Rope Slipping in Pulley System: Does ##\ddot{l} = 0##?

[PFuser1232]

In this derivation (see attached image), does the author assume that the rope doesn't slip? Because if it does, then there is no guarantee that $\ddot{l} = 0$, right?
Also, does this equation apply to all pulley systems where one pulley is fixed and the other is allowed to move? For example, would this equation still hold for an arrangement where $|h| > |X|$?
Also, would this equation hold if we were to choose a coordinate system where $x$ and $X$ are both negative?

 

[Doc Al]

In this derivation (see attached image), does the author assume that the rope doesn't slip? Because if it does, then there is no guarantee that l¨=0\ddot{l} = 0, right?

What the author assumes is that the rope doesn't stretch. I don't see how slipping is relevant.

Also, does this equation apply to all pulley systems where one pulley is fixed and the other is allowed to move? For example, would this equation still hold for an arrangement where |h|>|X||h| > |X|? Also, would this equation hold if we were to choose a coordinate system where xx and XX are both negative?

Rather than try to apply this equation to other situations, just understand the logic of the derivation--then you can derive your own equation.

 

[PFuser1232]

What the author assumes is that the rope doesn't stretch. I don't see how slipping is relevant.Rather than try to apply this equation to other situations, just understand the logic of the derivation--then you can derive your own equation.

If the rope slips then the length of the section of the rope which terminates at the point where the rope is pulled is no longer constant.

 

[Doc Al]

If the rope slips then the length of the section of the rope which terminates at the point where the rope is pulled is no longer constant.

I'm not sure what you mean. That section is labeled as "x" in the diagram; it's not constant.

 

[PFuser1232]

I'm not sure what you mean. That section is labeled as "x" in the diagram; it's not constant.

If the grip isn't firm enough, the rope would begin to slip. This change in $x$ isn't accounted for in the constraint equation, right? For example, if someone were to move their hand up and down the rope without actually pulling it we would end up with a change in $x$ without a corresponding change in $X$.

 

[Doc Al]

If the grip isn't firm enough, the rope would begin to slip. This change in $x$ isn't accounted for in the constraint equation, right? For example, if someone were to move their hand up and down the rope without actually pulling it we would end up with a change in $x$ without a corresponding change in $X$.

Ah... you're talking about slipping of the rope with respect to the hand! (I thought you meant slipping along the pulleys.) Just think of x as the position of the rope, not the hand. Sure, if you want x to represent the position of the hand, then you'll have to assume no slipping. But the relationship really has to do with the position of the rope, not the hand.

 

[PFuser1232]

If the grip isn't firm enough, the rope would begin to slip. This change in $x$ isn't accounted for in the constraint equation, right?

Ah... you're talking about slipping of the rope with respect to the hand! (I thought you meant slipping along the pulleys.) Just think of x as the position of the rope, not the hand. Sure, if you want x to represent the position of the hand, then you'll have to assume no slipping. But the relationship really has to do with the position of the rope, not the hand.

This is clear now, thanks!
I'm still curious as to whether reversing the coordinate system used would make a difference though. I know we chose a coordinate system where the coordinates of all parts of the pulley system are positive for convinience, but what happens when we use a coordinate system where the coordinates are negative? Also, should I even bother? I always get bogged down in trying to understand what would happen if a fairly simple problem were much more complicated. Will doing so improve my understanding of the subject, or will it just backfire and prevent me from making progress?

 

[Doc Al]

but what happens when we use a coordinate system where the coordinates are negative?

The physical constraint will be the same: the length of the rope is a constant.

Also, should I even bother? I always get bogged down in trying to understand what would happen if a fairly simple problem were much more complicated. Will doing so improve my understanding of the subject, or will it just backfire and prevent me from making progress?

It's hard to generalize. Often it helps, as long as you get the physics right. Other times it's a waste.

Perhaps a better strategy might be to actually try a more complicated problem and see if you can apply the same principles as in the simple one.

 

[PFuser1232]

The physical constraint will be the same: the length of the rope is a constant.It's hard to generalize. Often it helps, as long as you get the physics right. Other times it's a waste.

Perhaps a better strategy might be to actually try a more complicated problem and see if you can apply the same principles as in the simple one.

If we wish to apply the physical constraint (that is constant length) in a coordinate system where all parts of the pulley system have negative coordinates we would have to use absolute values, but since dealing with absolute values is annoying, we're better off sticking to a coordinate system where each part has positive coordinates, right?

 

[Doc Al]

we're better off sticking to a coordinate system where each part has positive coordinates, right?

That would be my choice. Choose a coordinate system that makes the problem as easy as possible.

Regardless of coordinate system chosen, the length will still be a positive quantity. (Though expressing it might be cumbersome.)

 

[PFuser1232]

That would be my choice. Choose a coordinate system that makes the problem as easy as possible.

Regardless of coordinate system chosen, the length will still be a positive quantity. (Though expressing it might be cumbersome.)

Thanks Doc!