https://r2.erweima.ai/mathbot/image/1724843608328u3ujfhz6.jpg

If the maximum resultant force that the wire can exert on each pulley is 130  N
  , determine the greatest force  P
  that can be applied to the wire as shown.


To solve this problem, we need to find the greatest force \( P \) that can be applied to the wire given that the maximum resultant force that the wire can exert on each pulley is \( 130 \, \text{N} \).

### Problem Analysis

1. The force \( P \) is applied at a 30° angle to the horizontal.
2. The force is transmitted through the wire, creating tension forces in the sections of the wire attached to pulleys A and B (or D and C, depending on the wire's configuration).
3. The resultant tension on each pulley should not exceed \( 130 \, \text{N} \).

### Step-by-Step Solution

#### Step 1: Resolve Force \( P \) into Components
- The force \( P \) can be resolved into two components:
  - Horizontal component: \( P_x = P \cos(30^\circ) \)
  - Vertical component: \( P_y = P \sin(30^\circ) \)

#### Step 2: Apply the Equilibrium Conditions
For each pulley, the forces must balance. Assuming symmetry, let's consider one half of the wire.

For example, for pulley A:
- The horizontal force \( P_x \) will be balanced by the horizontal components of the tensions in the sections of the wire.
- The vertical component \( P_y \) will be balanced by the vertical components of the tensions.

#### Step 3: Use the Given Constraint
The maximum force that the wire can exert on each pulley is \( 130 \, \text{N} \). Therefore, the magnitude of the tension force \( T \) in the wire sections connected to any pulley should satisfy:

\[
T \leq 130 \, \text{N}
\]

#### Step 4: Relate Force \( P \) to the Tensions
Considering that the wire is in tension and assuming that the tension is evenly distributed due to symmetry, each section of the wire from the point of force application will experience tension equal to the magnitude of \( P \).

Given the constraints:
\[
P \leq \frac{130}{2 \cos(30^\circ)}
\]

This formula comes from considering the balance of forces in one direction (either horizontal or vertical) and recognizing that the force in one direction is shared equally by two wires.

### Calculation

Let's compute \( P \):

\[
P \leq \frac{130}{2 \cos(30^\circ)}
\]

Substitute \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \):

\[
P \leq \frac{130}{2 \times \frac{\sqrt{3}}{2}} = \frac{130}{\sqrt{3}} \approx 75.1 \, \text{N}
\]

### Final Answer

The greatest force \( P \) that can be applied to the wire is approximately **75.1 N**.

---

Would you like more details on any part of the solution? 

Here are some related questions you might consider:
1. How do you resolve a force vector into its components?
2. What are the equilibrium conditions in mechanics?
3. How does symmetry simplify the analysis of forces in a system?
4. What are the trigonometric functions and their importance in physics?
5. How would the problem change if the angle were different?

**Tip:** When solving problems involving forces, always ensure that all components of the forces are correctly accounted for in both directions.

Discover how to calculate the maximum force that can be applied to a wire in a pulley system at a 30° angle, considering a maximum resultant force of 130 N per pulley. This problem involves vector analysis, trigonometric functions, and equilibrium conditions in mechanics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation