Describe the position of a pulley attached to a sling

[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?

 

[Jeroen Staps]

I meant to write

OK, I see

 

[Jeroen Staps]

I meant to write

But then there are still two unknowns, namely θ and φ.

 

[haruspex]

But then there are still two unknowns, namely θ and φ.

Doesn’t matter. At this stage we are just trying to find a relationship between them. Next step will be to involve them in force balance equations.
In fact, if you are stuck on getting that relationship, set that aside and move onto the other equations.

 

[Jeroen Staps]

Doesn’t matter. At this stage we are just trying to find a relationship between them. Next step will be to involve them in force balance equations.
In fact, if you are stuck on getting that relationship, set that aside and move onto the other equations.

Alright I'll try to find the relationship then.

 

[Jeroen Staps]

Still stuck on finding the relationships :(

 

[Jeroen Staps]

Now I have that: AD²=AC²+CD²-2*AC*CD*cos(γ-θ), with γ the angle ∠ACB. But still don't understand the force balance equations

 

[BvU]

Been a while. Remind us what force balance equations you have gathered so far and which one you don't understand

 

[Jeroen Staps]

Been a while. Remind us what force balance equations you have gathered so far and which one you don't understand

Was on holiday ;) , I just don't understand how I can solve this problem when there are more than one unknowns

 

[haruspex]

Still stuck on finding the relationships :(

First, find all the angles of triangle ACD in terms of α, θ, φ.

 

[Jeroen Staps]

Been a while. Remind us what force balance equations you have gathered so far and which one you don't understand

Does this make sense?

 

[Jeroen Staps]

First, find all the angles of triangle ACD in terms of α, θ, φ.

Need more?

 

[Jeroen Staps]

Now I have that: AD²=AC²+CD²-2*AC*CD*cos(γ-θ), with γ the angle ∠ACB. But still don't understand the force balance equations

And I have that: BD²=AB²+AD²-2*AB*AD*cos(α₂) when you substitute AD from the earlier post in this equation then you have an equation to solve α₂ with θ as unknown and α₂ is equal to α-(180-90-φ). So I can express θ in φ

 

[haruspex]

View attachment 239863

Need more?

That’s good. You know the lengths of two sides of ACD. What equation can write connecting those with two of its angles?

 

[Jeroen Staps]

That’s good. You know the lengths of two sides of ACD. What equation can write connecting those with two of its angles?

I now have the following equation that seems to be correct:

φ = cos^-1((AC-CD*cos(γ-θ)) / √(AC²+CD²-2*AC*CD*cos(γ-θ))) + 90 - α

 

[Jeroen Staps]

I now have the following equation that seems to be correct:

φ = cos^-1((AC-CD*cos(γ-θ)) / √(AC²+CD²-2*AC*CD*cos(γ-θ))) + 90 - α

But how do I use this to describe the location of D when there is a certain mass hanging at D and there is a certain pulling force in AD?

 

[haruspex]

I now have the following equation that seems to be correct:

φ = cos^-1((AC-CD*cos(γ-θ)) / √(AC²+CD²-2*AC*CD*cos(γ-θ))) + 90 - α

I was thinking of something simpler, though it may be equivalent: the sine rule.

 

[haruspex]

But how do I use this to describe the location of D when there is a certain mass hanging at D and there is a certain pulling force in AD?

As I posted, you can use angles θ and φ to write a force balance equation.

 

[Jeroen Staps]

I was thinking of something simpler, though it may be equivalent: the sine rule.

I used the cosine rule twice to come up with this

 

[Jeroen Staps]

As I posted, you can use angles θ and φ to write a force balance equation.

So does my post #45 make sense?

 

[BvU]

Yes, at least for equilibrium. Adding constant acceleration should be easy.

But I wished you wouldn't change notation every time.
And you can easily simplify $\ \sin(\pi/2-\pi/4-\phi)\$ and $\ \cos(\pi/2-\pi/4-\phi)\$ I should hope.

So how many equations with how many unknowns do you now have altogether ?

 

[haruspex]

Adding constant acceleration should be easy.

I believe it is a statics question, how the equilibrium position depends on the applied tension.

 

[Jeroen Staps]

I believe it is a statics question, how the equilibrium position depends on the applied tension.

The question is: What is the position of the pulley when there is a given mass of the load and a given ratio between the force of gravity and the pulling force.

 

[haruspex]

The question is: What is the position of the pulley when there is a given mass of the load and a given ratio between the force of gravity and the pulling force.

Which is the same as I wrote.