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

Question

Thanks!

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 T with respect to the base P of the tree. The distances in the figure are: d = 14.7 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 + (4.00 m)² + (-14.7 + 1.45 m)²

L = 17.457 m

Now, consider torques and write force equations

F/L = (Fx)/r = (Fy)/s = (Fz)/(-d + h)

Fx = (10.64 m) (30.0 N)/(17.457 m) = 18.28 N

Fy = (4.00 m)(30.0 N)/(17.457 m) = 6.87 N

Fz = (-14.7 + 1.45 m)(30.0 N)/(17.457 m) = -22.76 N

Now, find the torque about point P:

= d x F

= <0, 0, 14.7 m> x <18.28 N, 6.87 N, -22.76 N >

= < -(14.7 m)(6.87 N), (14.7 m)(18.28 N), 0 >

= <-101 N-m, 269 N-m, 0 >

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