How do I calculate speed in a pulley system when all ropes are connected?

[k_squared]

Homework Statement

I have a problem when doing math for pulley problems - if not at least one of the rope segments in independent, I do not seem to be able to compute the speed of given objects. For instance:

This is a very easy problem. S_a+3S_b=l and derive for speed; you don't really have to even go that far because you can see 3 lengths of rope are moved for motion in A, so the motion of d is 1/3 the speed of the motion of A.

This problem, however, confuses me:

Because none of the ropes are independent, as A was in the beginning, I can't set up enough equations to solve this formally!

Homework Equations

Position derives to speed. No real governing equations.

The Attempt at a Solution

I see that 4 lengths of rope move and area all directly connected to the cart, so it makes sense that the motion of A is 1/4 the motion of P. However, I can't show this formally.

 

[SteamKing]

Homework Statement

I have a problem when doing math for pulley problems - if not at least one of the rope segments in independent, I do not seem to be able to compute the speed of given objects. For instance:

This is a very easy problem. S_a+3S_b=l and derive for speed; you don't really have to even go that far because you can see 3 lengths of rope are moved for motion in A, so the motion of d is 1/3 the speed of the motion of A.

This problem, however, confuses me:

Because none of the ropes are independent, as A was in the beginning, I can't set up enough equations to solve this formally!

Homework Equations

Position derives to speed. No real governing equations.

The Attempt at a Solution

I see that 4 lengths of rope move and area all directly connected to the cart, so it makes sense that the motion of A is 1/4 the motion of P. However, I can't show this formally.

The presence of the motor M in Fig. F12-43 should not make a difference in analyzing the speed of the pulley. Fig. F12-43 is similar to Fig. F12-39 but uses a motor to wind up the hauling rope instead of a couple of hands. If you cut the hauling rope in Fig. F12-43 at point P and pulled on the end with your hands instead of using a motor, what would be different in analyzing that setup versus analyzing the other rig?

 

[k_squared]

The difference is the rope that is being pulled on counts for our division, whereas in the other ones, they do not!

Moreover, in the first one, I can define the rate at which the object is traveling as a single v_something and a coefficient, in this case, ALL of the ropes are pulling on the object.

 

[SteamKing]

The difference is the rope that is being pulled on counts for our division, whereas in the other ones, they do not!

It's not clear what you mean by 'the other ones'. You have presented two examples; one is being pulled by hand, the other is being pulled by a motor.

Moreover, in the first one, I can define the rate at which the object is traveling as a single v_something and a coefficient, in this case, ALL of the ropes are pulling on the object.

Why can't you define how fast the haul line is being pulled in the second example? There's nothing stopping you from saying the motor is pulling the rope, such that the velocity at point P is 1 m/s.

It's no different when a motor is pulling on the haul rope than when a couple of hands are doing it. If the point P is moving at 1 m/s, that just implies that the motor is turning at a certain number of RPMs when it is doing the hauling.

You're letting a minor difference in a couple of pictures mess with your head.