# Solving equations simultaneously

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:
ƩFy=Mg-T=Macm-y
Ʃτz=TR=Iαz
I know that I=1/2MR2
and have determined that acm-y=Rαz

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!!

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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:
ƩFy=Mg-T=Macm-y
Ʃτz=TR=Iαz
I know that I=1/2MR2
and have determined that acm-y=Rαz

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.

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