A 3.00-m-long, 230-N, uniform rod at the zoo is held in a horizontal position by

Question

A 3.00-m-long, 230-N, uniform rod at the zoo is held in a horizontal position by two ropes at its ends (the figure (Figure 1) ). The left rope makes an angle of 150 ? with the rod and the right rope makes an angle ? with the horizontal. A 97-N howler monkey (Alouatta seniculus) hangs motionless 0.50 m from the right end of the rod as he carefully studies you.

Part A

Calculate the tension in the left rope.

Express your answer using two significant figures.

Part B

Calculate the tension in the right rope.

Express your answer using two significant figures.

Part C

Calculate the angle ?.

Explanation / Answer

L = length of the rod = 3 m
W = weight of the rod = 230 N
w = weight of the monkey = 97 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
(97 N)×(0.5 m) + (230 N)×(3 m)/2 = Tv×(3 m)
Tv = 131 N

The tension in the left rope is:
T×sin() = Tv
T×sin(150°) = 131 N
T = 262 N

The horizontal component of T is:
Th = T×cos()
Th = (262 N)×cos(150°)
Th = -201 N
(negative, because pointing to the left; so its magnitude is Th = 201 N)

The horizontal component of T is:
Th + Th = 0
Th - (201 N) = 0
Th = 201 N

The vertical component of T is:
w×d + W×L/2 = Tv×L
(97 N)×(2.5 m) + (230 N)×(3 m)/2 = Tv×(3 m)
Tv = 196 N

The tension in the right rope is
T = ( Th² + Tv² )
T = ( (201 N)² + (196 N)² )
T = 281 N

The angle of the right rope is:
= arctan( Tv / Th )
= arctan( (196 N) / (201 N) )
= 44.27 °

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