Finding the relationship between accelerations in a pulley system

[Juan Pablo]

I have been trying to find the relationship between the accelerations in a pulley system. My book doesn't explain it.

My first question: Which object moves faster, the one attached to the pulley or the other? I'm guessing the one that isn't attached to the pulley.

My second question: Does the relationship between the accelerations can be found by looking at the forces? For example, the tension on object one is twice the tension in object two. Does this mean the acceleration of one is the double of the acceleration of two?


My prof did explain it, but I didn't understand. He used derivatives of the length of the rope between the pulleys. Can anyone explain this method?

I'm not asking for straight answers, more for some guidance. Sorry for not using the template, it didn't fit my question since it isn't a specific problem.

Thanks!

 

[semc]

Lets see the free body diagram of the 2 object. T-m₁g=m₁a₁ T-m₂g=m₂a₂ in this case its just a simple pulley system so you can see that the tension throughout the string is the same. For your question i believe the 2nd pulley is resting on the string hence different tension correct? Well let's derive the acceleration in your case. Let's say block 1 is directly connected to the string and block 2 is connected to a pulley that is resting on the string connecting block 1. We can say that when block 1 moves a distance of x, block 2 will move a distance of \frac{x}{2} am i right? Hence we can relate the displacement of block 1 and 2 by this equation x₁=0.5a₂. Well from here it seem pretty obvious that the 2nd derivative with respect to time will give you the the acceleration of block 1 and block 2 which in this case a₁=0.5a₂.

 

[Juan Pablo]

I think I understand what you mean. Anyway, the diagram in question:

.

Both of them are pulleys, I made a mistake in the drawing of the first one.