Differential Pulley: Force to Balance Weight W

[LCSphysicist]

Homework Statement

Theoric doubt about differential pulley.

Relevant Equations

All below.

I am trying to deal with this problem, the question is what is the force to balance the weight W, where the rope don't have weight. The bigger pulley at the top has radius a, and the other, attached to the same axis, has radius 0.9a. The force is applied in one side of the freeling rope.

I could find it:

I am not sure about the torques.

 

[Halc]

There seems to be nothing holding up the weight W. The rope to the right of Q is slack, be there force at T or not. So that loops over the small pulley and the rope on the left of W is slack (tension T2 is zero), allowing W to just freefall until the slack is entirely taken up between P and R.

 

[LCSphysicist]

There seems to be nothing holding up the weight W. The rope to the right of Q is slack, be there force at T or not. So that loops over the small pulley and the rope on the left of W is slack (tension T2 is zero), allowing W to just freefall until the slack is entirely taken up between P and R.

I am not sure i get, if it helps, the answer is W/20

 

[LCSphysicist]

I tried by all ways that i could think, i am not sure if all the three pulley will rout, i just came to two equations and three incognits, i can't see any constraint that would help me too, so gave up by this way.

 

[Delta2]

The two upper pulleys are constrained to move at the same angular frequency $\omega(t)$? at any time instant t?

 

[Lnewqban]

Can you see the pulleys as levers?
The top pulley gives you certain amount of mechanical advantage (1/0.9).
The bottom pulley always gives you a mechanical advantage of 2.

 

[LCSphysicist]

The two upper pulleys are constrained to move at the same angular frequency $\omega(t)$? at any time instant t?

I think not, that was a doubt that i had too, but i think what is the question say by the same axis, maybe want to say the same direction. That can be a case, i will try with the same angular frequency.

 

[LCSphysicist]

Can you see the pulleys as levers?
The top pulley gives you certain amount of mechanical advantage (1/0.9).
The bottom pulley always gives you a mechanical advantage of 2.

The top you say the pulley with radius a?

 

[Lnewqban]

I think not, that was a doubt that i had too, but i think what is the question say by the same axis, maybe want to say the same direction. That can be a case, i will try with the same angular frequency.

For the contraption to work, the top pulleys must be solidly joined by a common shaft.
Please, see:
https://en.wikipedia.org/wiki/Differential_pulley

 

[Halc]

For the contraption to work, the top pulleys must be solidly joined by a common shaft.

In which case it becomes a problem involving statics and levers, per your prior post.
The ropes cannot slip through the pulleys, which is perhaps what the dots P and R are trying to convey.

 

[Lnewqban]

In which case it becomes a problem involving statics and levers, per your prior post.
The ropes cannot slip through the pulleys, which is perhaps what the dots P and R are trying to convey.

As you properly explained before, if both top pulleys were not one solid piece, the weight would slide all the way down until the reach of the rope's loop.

This arrangement is commonly used with sprockets and chain (and some internal gears sometimes) to lift heavy loads in mechanical shops.

 

[LCSphysicist]

Now i could understand, actually i am really frustrated because is so easy and i was using MOI
thank you all