account the weighted average of both the masses and their positions.
If the masses are evenly distributed (m = 2m = 3m = ... = nm), the COM will be exactly in the middle:
```
x_COM = (1 + n) / 2
```
However, for a general case where masses are not equal, the COM will be closer to the particles with larger masses. The specific location depends on the relative values of the masses.
