Calculating Speed of Falling Mass on Pulley System

[Xels]

Homework Statement

A string is wrapped tightly around a fixed pulley that has a moment of inertia of 0.0352kg/m²; and a radius of 12.5cm. A mass of 423g is attached to the free end of the string. The mass is allowed to fall under the influence of gravity. As the mass falls the string causes the pulley to rotate. What is the speed of the mass after is has fallen through 1.25m?

Homework Equations

T-mg=ma
T=-I(a/r²)
x=(1/2)at²

The Attempt at a Solution

T-mg=ma
-I(a/r²)-mg=ma

a=-(g/(1+(I/mr²))) <---- this step is given by my prof; and honestly I'm not sure how he isolated 'a' here. I'd really appreciate any insights here. Rationally I understand that the statement means that net acceleration is equal to gravity divided by linear kinetic energy of the falling mass plus the kinetic energy of the pulley.

x=(1/2)at²
t=((2x)/a)^(1/2)
v=|a|t
v=(2xa)^(1/2)

So my questions essentially are the above where we jump from I(a/r²)-mg=ma to a=-(g/(1+(I/mr²)))
and
If my moment of inertia is given as 0.0352 kg/m² then in (1+(I/mr²))
how do I incorporate I?

 

[ehild]

The Attempt at a Solution

T-mg=ma
-I(a/r²)-mg=ma

Collect both terms containing 'a' at one side of the equation,

-mg=ma+Ia/r²

divide both sides with 'm'

-g=a+Ia/(mr₂)

then factor out 'a'.

-g=a(1+I/(mr²))

Divide both sides of the equation by the factor
(1+I/(mr²)):

a=-g/(1+I/(mr²))

ehild

 

[Xels]

Thanks; that makes total sense.