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Chapter 5: Problem 35

A 15 -kg monkey hangs from the middle of a massless rope, each half of which makes an \(8^{\circ}\) angle with the horizontal. What's the rope tension? Compare with the monkey's weight.

Short Answer

Expert verified
The tension in the rope is greater than the monkey's weight.

Step by step solution

01

Calculate the Weight of the Monkey

Find the weight of the monkey by multiplying the mass of the monkey \( m = 15 \ kg \) by the acceleration due to gravity \( g = 9.8 \ m/s^{2} \). Thus, the weight \( W_{m} = m.g = 15 * 9.8 = 147 \ N \)
02

Resolve the Weight in the Vertical and Horizontal Directions

The weight of the monkey acts vertically downward. We can resolve this weight into two components: one acting along the direction of the rope (parallel to it) and one acting perpendicular to it. Considering the rope is making an angle \( 8^{\circ} \) with the horizontal, and using trigonometric functions, we calculate the weight's component parallel to the rope, \( W_{p} = W_{m} * cos(8^{\circ}) \) and its component perpendicular to the rope, \( W_{t} = W_{m} * sin(8^{\circ}) \)
03

Calculate the Tension in the Rope

The rope is in equilibrium, so the sum of the forces acting along it and perpendicular to it must be zero. In the perpendicular direction (horizontal), there were no forces at the beginning. Therefore, the sum of the forces in the direction of the rope (vertical) must also be zero. We have two forces acting downwards (the weight and the component of the weight along the rope), so the tension \( T \) in the rope should balance them, which equals the sum of the weight of the monkey and its vertical component: \( T=W_{m}+W_{p} \)
04

Compare the Tension with the Monkey's Weight

Divide the calculated tension \( T \) by the weight of the monkey \( W_{m} \). If \( T/W_{m} > 1 \), the tension is greater than the weight. If \( T/W_{m} = 1 \), the tension equals the weight. If \( T/W_{m} < 1 \), the tension is less than the weight.

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