# The yo-yo

1. Oct 11, 2009

### kudoushinichi88

1. The problem statement, all variables and given/known data
A yo-yo is made from two uniform disks, each with mass m and radius R, connected by a light axle of radius b. A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds. Find the linear acceleration of the yo-yo.

2. Relevant equations
$\tau=TR=I\alpha$

$F=ma$

3. The attempt at a solution
$\tau=TR=I\alpha$

$Tb=2\left(\frac{1}{2}mR^2\right)\alpha$

since $a_{tan}=r\alpha$,substituted into the equation above and simplified,

$Tb=mRa$ ...1

The yo-yo is accelerating downwards linearly, so

$2mg-T=2ma$ ...2

Solving for T in eq.1 and substituting into eq.2,

$2mg-\frac{mRa}{b}=2ma$

Solving for a, I got

$a=\frac{2g}{2+R/b}$

$a=\frac{2g}{2+(R/b)^2}$

what did I do wrong??

2. Oct 11, 2009

### tiny-tim

Hi kudoushinichi88!

(I haven't actually checked your equations, but …)

wouldn't it be easier to use conservation of energy?

3. Oct 11, 2009

### rl.bhat

String unwinds around the cylinder of radius b. So a(tan) = b

4. Oct 11, 2009

### willem2

You substituted the wrong value for r. If the yoyo unwinds with angular speed $$\omega$$
the vertical speed of the yoyo is $$b \omega$$ and not $$R \omega$$

5. Oct 13, 2009

### kudoushinichi88

ah... so that's why!
Thank you all! This has also helped me to find the angular acceleration and tension in the string...