https://file.aiquickdraw.com/mathbot/file/1730160349789qeeg1a18.jpg

To solve this problem, we need to analyze the forces and apply the conditions of equilibrium for the rod, specifically focusing on the torques and forces along the horizontal and vertical directions.

### Given Information
1. **Tension in the cable holding the arm**: \( 185 \, \text{N} \).
2. **Angle of the support cable with the rod**: \( 50^\circ \).
3. **Point of attachment of the tension force**: At \( \frac{3}{5} \) of the rod’s length from the hinge.

### Problem Breakdown
Let's solve each part step-by-step.

#### Part (a): Determine the Tension \( T \) in the Horizontal Support Cable

1. **Choose a Pivot Point**: For calculating torques, we’ll use the hinge as the pivot point to eliminate the unknown reaction force in the hinge from the torque equation.
   
2. **Calculate the Torque due to the 185 N Tension Force**: Since the tension force is applied at \( \frac{3}{5} \) of the length \( L \) of the rod, the perpendicular distance to the hinge is:
   \[
   d = \frac{3}{5} L
   \]
   The torque caused by the 185 N tension force, \( F_T = 185 \, \text{N} \), is:
   \[
   \tau_T = F_T \cdot d \cdot \sin(50^\circ) = 185 \cdot \frac{3}{5} L \cdot \sin(50^\circ)
   \]

3. **Calculate the Torque due to the Horizontal Support Cable Tension \( T \)**: The tension \( T \) in the horizontal support cable exerts a torque in the opposite direction. Since the support cable is horizontal, the perpendicular distance to the hinge is \( L \). Thus:
   \[
   \tau_{support} = T \cdot L
   \]

4. **Set Up the Torque Equilibrium Equation**: Since the rod is in equilibrium, the sum of the torques around the hinge must be zero:
   \[
   T \cdot L = 185 \cdot \frac{3}{5} L \cdot \sin(50^\circ)
   \]

   Simplifying by dividing through by \( L \):
   \[
   T = 185 \cdot \frac{3}{5} \cdot \sin(50^\circ)
   \]

5. **Calculate \( T \)**:
   - Using \( \sin(50^\circ) \approx 0.7660 \):
   \[
   T = 185 \cdot \frac{3}{5} \cdot 0.7660
   \]
   \[
   T \approx 85 \, \text{N}
   \]

#### Part (b): Determine the Magnitude of the Reaction Force at the Hinge

1. **Force Equilibrium in the Horizontal Direction**:
   - The hinge reaction force \( F_H \) has a horizontal component \( F_{H_x} \) that counteracts the horizontal component of the 185 N tension.
   \[
   F_{H_x} = T = 85 \, \text{N}
   \]

2. **Force Equilibrium in the Vertical Direction**:
   - The vertical component of the reaction force \( F_H \) must balance the vertical component of the 185 N tension.
   \[
   F_{H_y} = 185 \cdot \sin(50^\circ)
   \]
   - Calculating \( F_{H_y} \):
   \[
   F_{H_y} = 185 \cdot 0.7660 \approx 142 \, \text{N}
   \]

3. **Magnitude of the Reaction Force**:
   - Using the Pythagorean theorem to find the resultant reaction force:
   \[
   F_H = \sqrt{F_{H_x}^2 + F_{H_y}^2} = \sqrt{85^2 + 142^2}
   \]
   \[
   F_H \approx 164 \, \text{N}
   \]

### Final Answers
- **(a)** The tension in the support cable \( T \) is approximately **85 N**.
- **(b)** The magnitude of the reaction force at the hinge \( F_H \) is approximately **164 N**.

Would you like further details on any part of this solution?

### Related Questions:
1. How would the solution change if the angle of the cable was different?
2. What would happen if the arm's weight was applied at a different location on the rod?
3. How does the negligible mass of the rod simplify our calculations?
4. What would be the impact if the rod had significant mass?
5. How does using a hinge as a pivot point help in calculating torques?

### Tip:
In equilibrium problems, always use the pivot point strategically to eliminate unknown forces from torque equations.

In the hospital, Shadbraw's arm is placed in traction. The weight of his arm causes a tension of 185 N in the cable hanging from the end of the traction rod. The traction rod is held by a horizontal support cable that attaches 3/5 of its length from a hinge. The hinge and the support cable are both attached to a wall. The rod has negligible mass.

a. What is the tension in the support cable?
b. What is the magnitude of the reaction force in the hinge?

Learn how to calculate the tension in a support cable and the reaction force in a hinge for a traction system in equilibrium. Given a 185 N force acting at an angle, and using torque and equilibrium principles, this problem demonstrates key concepts in statics. Suitable for high school physics students and above.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation