Is the Mass of My Birthday Pulley Evenly Distributed?

[bcjochim07]

Homework Statement

You have been given a pulley for your birthday. It is 12 cm in diameter and has a mass of 2 kg. You get to wondering whether the pulley is uniform. That is, is the mass evenly distributed, or is concentrated toward the center of near the rim? To find out, you hang the pulley on a hook, wrap a string around it, and suspend you 1 kg physics book 1 m above the floor. With your stopwatch you find that it takes .71 s for the book to hit the floor. What can you conclude about the pulley

Homework Equations

Moment of Inertia for a disk I=.5MR^2
Torque=rFt=I * angular acceleration(alpha)
x=xo + volt+ .5at^2

The Attempt at a Solution

First of all, Is my work correct?


Okay so I calculated the acceleration of the book:

0m=1m + .5*a*(.71s)^2
a= -3.97 m/s^2

Then I decided to calculate the acceleration of the book as if the pulley were uniform:

I=.5(2kg)(.06m)^2 I= .0036 kgm^2
Torque=(.06m)*(T) = (.0036)*alpha
T=.06*alpha
alpha=-a/r
T=(.06)*(-a/.06) = -a


Sum of Forces on y for book = T-mg=ma
Making the substitution for T : -a-mg=ma
-mg= ma +a
-(1.0kg)(9.80)= a(1kg +1)
a= -9.80 m/s^2

Since the two calculated accelerations are different, the pulley must not be uniform. But I don't know if I can make any conclusions further than this about where the mass must be concentrated.

 

[foxjwill]

First of all, if the disk isn't uniform, the moment of inertia won't be $$.5MR^2$$. Beyond that, I'm about to fall over in my chair from being really tired, so I'll try and get back to you on the rest tomorrow. >_<