A tree contractor has been hired by a homeowner to remove a damaged tree from a

Question

A tree contractor has been hired by a homeowner to remove a damaged tree from a previous storm. The top of the tree broke off leaving the trunk left in the ground. The two arborists will remove the tree by attaching a rope near the top and cutting the base close to the ground. The scenario is shown in the figure below. If the arborist pulls on the rope with a force of magnitude F = 38.5 N calculate the components of the torque with respect to the base P of the tree. The distances in the figure are: d = 12.9 m, h = 1.45 m, s = 4.00 m and r = 10.64 m.

Explanation / Answer

Find the vector components of the force.
Let L be the length of rope.
L² = r² + s² + (-d + h)²
L² = (10.64 m)² + (4 m)² + (-12.9 + 1.45 m)²
L = 16.13 m

F/L = (Fx)/r = (Fy)/s = (Fz)/(-d + h)
Fx = (10.64 m)(38.5 N)/(16.13 m) = 25.39 N
Fy = (4 m)(38.5 N)/(16.13 m) = 9.54 N
Fz = (-12.9 + 1.45 m)(38.5 N)/(16.13 m) = -27.32 N

The torque about point P would be
= d x F
= <0, 0, 12.9 m> x <25.39 N, 9.54 N, -27.32 N>
= <-(12.9 m)(9.54 N), (12.9 m)(25.39 N), 0>
= <-123 N-m i, 327.5 N-mm j, 0>

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