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Question: A monkey of mass 40kg climbs on a rope which can stand a maximum tension of 600N. In which of the fo...

A monkey of mass 40kg climbs on a rope which can stand a maximum tension of 600N. In which of the following cases will the rope break. When the monkey
A. climbs up with an acceleration of 6ms−26m{{s}^{-2}}
B. climbs down with an acceleration of 4ms−24m{{s}^{-2}}
C. climbs up with a uniform speed of 5ms−15m{{s}^{-1}}
D. falls down the rope nearly freely under gravity

Explanation

Solution

Hint: Newton’s second law of motion is used to write the equation of tension. All the options are substituted in the formula because only then it is possible to compare the tension and find the correct answer.

Formula used: F=T−mgF=T-mg

Complete step by step answer:
Newton’s second law states that force is equal to the change of momentum per unit time. It can also be defined as force is equal to the product of mass and acceleration. Which can be written as F=maF=ma. On substituting this in the formula of tension, we get F=T−mgF=T-mg

Mass of the monkey, m=40kgm=40kg
Acceleration due to gravity, g=10ms−1g=10m{{s}^{-1}}
Maximum tension of the rope, Tmax⁡=600N{{T}_{\max }}=600N

(A) Consider the first option, acceleration of the monkey, a=6ms−2a=6m{{s}^{-2}}
Substituting the value of ‘a’ in the formula,
F=T−mgF=T-mg
ma=T−mgma=T-mg(since F=maF=ma)
which becomes, T=ma+mgT=ma+mg
T=m(a+g)T=m\left( a+g \right)
T=40(6+10)T=40\left( 6+10 \right)
T=640NT=640N
This shows, T>Tmax⁡T > {{T}_{\max }}. That means, it is greater than the maximum tension and the rope will break.

(B) Consider the second option, acceleration of the monkey, a=4ms−2a=4m{{s}^{-2}}
Substituting the value of ‘a’ in the formula,
F=T−mgF=T-mg
T=ma+mgT=ma+mg
T=m(a+g)T=m\left( a+g \right)
T=40(10−4)T=40\left( 10-4 \right)
(negative because the acceleration is downwards)
T=240NT=240N
This shows, T<Tmax⁡T < {{T}_{\max }} . That means, it is less than the maximum tension and the rope will not break.

(C) Consider the third option, it is moving with uniform speed. Therefore, acceleration is zero.
Substituting the value of a = 0 in the formula,
F=T−mgF=T-mg
T=ma+mgT=ma+mg
T=m(a+g)T=m\left( a+g \right)
T=40(0+10)T=40\left( 0+10 \right)
T=400NT=400N
This shows, T<Tmax⁡T < {{T}_{\max }} . That means, it is less than the maximum tension and the rope will not break.

(D) Consider the fourth option. If the monkey falls freely under gravity, its acceleration will be equal to the acceleration due to gravity.
Substituting the value of a=ga=g
F=T+mgF=T+mg
mg=T+mgmg=T+mg
T=m(g−g)T=m\left( g-g \right)
T=0T=0
This shows, T<Tmax⁡T < {{T}_{\max }}. That means, it is less than the maximum tension and the rope will not break.
Therefore, the correct answer to this question is option (A).

Note: Tension is known as the pulling force transferred through a string or object. It is also known as the action-reaction forces acting on the object’s end. Tension is the opposite of compression force. The unit of tension force is Newton (N).