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 = 30.0 N calculate the components of the torque with respect to the base P of the tree. The distances in the figure are: d = 12.0 m, h = 1.45 m, s = 5.66 m and r = 11.96 m.

Calculate torque in the x, y, and z direction

Explanation / Answer

Calculate torque in the x, y,
and z direction
Find the vector components of the force.
Let L be the length of rope.
L² = r² + s² + (-d + h)²
L² = (11.96 m)² + (5.66 m)² + (-12.0 + 1.45 m)²
L = 16.92m
F/L = (Fx)/r = (Fy)/s = (Fz)/(-d + h)
Fx = (30.0 N)(11.96 m)/(16.92 m) = 21.20 N
Fy = (30.0 N)(5.66 m)/(16.92 m) = 10.035 N
Fz = (30.0 N)(-12.0 + 1.45 m)/(16.92 m) = -18.70 N

The torque about point P would be
= d x F
= <0, 0, 12.0 m> x <21.20 N, 10.035 N, -18.70 N
= <-(12.0 m)(10.035 N), (12.0 m)(21.20 N),(12.0 m)(-18.70) 0>
= <-120.42 N-m, 254.40 N-m,224.4 N-m 0>

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