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We are given a pulley system supporting a 160-kg load, and we need to determine the magnitude and direction of the force \( P \) that must be exerted on the free end of the rope to maintain equilibrium. The following information is provided:

- The weight of the load \( W = 160 \, \text{kg} \).
- The angle \( \beta = 20^\circ \).
- The tension in the rope is the same on both sides of the pulley (since it's a simple pulley system).
- We need to find the magnitude and direction of the force \( P \).

### Steps:

1. **Calculate the weight force \( W \):**
   \[
   W = mg = 160 \times 9.81 = 1569.6 \, \text{N}
   \]
   This is the downward force exerted by the load.

2. **Tension in the rope:**
   Since the pulley is ideal (massless and frictionless), the tension in the rope is the same on both sides of the pulley. Let \( T \) represent the tension in the rope.

   The vertical component of the tension must balance the weight of the load. Thus:
   \[
   2T \sin(\beta) = W
   \]
   Substituting the known values:
   \[
   2T \sin(20^\circ) = 1569.6
   \]
   Solving for \( T \):
   \[
   T = \frac{1569.6}{2 \sin(20^\circ)}
   \]

3. **Calculate \( T \):**
Let me calculate the value of \( T \).The tension in the rope is approximately \( T = 2294.6 \, \text{N} \).

4. **Horizontal Force \( P \):**
   The force \( P \) that must be exerted at the free end of the rope has two components: a horizontal component and a vertical component. Since we know the angle \( \alpha \) at which \( P \) is applied is not given, we assume it's determined by the geometry of the system.

   Since the tension \( T \) is acting along the rope, and the vertical component of the tension is balancing the weight, the force \( P \) must account for the horizontal component of the tension. Using the angle \( \alpha \) for the direction of \( P \), we would typically resolve \( P \) in terms of \( T \), using equilibrium conditions.

Would you like to solve for the exact magnitude of \( P \) given specific angles or get further clarification on the assumptions for \( \alpha \)? Let me know if you need more details or have any questions.

### Related questions:
1. What are the assumptions behind the ideal pulley system?
2. How would friction in the pulley affect the tension calculations?
3. Can the direction of the force \( P \) be found using vector decomposition?
4. How is the equilibrium condition applied in such systems?
5. What if the pulley was not frictionless, how would the tension vary?

**Tip:** Always ensure to decompose forces into their components when solving pulley problems.

A 160-kg load is supported by a rope-and-pulley arrangement. Knowing that β = 20°, determine the magnitude and direction of the force P that must be exerted on the free end of the rope to maintain equilibrium.

Learn how to calculate the magnitude and direction of force P in a rope-and-pulley system supporting a 160-kg load. This statics problem covers key concepts such as force equilibrium, tension, and trigonometric relationships. Suitable for university engineering students.

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