Rotational accel to linear accel

[chloechloe]

Homework Statement

I do not understand how a was solved for. the picture is a sphere rolling down a hill. Asked to solve for a.

Homework Equations

solve for a, use alpha = a/r
ma = mg sin theta - fs; T = fs*r = I*alpha = (2/5)mr^2; fs = (2/5)ma
a = (5/7)g sin theta

I can't see the progression through these steps

The Attempt at a Solution

 

[Doc Al]

use alpha = a/r

(1) That's the condition for rolling without slipping.

ma = mg sin theta - fs;

(2) That's Newton's 2nd law applied to translation.

T = fs*r = I*alpha = (2/5)mr^2;

This is an attempt to apply Newton's 2nd law to rotation, but alpha was left out of the right hand term. It should be:
(3) T = fs*r = I*alpha = (2/5)mr^2*alpha

fs = (2/5)ma

(4) Combine 1 and 3 to get this.

a = (5/7)g sin theta

Combine 4 with 2 to get this.

 

[chloechloe]

Thank you. Makes sense now. I was getting stuck at combining things at the end. The missing alpha didn't help either.