Question

A window washer of mass M is sitting on a platform suspended by system of cables and pulleys as shown(Intro1 figure). He is pulling on the cable with a force of magnitude F. The cables and pulleys are ideal (massless and frictionless), and the platform has negligible mass.


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Part A

Find the magnitude of the minimum force F that allows the window washer to move upward. Express your answer in terms of the mass M and the magnitude of the acceleration due to gravity g.

Answer 1

Concepts and reason

The required concept to solve the given question is Newton’s second law of motion.

First, consider the forces acting on the system. Find an expression for the net force acting on the system. Then consider the condition for minimum force to be act on the system, apply the condition in the expression of net force acting on the system. Rearrange and simplify the above expression to obtain the minimum force acting on the system.

Fundamentals

Force acting on a body may be defined as the sudden push or pull which is capable of changing the direction of motion of the object if it is unopposed.

Newton’s second law states that “the acceleration of an object as produced by a net force is directly proportional to the magnitude of the applied force and inversely proportional to the mass of the object”. It is in the same direction of the applied force.

The expression for force acting on an object is,

$F = ma$

Here, F is the force acting on the body, m is the mass of the object and a is its acceleration.

Weight of an object is the “force of gravity”. It is defined as the mass times the acceleration due to gravity.

Weight of the object is,

$W = mg$

Here, W is the weight of the object, m is the mass of the object and g is the acceleration due to gravity.

Normal force acting on the object is equal and opposite to the gravitational force applied on the object.

Net force acting on the system:

Here the person pulls the rope with a force F in the downward direction. The normal force N is acting along the upward direction. It is equal to the force F.

Tension acting on the rope that supports the pulley,

$T = 2F$

Weight of the window washer W is acting downwards.

Net force acting on the system,

$\begin{array}{c}\\{F_{net}} = F + 2F - W\\\\ = 3F - Mg\\\end{array}$

Minimum force acting on the system:

For the minimum force, the net force acting on the system is zero.

Minimum force acting on the system is,

$\begin{array}{c}\\{F_{net}} = 0\\\\0 = 3{F_{\min }} - Mg\\\\3{F_{\min }} = Mg\\\\{F_{\min }} = \frac{{Mg}}{3}\\\end{array}$

Ans:

Magnitude of minimum force that allows that allows the window washer to move upward is $\frac{{{\bf{Mg}}}}{{\bf{3}}}$.

Answer 2

This is a simple machine. Since the mass M is supported by not1, but 2 strings, then the worker essentially has to pull with halfthe force to move itup. Conversely, if it were supported by 1string, he would need to pull with force F.So we find that each string lessens the force he mustapply.F = .5mg [2 strings.]However, you should note that he would need to pull the cordtwice as far in order to move it the same distance as he would ifit were only supported by1 cord.So for 1 string, naturally pull:

F = mg2 strings:

F = .5mg3 strings:
F =1/3mg4 strings:

F = 1/4mg....etc.