If the maximum resultant force that the wire can exert on each pulley is 134 N, determine the greatest force P that can be applied to the wire as shown. 

To solve this problem, we'll assume that the setup involves a wire passing over multiple pulleys, with force \( P \) being applied to the wire. The maximum resultant force that the wire can exert on each pulley is given as 134 N. The objective is to determine the greatest force \( P \) that can be applied to the wire without exceeding this limit on any pulley.

### Solution Steps

1. **Understanding the Setup:**
   - Each pulley can only handle a maximum force of 134 N.
   - The force \( P \) is applied to the wire, and depending on how the pulleys are arranged, this force will be distributed across them.

2. **Force Distribution:**
   - If the pulleys are arranged such that the wire loops through them, the force on each segment of the wire would typically depend on the number of times the wire loops through the pulley system.

3. **Determine the Relation Between \( P \) and Force on Pulley:**
   - If the wire runs over two pulleys (a typical scenario), the force in each segment of the wire might equal the force \( P \) applied.
   - Given that the wire passes over the pulley, the tension in the wire on either side of the pulley would be the same. So if \( P \) is the applied force, the tension in the wire will also be \( P \).

   - However, due to the way forces are distributed in pulley systems, the effective force on the pulley might be doubled. This is a key point in pulley systems with two or more pulleys.

4. **Set Up the Inequality:**
   - If each pulley can handle a maximum force of 134 N, and the effective force on each pulley is twice the applied force \( P \), we get the inequality:
     \[
     2P \leq 134 \, \text{N}
     \]

5. **Solve for \( P \):**
   \[
   P \leq \frac{134 \, \text{N}}{2} = 67 \, \text{N}
   \]

### Conclusion:
The greatest force \( P \) that can be applied to the wire without exceeding the maximum force of 134 N on any pulley is **67 N**.

Would you like more details on pulley systems, or do you have any other questions related to this topic?

### Relative Questions:

1. How does the number of pulleys affect the force that can be applied to the system?
2. What happens to the tension in the wire if the angle between the wire segments changes?
3. Can you calculate the force applied in a system with three pulleys?
4. How does friction in the pulley affect the maximum force that can be applied?
5. What other factors could limit the maximum force applied in a pulley system?

### Tip:
When dealing with pulley systems, always consider the number of pulleys and how the tension is distributed across the wire. This will help you accurately determine the forces in the system.

Explore how to determine the greatest force P that can be applied to a pulley system without exceeding the maximum force limit on any pulley. This problem involves mechanics and force distribution in pulley systems, crucial for understanding mechanical systems.

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