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 8.32x10^2 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.

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

where did the cos come from please

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

Torque = Force x Perpendicular Distance

In this case, the force is the tension in the rope, which is given as 8.32x10^2 N, and the perpendicular distance is the vertical distance from the base of the tree to the point where the rope is attached to the tree trunk, which is 3.92 m.

Therefore, substituting these values into the formula:

Torque = (8.32x10^2 N) x (3.92 m)

Torque = 3278.4 Nxm

Hence, the magnitude of the torque on the tree due to the tension in the rope is 3278.4 Nxm.

To compute the magnitude of the torque on the tree due to the tension in the rope, we need to multiply the tension force by the perpendicular distance from the point of rotation (base of the tree) to the line of action of the force.

In this case, the perpendicular distance can be determined using the Pythagorean theorem, as the truck is positioned 0.506 m above the ground and the rope is tied to the tree at a height of 3.92 m.

Step 1: Find the perpendicular distance
Using the Pythagorean theorem, we can find the perpendicular distance using the formula:

perpendicular distance = √(height^2 + distance^2)

distance = 0.506 m
height = 3.92 m

perpendicular distance = √(3.92^2 + 0.506^2)
perpendicular distance = √(15.3664 + 0.256036)
perpendicular distance = √(15.622436)
perpendicular distance ≈ 3.9531 m

Step 2: Calculate the torque
The torque can be determined by multiplying the tension force by the perpendicular distance. Given that the tension is 8.32x10^2 N, we have:

torque = tension force x perpendicular distance
torque = (8.32x10^2 N) x (3.9531 m)
torque = 3297.28 N·m

Therefore, the magnitude of the torque on the tree due to the tension in the rope is approximately 3297.28 N·m.