'Force on pulley by a string' conception

[Oklid]

Homework Statement

In the figure below is shown the system below are shown two blocks linked by a string through a pulley, where the block of mass m1 slides on the frictionless table. We assume that the string is massless and the pulley is massless and frictionless.

Show the forces ( as vectors ) on pulley by the string.

Homework Equations

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The Attempt at a Solution

[/B]
First things first, I know that is really easy problem to solve but when I though about that, I couldn't handle it by Newton's third law or whatever. Yes, I've solved it. But in my head, still there is a shortage.

There is a tension on the string. Yes, we know that.

But these forces affect pulley but which direction?

 

[kuruman]

There are more forces acting on the pulley. If what you show were the only ones, the pulley would accelerate, no?

 

[Oklid]

There are more forces acting on the pulley. If what you show were the only ones, the pulley would accelerate, no?

I didn't get quitely. Why would the pulley accelerate? It is fixed to the floor.

And which forces are they that acting on the pulley, I'd like to know. Second things, thanks for answer.

 

[kuruman]

I didn't get quitely. Why would the pulley accelerate? It is fixed to the floor.

And which forces are they that acting on the pulley, I'd like to know. Second things, thanks for answer.

You show two forces acting on the pulley. They have equal magnitudes and one is horizontal while the other is vertical. They are correctly drawn and their resultant is at 45^o down and to the left. If that's all there is, the net force on the pulley is down and to the left and since $\vec F_{net}=m\vec a,$ the pulley will have a non-zero acceleration directed down and to the left. If you know that the pulley does not accelerate, then you need at least one additional force to cancel the tensions. How big should that force be, what is its direction and where does it come from?

 

[Oklid]

You show two forces acting on the pulley. They have equal magnitudes and one is horizontal while the other is vertical. They are correctly drawn and their resultant is at 45^o down and to the left. If that's all there is, the net force on the pulley is down and to the left and since $\vec F_{net}=m\vec a,$ the pulley will have a non-zero acceleration directed down and to the left. If you know that the pulley does not accelerate, then you need at least one additional force to cancel the tensions. How big should that force be, what is its direction and where does it come from?

There is a misunderstanding.

I think, you're talking about the second image that I upload. It is not the question, the first one is.

The second image that I upload to show the correct answer behalf of you, if you have written the answer directly.

I've edited it.

 

[kuruman]

There is a misunderstanding.

I think, you're talking about the second image that I upload. It is not the question, the first one is.

The second image that I upload to show the correct answer behalf of you, if you have written the answer directly.

I've edited it.

I do not see an image that shows correctly the forces acting on the pulley. Please draw and post a separate free body diagram of just the pulley drawn as a circle and include all the forces that you think are acting on it.

 

[haruspex]

There are more forces acting on the pulley. If what you show were the only ones, the pulley would accelerate, no?

The question specifies only the forces the string exerts on the pulley.

 

[haruspex]

these forces affect pulley but which direction?

Tension (likewise compression) is not so much a force as a pair of equal and opposite forces.

In string-over-pulley questions, it often helps to consider the portion of string in contact with the pulley as being part of the pulley.
With that in mind, consider the vertical segment of string. The suspended weight exerts a force T on that. If we take the string to be massless, what force must the pulley exert on it?

 

[kuruman]

The question specifies only the forces the string exerts on the pulley.

Yes, I know. OP had correctly posted an initial drawing in which the string forces on the pulley were correctly drawn and I indicated as much in post #4. In that post I also tried to address OP's statement

But in my head, still there is a shortage.

which I assumed to mean that there ought to be more forces; maybe I was wrong. OP responded with #5 and replaced the original drawing in #1 with the current one. I don't know where this thread is headed any more.

 

[haruspex]

Yes, I know. OP had correctly posted an initial drawing in which the string forces on the pulley were correctly drawn and I indicated as much in post #4. In that post I also tried to address OP's statement which I assumed to mean that there ought to be more forces; maybe I was wrong. OP responded with #5 and replaced the original drawing in #1 with the current one. I don't know where this thread is headed any more.

