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Chapter 3: Problem 383

A rope which can withstand a maximum tension of \(400 \mathrm{~N}\) hangs from a tree. If a monkey of mass \(30 \mathrm{~kg}\) climbs on the rope in which of the following cases-will the rope break? (take \(g=10 \mathrm{~ms}^{-}{ }^{2}\) and neglect the mass of rope \()\) (A) When the monkey climbs with constant speed of \(5 \mathrm{~ms}^{-1}\) (B) When the monkey climbs with constant acceleration of \(2 \mathrm{~ms}^{-2}\) (C) When the monkey climbs with constant acceleration of \(5 \mathrm{~ms}^{-2}\) (D) When the monkey climbs with the constant speed of \(12 \mathrm{~ms}^{-1}\)

Short Answer

Expert verified
The rope will only break in case (C), when the monkey climbs with a constant acceleration of \(5 \mathrm{~ms}^{-2} \). In cases (A), (B), and (D), the tension in the rope is below its maximum limit of 400 N.

Step by step solution

01

Calculate Force due to weight

Begin with calculating the force due to gravity acting on the monkey when it is just hanging from the rope or moving with a constant speed (acceleration is zero). This force can be calculated using the formula for the force of gravity, which states that force equals mass times gravity. Here, the weight of the monkey is given as 30 kg, and the acceleration due to gravity is given as 10 ms\(^{-2}\). So, \(Force_{weight} = mass \times gravity = 30 \, kg \times 10 \, ms^{-2} = 300 \, N\).
02

Compare with maximum tension

The rope can withstand a maximum tension of 400 N. Since the force due to gravity (300 N) is smaller than this value, the rope won't break if the monkey is just hanging still or moving at a constant speed, regardless of what that speed is. Therefore, options (A) and (D) can be considered safe for the monkey.
03

Calculate additional tension due to acceleration for option (B) and option (C)

The force due to the monkey's acceleration is calculated using the formula for force, i.e. force equals mass times acceleration. For option (B), with an acceleration of 2 ms\(^{-2}\), we have \(Force_{acceleration} = mass \times acceleration = 30 \, kg \times 2 \, ms^{-2} = 60 \, N\). For option (C) where acceleration is 5 ms\(^{-2}\), \(Force_{acceleration} = mass \times acceleration = 30 \, kg \times 5 \, ms^{-2} = 150 \, N\).
04

Compare total tension with maximum tension for both options

The total force on the rope in each case is the sum of the force due to the monkey's weight and the force due to its acceleration. For option (B), the total force on the rope is \(300 \, N + 60 \, N = 360 \, N\), which is less than the maximum tension of 400 N. Therefore, the rope will not break in this case. However, for option (C), the total force applied on the rope is \(300 \, N + 150 \, N = 450 \, N\), which exceeds the rope's maximum tension of 400 N. In this case, the rope will break. The correct answer is (C).

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