A pulley with a mass on a string

[Biker]

Homework Statement

Problem: ABC is a cable with length 5 m attached to both a and c. By neglecting the dimensions of the pulley, and if the pulley is stationary, Calculate:
1) x
2) Tension[/B]

Homework Equations

F net = 0[/B]

The Attempt at a Solution

I have two problems with this question.
1) I fail to see why should we treat the pulley as a point and draw the free body diagram around it. I understand that the string exerts a force on the pulley as it goes around it because of it is shape but does treating it as a point apply to every situation no matter what the angle is of the rope is?2) as I treated it as a point, I defined y as the vertical distance between a and b... With that being defined I can make a system of 2 equation and 2 variables as following:

$(\sqrt{(y^2+x^2)} + \sqrt{((3.5-x)^2 +(y+0.75)^2)} = 5$
$\frac{x}{\sqrt{(y^2+x^2)}} = \frac{3.5-x}{\sqrt{((3.5-x)^2 +(y+0.75)^2)}}$

I solved for x and y and got x to be 1.383 but that needed a lot of steps. Is there is any shorter answer for this?

 

[kuruman]

... but does treating it as a point apply to every situation no matter what the angle is of the rope is?

It applies to situations in which the pulley is in static equilibrium as in this case.

Is there is any shorter answer for this?

Your solution is pretty short, only two lines, how much shorter do you want it to be? Seriously, if you ask this kind of question, then you need to show your solution in detail.

 

[Biker]

It applies to situations in which the pulley is in static equilibrium as in this case.

Is there is some kind of a proof for that?

It applies to situations in which the pulley is in static equilibrium as in this case.Your solution is pretty short, only two lines, how much shorter do you want it to be? Seriously, if you ask this kind of question, then you need to show your solution in detail.

Solving for x and y took me a page. I was just asking if there was another approach. Sorry...

 

[kuruman]

s there is some kind of a proof for that?

Just reason it out. Paint two dots on the pulley. They are at rest with respect to each other. The sum of all the forces on one dot is the same as on the other dot, zero. In fact all the points making up the pulley have zero net force acting on them. Now suppose you shrink the pulley to one dot. Does that change anything? If that still bothers you, draw a free body diagram of the knot that ties the hanging mass to the rope and proceed that way.

Solving for x and y took me a page. I was just asking if there was another approach.

I still need to see your approach before I can say that there is a shorter one. "One page" is not enough information. Is your handwriting small or large?

 

[Biker]

Just reason it out. Paint two dots on the pulley. They are at rest with respect to each other. The sum of all the forces on one dot is the same as on the other dot, zero. In fact all the points making up the pulley have zero net force acting on them. Now suppose you shrink the pulley to one dot. Does that change anything? If that still bothers you, draw a free body diagram of the knot that ties the hanging mass to the rope and proceed that way.

I was thinking of something like this https://physics.stackexchange.com/q...etween-an-ideal-pulley-and-an-ideal-rope?rq=1
where I know that the net force the quantity and the direction of it.

Never mind about the answer, Just arithmetic.
But I will try to make sense of it. Thanks for the help!

 

[kuruman]

But I will try to make sense of it.

Would it help if you modeled the problem as the 100 kg mass being attached to the cable with a frictionless knot instead of a pulley? It's the same thing. The knot will slide along the cable until equilibrium is reached.