I agree it is not clear what the OP is struggling with.
The question asks for the forces the string exerts on the pulley, but the diagram the OP has drawn shows the forces the string exerts on the masses, so I interpreted the question as being how to derive the one from the other.

 

[Oklid]

The question specifies only the forces the string exerts on the pulley.

Yes, that's right.

Tension (likewise compression) is not so much a force as a pair of equal and opposite forces.

In string-over-pulley questions, it often helps to consider the portion of string in contact with the pulley as being part of the pulley.
With that in mind, consider the vertical segment of string. The suspended weight exerts a force T on that. If we take the string to be massless, what force must the pulley exert on it?

The suspended weight doesn't exert a force T by the way. I had drawn this to show tension with initial (T).

Yes, I know. OP had correctly posted an initial drawing in which the string forces on the pulley were correctly drawn and I indicated as much in post #4. In that post I also tried to address OP's statement which I assumed to mean that there ought to be more forces; maybe I was wrong. OP responded with #5 and replaced the original drawing in #1 with the current one. I don't know where this thread is headed any more.

The pulley were correctly drawn but the answer I drew it in case anyone draws a something like that directly without explanation.

Yes that was true. But I can't get it.

I agree it is not clear what the OP is struggling with.
The question asks for the forces the string exerts on the pulley, but the diagram the OP has drawn shows the forces the string exerts on the masses, so I interpreted the question as being how to derive the one from the other.

I agree too, it is not clear what my struggling is.

 

[Oklid]

Maybe, it could be more clear.

The string has a tension, right?
So they say, the string exerts force on the pulley. But in which direction?

Tension (likewise compression) is not so much a force as a pair of equal and opposite forces.

In string-over-pulley questions, it often helps to consider the portion of string in contact with the pulley as being part of the pulley.
With that in mind, consider the vertical segment of string. The suspended weight exerts a force T on that. If we take the string to be massless, what force must the pulley exert on it?

Still don't get it.

The reason that I want to know this, or figure it out, I have another question.

Maybe it is better to open a new thread, But to solve my problem about string and pulley thing could be helpful.

This question is the main reason to open this thread.

The mass of the weight hanging off the side is greater than that resting on top of the cart. We have a pulley is set up on a cart, with a massless pulley and string. The mass hanging off the side is attached via a rail, and all surfaces & pulley are frictionless except between the tires and the ground (rolling).

When the system is released from rest, What about car's motion?

The only thing that I know is the string has tension and that affects the pulley or exerts force on the pulley. And the pulley is set up on a cart which is they are integrated.

If I find out the exerted force with vectors on the pulley by the string, I think I will be able to find car's motion direction.

I wanted to start with a basic problem to understand exerted force by a string on the pulley. But I am misunderstood.

This is the real reason.

 

[haruspex]

The suspended weight doesn't exert a force T by the way. I had drawn this to show tension with initial (T).

If the tension is T then the suspended mass does indeed exert a force T on the string. Action and reaction are equal and opposite.

the string exerts force on the pulley. But in which direction?

As I posted, it helps to consider that portion of the string in direct contact with the pulley as being part of the pulley. So you can ask yourself what direction is the force which the vertical part of the string exerts on the piece of string in contact with the pulley. Similarly for the horizontal segment of string.

 

[Oklid]

As I posted, it helps to consider that portion of the string in direct contact with the pulley as being part of the pulley. So you can ask yourself what direction is the force which the vertical part of the string exerts on the piece of string in contact with the pulley. Similarly for the horizontal segment of string.

Still can't imagine in my head, I am sorry for that.

And have you seen the another question (a cart and the pulley system) in #12? It explains my struggle.

 

[haruspex]

Still can't imagine in my head, I am sorry for that.

What if, instead of a pulley, it were just a horizontal rod with the vertical string tied to it? If the tension in the string is T, what is the force on the rod?

 

[Oklid]

What if, instead of a pulley, it were just a horizontal rod with the vertical string tied to it? If the tension in the string is T, what is the force on the rod?

The force magnitude on the rod must be T and the force vectors on the rod must be upwards.

 

[haruspex]

The force magnitude on the rod must be T and the force vectors on the rod must be upwards.

The string itself is in equilibrium. The weight pulls down on it with force T, so in which direction is the force the rod exerts on the string?

