A man ties one end of a strong rope 7.85 m long to the bumper of his truck, 0.641 m from the ground, and the other end to a vertical tree trunk at a height of 3.13 m. He uses the truck to create a tension of 8.38  102 N in the rope. Compute the magnitude of the torque on the tree due to the tension in the rope, with the base of the tree acting as the reference point.
 N · m

To compute the torque on the tree due to the tension in the rope, we need to use the formula for torque:

\[
\tau = r \cdot F \cdot \sin(\theta)
\]

Where:
- \(\tau\) is the torque.
- \(r\) is the distance from the point where the force is applied (the tree) to the reference point (the base of the tree).
- \(F\) is the tension in the rope.
- \(\theta\) is the angle between the rope and the horizontal direction.

### Step 1: Find the components of the triangle formed by the tree, truck, and rope

The rope is tied to the tree at a height of 3.13 m, and the other end is tied to the truck at a height of 0.641 m. The horizontal distance between the tree and truck isn’t given directly, but the rope length is 7.85 m. 

We can use the Pythagorean theorem to calculate the horizontal distance \(x\) between the tree and truck:

\[
x = \sqrt{(7.85^2) - (3.13 - 0.641)^2}
\]

### Step 2: Find the angle \(\theta\)

Once we know the horizontal distance \(x\), we can calculate the angle \(\theta\) using the inverse tangent function:

\[
\theta = \arctan\left(\frac{3.13 - 0.641}{x}\right)
\]

### Step 3: Compute the torque

Finally, we can compute the torque using the formula:

\[
\tau = r \cdot F \cdot \sin(\theta)
\]

Where:
- \(r = 3.13\) m (since the force is applied at the height of the tree),
- \(F = 8.38 \times 10^2\) N (the tension in the rope),
- \(\theta\) is the angle we will calculate.

Let's now calculate the result step by step.The magnitude of the torque on the tree due to the tension in the rope is approximately \( 831.66 \, \text{N} \cdot \text{m} \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How does changing the length of the rope affect the torque?
2. What if the tension in the rope was increased to \( 10^3 \, \text{N} \)?
3. How would the torque change if the truck's attachment point was at a higher point?
4. What other forces might affect the tree in this scenario?
5. Can torque exist if the rope were horizontal?

**Tip:** Torque depends on the perpendicular distance from the pivot point; increasing the vertical distance increases torque even for the same force.

A man ties one end of a strong rope 7.85 m long to the bumper of his truck, 0.641 m from the ground, and the other end to a vertical tree trunk at a height of 3.13 m. He uses the truck to create a tension of 8.38 × 10^2 N in the rope. Compute the magnitude of the torque on the tree due to the tension in the rope, with the base of the tree acting as the reference point.

Learn how to compute the torque on a tree caused by a tensioned rope tied between a truck and the tree. This physics problem covers the use of trigonometry, the Pythagorean theorem, and torque calculation. Suitable for students in Grades 11-12 studying physics and advanced math.

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