A 3.00-m-long, 140-N, uniform rod at the zoo is held in a horizontal position by

Question

A 3.00-m-long, 140-N, uniform rod at the zoo is held in a horizontal position by two ropes at its ends in (Figure 1). The left rope makes an angle of 150 degree with the rod and the right rope makes an angle with the horizontal. A 90-N howler monkey (Alouatta seniculus) hangs motionless 0.50 m from the right end of the rod as he carefully studies you. Calculate the tension in the right rope. Express your answer to two significant figures and include the appropriate units. Calculate the angle theta. Express your answer using two significant figures.

Explanation / Answer

Let

L = length of the rod = 3 m
W = weight of the rod = 140N
w = weight of the monkey = 90 N
d = distance of the monkey to the right end of the rod = 2.5 m
d = distance of the monkey to the right end of the rod = 0.5 m
= angle of the left rope = 150°

T = tension in the left rope
T = tension in the right rope
= angle of the right rope

The vertical component of T is
w×d + W×L/2 = Tv×L
(90 N)×(0.5 m) + (140 N)×(3 m)/2 = Tv×(3 m)
Tv = 85 N

The tension in the left rope is:
T×sin() = Tv
T×sin(150°) = 85 N
T = 170 N

The horizontal component of T is:
Th = T×cos()
Th = (170 N)×cos(150°)
Th = -147.22 N
(negative, because pointing to the left; so its magnitude is Th = 147.22N)

The horizontal component of T is:
Th + Th = 0
Th - (147.22 N) = 0
Th = 147.22 N

The vertical component of T is:
w×d + W×L/2 = Tv×L
(90 N)×(2.5 m) + (140 N)×(3 m)/2 = Tv×(3 m)
Tv = 145 N

The tension in the right rope is
T = ( Th² + Tv² )
T = ( (147.22 N)² + (145 N)² )
T = 206.64 N

The angle of the right rope is:
= arctan( Tv / Th )
= arctan( (145 N) / (147.22 N) )
= 44.56°

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