/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 65 A block \(20 \%\) more massive t... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 65

A block \(20 \%\) more massive than you hangs from a rope that goes over a frictionless, massless pulley. With what acceleration must you climb the other end of the rope to keep the block from falling?

Short Answer

Expert verified
To prevent the block from falling, you need to climb the rope with an acceleration of \(0.2g\) upwards.

Step by step solution

01

Identifying Forces

Remember that when you are hanging on to the rope, there are two forces acting on you: the force of gravity pulling you downwards and the tension force from the rope pulling you upwards. The force of gravity acting on you is your weight which is your mass \(m_{1}\) times the acceleration due to gravity \(g\). The force of gravity acting on the block is its weight which is its mass \(m_{2}\) times \(g\). The given information tells us that \(m_{2} = 1.2m_{1}\). The tension in the rope is the same throughout, so the tension on your side is equal to the tension on the block's side.
02

Applying Newton鈥檚 second Law

When climbing up the rope, you both - you and the block - should be stationary relative to each other. So, net force acting on each must be zero. So, for you, net force can be represented by the equation \(m_{1}g - T = m_{1}a_{1}\) where \(T\) is the tension in the rope and \(a_{1}\) is your acceleration. For the block, net force can be represented by the equation \(T - m_{2}g = m_{2}a_{2}\) where \(a_{2}\) is the block's acceleration. Since \(a_{1} = a_{2}\) (both of you must have the same acceleration), the two equations can be combined into one equation.
03

Solve for Acceleration

Combining the two equations we get: \(m_{1}g - m_{2}g = m_{1}a\). Solving this for \(a\) gives us \(a = \frac{m_{1}g - m_{2}g}{m_{1}}\). Substituting \(m_{2} = 1.2m_{1}\) into this equation and solving, we find that \(a = -0.2g\). The negative sign means the acceleration is directed downwards. But you must climb the rope with this acceleration upwards to match the acceleration of the block and prevent it from falling.

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