Find the aceleration of the disk when falling (rotation)

[Hernaner28]

Homework Statement

I have that system and the problem says:

A disk of mass M and radious R are posed are shown. The disk is hanging by an ideal rope which is coiled up. From a smooth axis through the center of the disk is haning a body m. Calclate the module of the aceleration of the center of mass of the disk.

Homework Equations

The Attempt at a Solution

I've done this excercise but without the small mass hanging there. What I did was to use the torque and Newton formulas, that simple. But here I've got two bodies and tensions are driving me crazy. Should I consider the bodies as a single one?

Thanks!

 

[tiny-tim]

Hi Hernaner28!

I've done this excercise but without the small mass hanging there. What I did was to use the torque and Newton formulas, that simple. But here I've got two bodies and tensions are driving me crazy. Should I consider the bodies as a single one?

Yes, since they're moving together, you can regard them as a single body …

add the m to the mass of the pulley, but not to its moment of inertia.

 

[Hernaner28]

Done! Thank you!

 

[tiny-tim]

yes!

here's a shortcut (i don't know whether your professor would approve of this, but it's certainly a good way of checking the answer) …

the force is (M+m)g, and the effective mass is the actual mass M+m plus the "rolling mass" I/r² = M/2,

so a = (M + m)/(3M/2 + m) ​