# Algebraic Manipulation of Equations

## Homework Statement

I have two equations. The first is for all of the forces on a hanging mass from a pulley. The second is for the sum of the torques about the pulley from which the mass hangs. I simply have to combine the equations to find the acceleration of the object. I have attempted every algebraic manipulation I can think of and keep coming out with the wrong answer. Please help.

## Homework Equations

T-mg=ma (for the sum of forces on the hanging mass)
Tr=I(-a/r) (for the torques about the pulley)

Here, T is tension, m is the mass of the hanging object, a is the acceleration, r is the radius of the pulley, I is the moment of inertia of the pulley.

I'm supposed to combine the two equations to eliminate T and solve for a.

## The Attempt at a Solution

OK, solve equation 1 for T.

T = ma + mg

Cool, now plug into equation 2.

(ma+mg)r=I(-a/r)
mr(a+g)=I(-a/r)
a+g=I(-a/mr^2)
1+g/a=I/mr^2
g/a=-I/mr^2-1
a = -(gmr^2)/(I)-g

I keep coming out with the same exact solution every time, but it is apparently wrong. Can someone tell me where I went wrong?

Related Introductory Physics Homework Help News on Phys.org
Simon Bridge
Homework Helper

a+g=I(-a/mr^2)
1+g/a=I/mr^2

I think you mislaid a minus sign there.
But you are also going about it the long way.

(ma+mg)r=I(-a/r)

That simplifies to ##mr^2a+mr^2g = -Ia##
...now get all terms involving "a" on the LHS and put everything else of the RHS.

a(mr2 I) = -mgr2

a = -(mgr2)/(mr2+I)

Thank you, that's correct. I could cry tears of joy.

Simon Bridge