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 = 31.7 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 = 6.49 m and r = 10.64 m.

Explanation / Answer

force on the rope, F=31.7 N

and

d=14.7 m,

h=1.45 m,

s=6.49 m,

r=10.64 m

here,

L^2=r^2+s^2+(h-d)^2

L^2=10.64^2+6.49^2+(1.45-14.7)^2

===> L=18.19 m

now,

Fx=(F/L)*r

Fx=(31.7/18.19)*10.64 = 18.54 N

and

Fy=(F/L)*s

Fy=(31.7/18.19)*6.49 = 11.31 N

and

Fz=(F/L)*(h-d)

=(31.7/18.190)*(1.45-14.7) = -23.09 N

now,

torque, T=dXF

=(0i+oj+12.9k)x(Fxi+Fyj+Fzk)

=(0i+oj+12.9k)x(18.54i+11.31j-23.09k)

=(12.9*18.54)j-12.9*11.31i+ok

=(-145.9)i+(239.17)j+(0)k ---> in N/m ---------->

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