Solving equations simultaneously

[jstevenson16]

Problem:
You make a primitive yo-yo by wrapping a massless string around a solid cylinder of mass M and radius R. You hold the freee end of the string stationary and release the cylinder from rest. The string unwinds but does not slip of stretch as the cylinder descends and rotates. Find the downward acceleration of the cylinder and the tension in the string.

Necessary Equations/Attempt
I have so far found the translational and rotational forces to be as follows:
ƩF_y=Mg-T=Ma_cm-y
Ʃτ_z=TR=Iα_z
I know that I=1/2MR²
and have determined that a_cm-y=Rα_z

The issue I'm having is with solving the two equations simultaneously to find the target variables. Can anyone offer a guided explanation of how to do this or walk me through it? I have a final tomorrow and just realized I still can't figure it out. Any help is much appreciated!

 

[ehild]

Problem:
You make a primitive yo-yo by wrapping a massless string around a solid cylinder of mass M and radius R. You hold the freee end of the string stationary and release the cylinder from rest. The string unwinds but does not slip of stretch as the cylinder descends and rotates. Find the downward acceleration of the cylinder and the tension in the string.

Necessary Equations/Attempt
I have so far found the translational and rotational forces to be as follows:
ƩF_y=Mg-T=Ma_cm-y
Ʃτ_z=TR=Iα_z
I know that I=1/2MR²
and have determined that a_cm-y=Rα_z

The issue I'm having is with solving the two equations simultaneously to find the target variables. Can anyone offer a guided explanation of how to do this or walk me through it? I have a final tomorrow and just realized I still can't figure it out. Any help is much appreciated!

Mg-T=Ma

TR=Iα

a=αR

You have these three equations. Replace α with a/R in the second equation, and divide by R. You get an expression for T. Substitute 1/2 MR² for I. Substitute the expression for T in the first equation.


ehild