A man ties one end of a strong rope 8.35 m long to the bumper of his truck, 0.506 m from the ground, and the other end to a vertical tree trunk at a height of 3.92 m. He uses the truck to create a tension of 8.32 102 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.

He uses the truck to create a tension of 8.32x10^2 N in the rope.

sinα=(3.92-0.506)/8.35 =0.409

α =sin⁻¹0.409 = 24.13º
F=832•cosα=832•0.913 =759.6 N
τ=759.6•3.92=2977.6 N•m

To compute the magnitude of the torque on the tree due to the tension in the rope, you need to know the angle between the rope and the lever arm (the perpendicular distance from the pivot point to the line of action of the force).

In this case, the lever arm is the horizontal distance from the base of the tree to the point where the rope is attached to the tree. Let's call this distance x.

To find x, we can use the Pythagorean theorem:

x^2 = (8.35 m)^2 - (0.506 m)^2

x^2 = 69.8224 m^2

x ≈ 8.35 m

Now that we know the lever arm, we can calculate the torque by multiplying the tension in the rope by the lever arm:

Torque = Tension * Lever Arm

Torque = (8.32 x 10^2 N) * (8.35 m)

Torque ≈ 6943.2 N·m

So, the magnitude of the torque on the tree due to the tension in the rope is approximately 6943.2 N·m.