 

[Oklid]

The string itself is in equilibrium. The weight pulls down on it with force T, so in which direction is the force the rod exerts on the string?

Upwards.

 

[haruspex]

Upwards.

Right, so which way is the force the string exerts on the rod?

 

[Oklid]

Right, so which way is the force the string exerts on the rod?

Downward.

Because of 3rd law of Newton.

 

[haruspex]

Downward.

Because of 3rd law of Newton.

Right.
So going back to the pulley, what force does the vertical section of string exert on it?
What about the horizontal section?

 

[Oklid]

Right.
So going back to the pulley, what force does the vertical section of string exert on it?
What about the horizontal section?

Wait a minute. The string was tied to the rod. But there is no connection between string and the pulley? Am I wrong?

 

[Oklid]

And there is no friction.

 

[Chestermiller]

Throughout all of this discussion, I still don't see a free body diagram of just the pulley. Do you not believe in using free body diagrams?

 

[Oklid]

Throughout all of this discussion, I still don't see a free body diagram of just the pulley. Do you not believe in using free body diagrams?

I believe, of course. But here I have no problem with the free body diagram. The masses are okay but I can't show the pulley as using a free body diagram. As I said I have a confusion.

 

[Chestermiller]

I believe, of course. But here I have no problem with the free body diagram. The masses are okay but I can't show the pulley as using a free body diagram. As I said I have a confusion.

Well then, to help us understand your difficulty, please provide at least your best estimate of a free body diagram for the pulley alone.

 

[kuruman]

I wanted to start with a basic problem to understand exerted force by a string on the pulley. But I am misunderstood.

Maybe this will help you visualize the forces. A string force, also known as tension, is directed always away from the mass (system) it is acting on. That's because you can only pull with a string. Try pushing with a string and see what happens.

 

[Oklid]

Well then, to help us understand your difficulty, please provide at least your best estimate of a free body diagram for the pulley alone.

I will draw it in a few hours.

Maybe this will help you visualize the forces. A string force, also known as tension, is directed always away from the mass (system) it is acting on. That's because you can only pull with a string. Try pushing with a string and see what happens.

I get that. But If you have read the conversation between me and haruspex, can you explain the below question?

Wait a minute. The string was tied to the rod. But there is no connection between string and the pulley? Am I wrong?

Thank you all by the way.

 

[kuruman]

If the string is touching the pulley, the string exerts a force on the pulley. I finally start to see now what your difficulty is. For simplicity, let's assume that the contact between the pulley and the string is frictionless. Then the force exerted by an element of length $ds$ of the string on the pulley can only be in the radial direction and point towards the center of the pulley. The net force exerted by the string on the pulley is distributed radially along the length of the arc of the string that is in contact with the pulley.

What if there is friction? Usually there is, but the bearing of the pulley is assumed frictionless (the pulley is free to rotate) in which case you only have to consider static friction (the string does not slip on the pulley). In that case there is also a tangential force distributed along the length of the arc in the direction of the acceleration of the string; if the string does not accelerate, this force is zero. If the bearing is not frictionless and the string slips on the pulley, the situation becomes complicated and you will have a combination of radial and tangential forces acting on the pulley. In the special case when the pulley is fixed and not free to rotate while the rope slides around it, you have essentially a capstan (see link below).

https://en.wikipedia.org/wiki/Capstan_equation

 

[haruspex]

Wait a minute. The string was tied to the rod. But there is no connection between string and the pulley? Am I wrong?

That is true, but you can choose to define a subsystem however you like. As I wrote, it is simplest if you define the pullley together with that part of the string currently in contact with it as a subsystem. They will move as a rigid body (see below). The forces on this from the rest of the string are the horizontal and vertical tensions.

The alternative approach is to consider all the little segments of string length ds in contact with the pulley and integrate. I see @kuruman is taking you that route, and maybe that is what you need, but it is more complicated,

And there is no friction.

When a problem describes a pulley as frictionless it means there is no friction at the pulley's axle. You should generally assume no slipping between the pulley and the string.
Where the pulley is massless, as here, it makes no difference. It might as well be a string sliding over a smooth fixed curve.