# The yo-yo

## Homework Statement

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.

## Homework Equations

$\tau=TR=I\alpha$

$F=ma$

## 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??

Related Introductory Physics Homework Help News on Phys.org
tiny-tim
Homework Helper
Hi kudoushinichi88! (I haven't actually checked your equations, but …)

wouldn't it be easier to use conservation of energy? rl.bhat
Homework Helper
String unwinds around the cylinder of radius b. So a(tan) = b

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

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

$Tb=mRa$ ...1
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 $$R \omega$$

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