Final angular velocity of dropped yoyo

[clementc]

Homework Statement

Hey guys,
I just read Physics, by Halliday, Resnick and Krane. There's a chapter on rotational dynamics, and the example goes like this:
A yo-yo of total mass M = 0.24kg consists of two disks of radius R = 2.8cm connected by a thin shaft of radius R₀ = 0.25cm. A string of length L = 1.2m is wrapped around the shaft. If the yo-yo is thrown downward with an initial velocity of v₀ = 1.4m/s, what is its rotational velocity when it reaches the end of the string?

Homework Equations

$$KE = \frac{1}{2} mv^2$$
$$Rotational KE = \frac{1}{2} I{\omega}^2$$
$$I of a cylinder = \frac{1}{2} mr^2$$
Conservation of energy

The Attempt at a Solution

The way the book did it was really strange and long, so I tried doing it using conservation of energy, but got the wrong answer. Can anyone see what's wrong here?
[PLAIN]http://img151.imageshack.us/img151/6251/rotational.png
Their answer was 610 rad/s, but I really don't know where I went wrong here. Any help would be very greatly appreciated. Thanks!

 

[gneill]

You forgot to add the initial rotational energy when you summed up for the initial kinetic energy.

The rotational inertia of the disk(s) is I = (1/2)MR², and the rotational energy is KEr = (1/2) I ω². So there's two factors of 1/2 in there.

 

[clementc]

Oh oops yeah forgot about the initial rotational KE
And sorry my working was a bit unclear - i actually multiplied across all terms by 2 so one of those factors of 1/2 disappeared

This is my new working, but its still giving me a different answer of 433 rad/s =(
Sorry about this, but could you please please look through this new working? i really can't find anything wrong with it but the answers 610 rad/s
Thank you very very much. its just been nagging me all day and i have a really really big exam coming up on saturday. thanks a lot

[PLAIN]http://img851.imageshack.us/img851/3142/rotation2.png

 

[gneill]

You're still missing a (1/2) on the rotational energy term on the RHS. This is before you multiplied through by 2.

 

[clementc]

OH OF COURSE! thank you so so much sir =DD thank you thank you thank you!
im such an idiot ==; i treated the two disc separately at first and forgot that the mass in 1/2mr^2 was the mass of the entire yoyo.
the answer did indeed come out to 610 rad/s =) thanks so much!