Find Min Force for Window Washer: Ma = F + 2T_1 - Mg

In summary, the problem involves a window washer of mass M sitting on a platform suspended by cables and pulleys. The person is pulling on a cable with a force of magnitude F, and the goal is to find the minimum force needed for the washer to move upward. By applying Newton's second law and considering the tension in the cables, the minimum force is found to be 1/3 times the weight of the washer, or (1/3)Mg.
  • #1
clarineterr
14
0

Homework Statement


A window washer of mass M is sitting on a platform suspended by a system of cables and pulleys as shown . 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.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.
MFS_1l_18_001.jpg



Homework Equations



Newton's second Law

The Attempt at a Solution



For the person: Ma = F + T -Mg where T is the tension in the cable holding the platform and F is the force of the rope on the washer.

Then from the lower washer T = 2T[tex]_{1}[/tex], where T[tex]_{1}[/tex] is the tension in the rope

So Ma = F + 2T[tex]_{1}[/tex] - Mg

The block is about to move so Ma = 0 and I think T[tex]_{1}[/tex]=F so then

0 = 3F - Mg

I am not sure about this though
 
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  • #2
Looks good to me. :smile:
 
  • #3
.

I would first clarify the setup of the problem. It seems that the window washer is pulling on a rope to move the platform upward, and there is another rope attached to the platform that is also being pulled. The tension in this second rope is equal to twice the tension in the first rope. Additionally, the platform has negligible mass and the cables and pulleys are ideal.

Next, I would use Newton's second law to analyze the forces acting on the window washer. The only forces acting on the washer are the force of the rope he is pulling (F), the tension in the rope holding the platform (T), and the force of gravity (Mg). Therefore, the equation would be Ma = F + T - Mg, where M is the mass of the washer and a is the acceleration.

Since we are looking for the minimum force that allows the washer to move upward, we can assume that the washer is on the verge of moving, meaning that the acceleration is zero. Therefore, the equation simplifies to 0 = F + T - Mg.

We know that the tension in the second rope is equal to twice the tension in the first rope, so we can substitute 2T for T in the equation. This gives us 0 = F + 2T - Mg.

Since the platform has negligible mass, the tension in the first rope (T) can be assumed to be equal to the force of the rope on the washer (F). This means that T = F.

Substituting this into the equation, we get 0 = 3F - Mg. Solving for F, we get F = Mg/3.

Therefore, the minimum force that allows the window washer to move upward is Mg/3, where M is the mass of the washer and g is the magnitude of the acceleration due to gravity.
 

1. What does the equation "Ma = F + 2T_1 - Mg" represent?

The equation represents the forces acting on a window washer, where Ma is the mass of the window washer, F is the force applied by the washer, T_1 is the tension in the rope, and Mg is the force of gravity.

2. How do you find the minimum force required for a window washer?

To find the minimum force required for a window washer, you would set the equation "Ma = F + 2T_1 - Mg" equal to zero and solve for F. This would give you the minimum force needed to keep the window washer in equilibrium.

3. Why is the tension in the rope multiplied by 2 in the equation?

The tension in the rope is multiplied by 2 because there are two ropes supporting the window washer. This is known as the pulley system, where the tension is doubled when the rope is wrapped around a pulley.

4. How does the mass of the window washer affect the minimum force required?

The mass of the window washer, represented by Ma, directly affects the minimum force required. As the mass increases, the minimum force required also increases, as seen in the equation "Ma = F + 2T_1 - Mg". This is because a heavier window washer would require a greater force to keep it in equilibrium.

5. What other factors can affect the minimum force required for a window washer?

Other factors that can affect the minimum force required for a window washer include the angle of the rope, the coefficient of friction between the washer and the window, and external forces such as wind. These factors can alter the forces acting on the window washer and therefore change the minimum force required for equilibrium.

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