Example Problem Involving Forces

[Bashyboy]

I am reading in my textbook an example problem. In this problem, there is a rope tied to a boulder on one end, and on the other it is tied to a car stuck in the mud. A person then applies a force at the midpoint of the rope. I was wondering if someone could explain to me why the "person pushing on the rope was able to magnify their effort almost six times using this technique"? I ask, because the author didn't do an actual good job at explaining the physics behind this technique, but rather employed themselves in explaining the mathematics--which is the part I don't really have any trouble with. I was just wondering if someone could explain this technique from a conceptual standpoint.

 

[tiny-tim]

Hi Bashyboy!

It's because what is important is not force, but work …

you're probably familiar with levers and pulleys, where you move your end, say six metres, but the other end moves only one metre …

look at the geometry here … when the angle is θ, you move the middle of the rope, and the end moves tanθ as far, exactly like a lever

 

[Bashyboy]

Actually, I don't really know how pulleys and levers work, and I have not learned about the concept of work yet. I uploaded a screen shot of the problem. Its from a chapter bout Dynamics. To me, the way they explain the example is almost entirely mathematical. I was just wondering if there was a conceptual way of explaining the problem, especially why the person in the problem was able to magnify their force almost by six times by using this technique.

 

[azizlwl]

You can replace(resolve) a force with equivalent sum of forces.
Here the force exerted by the girl can be replaced by 2 forces at an angle of (90-5)°.

If we add the 2 forces, the resultant should be the same as applied by the girl which 300N vertically.

Let the value=x
xCos(90-θ)+xCos(90-θ)=y
2Sinθ=y
2xSin5°=300N
x=1721N

Evaluation.
It depends on the angle. Greater the angle the more resultant force.

 

[Steely Dan]

Actually, I don't really know how pulleys and levers work, and I have not learned about the concept of work yet. I uploaded a screen shot of the problem. Its from a chapter bout Dynamics. To me, the way they explain the example is almost entirely mathematical. I was just wondering if there was a conceptual way of explaining the problem, especially why the person in the problem was able to magnify their force almost by six times by using this technique.

On a very qualitative level, what is pulling on the car is not the person, but the tension in the rope. Now, if the person just pulls horizontally with some force on the rope, that force is just transmitted to the car via the tension in the rope. The cleverness of tying the rope to the boulder is that the rock exerts a force on the rope, increasing its tension (whereas without the rock, only the person is contributing to any forces on the car).