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To solve this problem, we need to determine the tension \( T \) in the rope that supports the horizontal rod. Here’s a step-by-step approach:

### Given Data:
1. Length of the rod, \( L = 3.85 \, \text{m} \).
2. Torque exerted by the rope, \( \tau = 1470 \, \text{N} \cdot \text{m} \).
3. Angle between the rope and the rod, \( \theta = 28.4^\circ \).

### Relevant Equations:
The torque (\( \tau \)) generated by the tension in the rope is given by:
\[
\tau = T \cdot L \cdot \sin(\theta)
\]

Where:
- \( T \) is the tension in the rope.
- \( L \) is the distance from the axis to the point where the rope is attached.
- \( \theta \) is the angle between the rod and the rope.

We can rearrange this formula to solve for \( T \):
\[
T = \frac{\tau}{L \cdot \sin(\theta)}
\]

### Calculations:
1. Substitute the values into the equation.
   \[
   T = \frac{1470}{3.85 \cdot \sin(28.4^\circ)}
   \]

2. Calculate \( \sin(28.4^\circ) \).

3. Plug in all values to get \( T \).

Let’s calculate:

1. \( \sin(28.4^\circ) \approx 0.4764 \).
   
2. Now, substitute these values:
   \[
   T = \frac{1470}{3.85 \times 0.4764}
   \]

3. Solving this gives:
   \[
   T \approx 795.7 \, \text{N}
   \]

### Answer:
The tension in the rope is approximately:
\[
T = 795.7 \, \text{N}
\]

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

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Here are some related questions for further exploration:

1. How would the tension change if the angle of the rope was increased?
2. What would be the impact on the torque if the length of the rod was doubled?
3. How does changing the angle affect the amount of force needed to hold the rod steady?
4. Can this setup be analyzed using vector components of force?
5. How would you calculate the force components along the rod due to tension?

**Tip:** When dealing with torque problems, always check the perpendicular distance (lever arm) between the force and the pivot point for accurate calculations.

A horizontal rod of length 3.85 m is free to rotate about a hinge at one end and is supported by a rope at the other. The torque exerted by the rope about the hinge is 1470 N·m. The rope makes an angle of 28.4° with the rod. What is the tension in the rope in units of newtons?

Learn how to calculate the tension in a rope supporting a rotating horizontal rod with torque. Given a rod length of 3.85 m, a torque of 1470 N·m, and an angle of 28.4° between the rope and rod, find the tension using statics and trigonometric principles.

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