How Do You Relate the Accelerations in a Disk and Pulley System?

[Avi Nandi]

Homework Statement

A disk of mass M and radius R unwinds from a tape wrapped around it. The tape passes over a frictionless pulley and mass m is suspended from the other end. Assume that the disk drops vertically.

a. relate the accelerations of mass m and disk ,a and A, respectively to the angular acceleration of the disk.

b. Find a, A and α.

Homework Equations

The Attempt at a Solution

equations of motion : T - mg = ma
T - Mg = MA

-TR = [itex]\frac{1}{2}[/itex]MR[itex]^{2}[/itex]α

Are these equations of motion wrong?

 

[haruspex]

equations of motion : T - mg = ma
T - Mg = MA

Which direction are you taking as positive, up or down? If up, I would write that as ma = T + mg etc., where g takes a negative value. But it's maybe a matter of personal preference.

-TR = [itex]\frac{1}{2}[/itex]MR[itex]^{2}[/itex]α

Are these equations of motion wrong?

They look ok to me, but you've not written out the relationships between A, a, α and R. That is where this question fools some people.

 

[Avi Nandi]

I took up as the positive direction. But I think T and g are always opposing each other. I have written a constraint equation A + a = Rα. But the clue given in the book is " if A= 2a then α= 3A/R"

 

[Saitama]

" if A= 2a then α= 3A/R"

That's a typo in the book. There was a thread on the same problem recently.

 

[haruspex]

I took up as the positive direction. But I think T and g are always opposing each other. I have written a constraint equation A + a = Rα. But the clue given in the book is " if A= 2a then α= 3A/R"

The consensus is that it's an error in the book. It should say α= 3a/R. See https://www.physicsforums.com/showthread.php?t=722327