Describe the position of a pulley attached to a sling

[Jeroen Staps]

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?

 

[BvU]

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]

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]

I asked a mentor to move the thread, so: patience.

But do you have an idea

Yes: make the sketches I mentioned

 

[Jeroen Staps]

I asked a mentor to move the thread, so: patience.
Yes: make the sketches I mentioned

 

[.Scott]

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.

 

[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]

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]

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]

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]

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]

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]

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]

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...

 

[Jeroen Staps]

I gave it to you already :$$
m\vec a = \sum \vec F_i$$can't do your work for you - you read the guidelines

The problem with dissolving the forces is that the location of the pulley is unknown.

 

[BvU]

The problem with dissolving the forces is that the location of the pulley is unknown.

Now is the time to make a better drawing and decompose the tension forces. First for a steady state (=balance, load hangs still). You 'll need to introduce one or two angles. And perhaps make a simplification if allowed (like: drum far enough away)

 

[haruspex]

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

That wouid mean when frictionless the percentage friction would be 100.
I suggest it means |T₁-T₂|=T₃(f/100), or maybe /200.

 

[haruspex]

The problem with dissolving the forces is that the location of the pulley is unknown.

Right, but you need an unknown for the angle of the sling. You can make it the angle to the vertical or to the line from the sling's anchor to the drum. You can use that to resolve the forces. You will need a bit of geometry to relate that to the angle of the line from the drum.
Is the drum's radius known? If not, you will have to take it as zero. Likewise the pulley.

 

[Jeroen Staps]

Right, but you need an unknown for the angle of the sling. You can make it the angle to the vertical or to the line from the sling's anchor to the drum. You can use that to resolve the forces. You will need a bit of geometry to relate that to the angle of the line from the drum.
Is the drum's radius known? If not, you will have to take it as zero. Likewise the pulley.

The radius of the drum and pulleys are negligible for now.
In the model they are represented by a point/dot.

 

[haruspex]

The radius of the drum and pulleys are negligible for now.
In the model they are represented by a point/dot.

Good.
Can you figure the relationship between the angles of the two strings?

 

[Jeroen Staps]

Good.
Can you figure the relationship between the angles of the two strings?

not really no, any ideas?

 

[Jeroen Staps]

That wouid mean when frictionless the percentage friction would be 100.
I suggest it means |T₁-T₂|=T₃(f/100), or maybe /200.

If the percentage friction is zero then F_pull causes the same tension in the two parts of the cable

 

[haruspex]

If the percentage friction is zero then F_pull causes the same tension in the two parts of the cable

Not according to what you wrote in post #9.
Also, in reality, the frictional torque would depend on the tension in the sling rather than on the tensions in the other rope. E.g. if the other rope were almost straight there would be little friction.

 

[haruspex]

not really no, any ideas?

Consider the triangle formed by the drum, the pulley and sling's support. You know two sides and the angle one of them makes to the vertical, you can create an unknown for the angle the sling makes to the vertical. In terms of those, what angle does the rope make to the vertical?

 

[Jeroen Staps]

Consider the triangle formed by the drum, the pulley and sling's support. You know two sides and the angle one of them makes to the vertical, you can create an unknown for the angle the sling makes to the vertical. In terms of those, what angle does the rope make to the vertical?

I don't fully understand. For example this is the situation with given parameters and CD has a length of 2.
alpha is the angle in ABC

 

[BvU]

situation with given parameters

?

I repeat #22: the sketch of #7 has T1 and T3 collinear, so that $\sum \vec F$ has a component tangent to the dashed circle (from T2). $\qquad$ For a constant position of the pulley (pseudo steady state: equilibrium or constant acceleration of the load -- up or down) that component has to be zero.

If grokking isn't, try starting with a very far away drum ($\alpha$ constant)

 

[haruspex]

View attachment 239501

I don't fully understand. For example this is the situation with given parameters and CD has a length of 2.
alpha is the angle in ABC

That's a helpful diagram.
Let CD make angle θ to the vertical (angle BCD) and AD make angle φ to the vertical.
Can you get an equation relating , θ, α and the lengths AC, CD?

 

[Jeroen Staps]

That's a helpful diagram.
Let CD make angle θ to the vertical (angle BCD) and AD make angle φ to the vertical.
Can you get an equation relating , θ, α and the lengths AC, CD?

I'll try, not sure if it is possible though.
Because α, AC and CD are fixed and θ can change.

 

[haruspex]

I'll try, not sure if it is possible though.
Because α, AC and CD are fixed and θ can change.

I meant to write

Can you get an equation relating , θ, φ, α and the lengths AC, CD?