# Describe the position of a pulley attached to a sling

#### Jeroen Staps

Moved from a technical forum, so homework template missing
Hi,

I am studying the behaviour of a pulley that is attached to a sling. The situation looks like this:

There is a drum that can give or take cable. Then there is a floating pulley with a cable through it. The pulley can move on the circle with a radius the same as the length of the sling. At the end of the cable a load is attached.

The following parameters are known:
- location of the drum
- length of the sling
- weight of the load (so you also know the gravity working on the system)
- location of the fixed point at the sling
- % friction in the pulley

The question is how the position of the floating pulley changes when a certain force is acting on the cable at the drum.

Anyone an idea on how to solve this problem?

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#### BvU

Homework Helper
Keep in mind that the force a cable exercises is tension and thereby in a direction along the cable.
how to solve
Make a sketch of the forces acting (free body diagram) for the relevant parts of the setup.

#### Jeroen Staps

Keep in mind that the force a cable exercises is tension and thereby in a direction along the cable.

Make a sketch of the forces acting (free body diagram) for the relevant parts of the setup.
There are three tensions in this systems each working on a part of a cable attached to the pulley

#### BvU

Homework Helper
Hallo Jeroen, $\qquad$ $\qquad$ !

Please read the guidelines . This should be in a homework forum and have the level tag B, not A.

#### Jeroen Staps

Hallo Jeroen, $\qquad$ $\qquad$ !

Please read the guidelines . This should be in a homework forum and have the level tag B, not A.
Alright I will post it there. But do you have an idea on how to describe the location of the pulley when there is a certain force pulling at the cable at the drum?

#### BvU

Homework Helper
But do you have an idea
Yes: make the sketches I mentioned

#### Jeroen Staps

Yes: make the sketches I mentioned

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#### .Scott

Homework Helper
If we ignore the "percent friction" part, then $F_{pull}$ will be the same force that is applied to the load. So the load would either accelerate upwards or downwards depending on whether $F_{pull}$ was sufficient to hold it up.

However, we have this "% friction" value that I will call $f$ - which apparently allows $F_{pull}$ and the force applied to the load ($F_{load}$) to differ. For a pulley, percent friction is the ratio in tension between the pulled end over the pulling end. So if the load is rising, we would have $F_{load} = f F_{pull}$ and if the load was dropping we would have $F_{pull} = f F_{load}$.

I would attack the problem by solving for the dropping case first. And I would assume that the position were such that the sling was at a steady angle with the load dropping exactly vertically.

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#### Jeroen Staps

If we ignore the "percent friction" part, then $F_{pull}$ will be the same force that is applied to the load. So the load would either accelerate upwards or downwards depending on whether $F_{pull}$ was sufficient to hold it up.
However, we have this "% friction" value - which apparently allows $F_{pull}$ and the force applied to the load to differ.
The problem is lacking in that there is no explanation of how this "percent friction" should be applied.

Normally kinetic friction would cause a pulley to exert some force on the cable proportional to the speed that the pulley was turning - or perhaps as a function of the stress on the pulley itself (a function of the tensions on the three lines attached to the pulley and their relative angles). But that doesn't sound like "percent friction" to me.

My guess is that the force applied to the load is only a certain percentage more or less than the $F_{pull}$ - acting to slow the pulley turning.
The percent friction means that when F_pull is +100N meaning the drum is rolling up the cable then the force in the part of the cable between the pulley and the load is F_pull * (100-percent friction)/100

#### BvU

Homework Helper
So this is for the pulley -- the most important one. I can see that there is no equilibrium component-wise, nor magnitude-wise.

The drum itself is outside the scope of your problem:
a certain force is acting on the cable at the drum
So you want to study the effect of a change in $|T_1|$.

And there is a complication:
- % friction in the pulley
Not sure how to interpret that one, but it should cause $|\vec T_1| \ne |\vec T_2|$.

