Angular Acceleration: Understanding the Relationship Between Connected Pulleys

[rashida564]

Hi everyone I had this argument with someone told. About angular acceleration.
His opinion: Since both pulleys are connected by a string then both of them must have the same angular acceleration.
My opinion: Since a2 "acceleration of the second mass" is half a1. Then angular acceleration of the second is half of the first.
Which one is correct

 

[berkeman]

Is the linear motion of the rope the same around the circumference of each pulley?

 

[rashida564]

No because acceleration I believe it's different which equal to a , and I think my friend was wrong

 

[berkeman]

Um, if you have v(t) at the edge of the pulley, do you now also have the angular acceleration as a function of time?

BTW, how much money is riding on this bet?

 

[berkeman]

Actually, let me do some calcs before I push the equality aspect too much. Back in a bit...

 

[rashida564]

I think considering It rolling without slipping then V(t)=w*r. Differeniating both sides we will get a=alpha*r.
10 pounds, we can split it 50 50

 

[berkeman]

It looks like the linear motion up for the 2nd pulley takes away from the angular motion of the rope around the 2nd pulley. Can you quantify that? You might win this bet after all...

 

[rashida564]

Sorry mate I couldn't understand your hint

 

[berkeman]

So start with the pulleys in a starting position. Put 4 dots on the rope where it comes into contact with the 2 pulleys. One dot each for where the rope first contacts the pulley, and one for where it leaves the pulley.

Then let the fixed pulley go through 1/2 rotation, so that the first dot on its rope rotates around to the postion of the second dot. That pulley has rotated through 180 degrees in some time T. Draw how the 2nd pulley has translated up some and rotated some. How many degrees has the 2nd pulley rotated in the same amount of time T?

 

[rashida564]

So part of the pulley will either go up or down depend on the direction of motion for the first pulley then from that the second pulley will rotate an angle of 90 degrees right

 

[berkeman]

So less linear motion of the rope leads to less angular motion on the 2nd pulley, right?

 

[rashida564]

yeah so I should get the money

 

[A.T.]

His opinion: Since both pulleys are connected by a string then both of them must have the same angular acceleration.

His argument would work, if the pulley centers were at rest relative to each other. But they aren't here.