Two skaters on frictionless ice hold opposite ends of a pole of negligible mass. An axis runs along it, with the origin at the center of mass of the two-skater system. One skater, Fred , weighs twice as much as the other skater, Ethel. Where do the skaters meet if (a) Fred pulls hand over along the pole so as to draw himself to Ethel , (b) Ethel pulls hand over hand to draw herself to Fred, and (c ) both skaters pull hand over hand ?

Answer

To solve the problem step by step, we need to analyze the situation involving the two skaters, Fred and Ethel, and their center of mass.

### Step-by-Step Solution:

1. **Identify the Masses**:
- Let the mass of Ethel be \( m \).
- Therefore, the mass of Fred, who weighs twice as much, will be \( 2m \).

2. **Determine the Center of Mass**:
- The center of mass (CM) of the system can be calculated using the formula:
\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
- Assume Ethel is at position \( -d \) and Fred is at position \( d \) (with the origin at the center of mass):
\[ x_{CM} = \frac{(m)(-d) + (2m)(d)}{m + 2m} = \frac{-md + 2md}{3m} = \frac{md}{3m} = \frac{d}{3} \]
- Since the center of mass is at the origin, we have:
\[ x_{CM} = 0 \]

3. **Analyze Each Scenario**:
- **(a) Fred pulls himself towards Ethel**:
- As Fred pulls himself towards Ethel, he moves closer to the center of mass. The center of mass remains unchanged (at the origin), so they will meet at the center of mass, which is at \( x = 0 \).

- **(b) Ethel pulls herself towards Fred**:
- Similarly, when Ethel pulls herself towards Fred, she also moves towards the center of mass. Again, they will meet at the center of mass, which is at \( x = 0 \).

- **(c) Both skaters pull hand over hand**:
- If both skaters pull themselves towards each other, they will still meet at the center of mass. The center of mass does not change, so they will meet at \( x = 0 \).

4. **Conclusion**:
- In all three scenarios, Fred and Ethel will meet at the center of mass, which is at the origin \( x = 0 \).

### Final Answer:
In all cases (a), (b), and (c), the skaters meet at the center of mass, which is at \( x = 0 \).