As you know, $m\vec a = \sum \vec F_i$, also for the pulley. Since it can only move over the dashed circle, the resultant $\vec a$ can only be tangent to that circle.
do you have an idea on how to describe the location of the pulley
Yes. You do too:
One extreme case is load mass $m=0$ -- then the pulley hangs ...

An equilibrium case (with $m \ne 0$) means ...
If then $|T_1|$ is increased, ....
But if $|T_1|$ is decreased from equilibrium, ....

Basically you will now write down the equation of motion, in terms of the known symbols, among which $|T_1|$

By the way,
- length of the sling
That this is a known seems a mistake to me. It is a consequence of the motion, not an input. It looks to me like an initial condition only.

#### .Scott

Homework Helper
The percent friction means that when F_pull is +100N meaning the drum is rolling up the cable then the force in the part of the cable between the pulley and the load is F_pull * (100-percent friction)/100
Yes. I looked that up and updated my post while you were posting.

#### Jeroen Staps

So this is for the pulley -- the most important one. I can see that there is no equilibrium component-wise, nor magnitude-wise.

The drum itself is outside the scope of your problem:
So you want to study the effect of a change in $|T_1|$.

And there is a complication:
Not sure how to interpret that one, but it should cause $|\vec T_1| \ne |\vec T_2|$.

As you know, $m\vec a = \sum \vec F_i$, also for the pulley. Since it can only move over the dashed circle, the resultant $\vec a$ can only be tangent to that circle.
Yes. You do too:
One extreme case is load mass $m=0$ -- then the pulley hangs ...

An equilibrium case means ...
If then $|T_1|$ is increased, ....
But if $|T_1|$ is decreased from equilibrium, ....

Basically you will now write down the equation of motion, in terms of the known symbols, among which $|T_1|$

By the way,
That this is a known seems a mistake to me. It is a consequence of the motion, not an input. It looks to me like an initial condition only.
In practice the pulley never hangs vertically because a cable runs through the pulley causing a certain angle.

The length of the sling is a known in fact. It is used to attach a floating pulley to a platform.

#### BvU

Homework Helper
In practice the pulley never hangs vertically because a cable runs through the pulley causing a certain angle
Can we not ignore the mass of the cable ?

The length of the sling is a known in fact. It is used to attach a floating pulley to a platform.
My mistake. I thought cable instead of sling.

#### Jeroen Staps

Yes. I looked that up and updated my post while you were posting.
I see, thanks. Any idea on how the equation of motion of the pulley would look like?

#### Jeroen Staps

Can we not ignore the mass of the cable ?

My mistake. I thought cable instead of sling.
Yes we can ignore the mass of the cable.

No problem :)

#### BvU

Homework Helper
we can ignore the mass of the cable
In that case the pulley hangs vertically when there is no load ($m=0$)

This claim can be attacked by saying that the pulley is considered massless too, but that is searching for nails at low tide (old dutch expression).

#### Jeroen Staps

In that case the pulley hangs vertically when there is no load ($m=0$)

This claim can be attacked by saying that the pulley is considered massless too, but that is searching for nails at low tide (old dutch expression).
The pulley is also considered masless haha. But I am making a model so I can assume that the pulley can hang vertically however this is never the case.

Any idea on how to get the equation of motion? This is mainly my struggle

#### .Scott

Homework Helper
I see, thanks. Any idea on how the equation of motion of the pulley would look like?
You will have the force and direction (vertical) of $F_{load}$. You will have the magnitude of $F_{pull}$. With no load, the pulley would end up directly between the drum and the attachment point for the sling. So you should be able to set up equations for how far down the pulley is pulled to balance all the forces at the pulley.

#### BvU

Homework Helper
how to get the equation of motion
I gave it to you already :$$m\vec a = \sum \vec F_i$$can't do your work for you - you read the guidelines

#### BvU

Homework Helper
Looks as if two threads were merged into one: I didn't see the posts @.Scott until now...
But I do get a 2 min ago alert from Jeroen without a post in the thread ?!

Well, that's what you get for double posting...

"Describe the position of a pulley attached to a sling"

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