# Change of rate in a wheel

Consider a wheel of radius $r$ (fixed) and an object $m$ tied to a rope of length $l$ (fixed). See Figure.

If the angular velocity is constant, calculate the speed at which you move the object m when $\alpha=\pi/2$

I dont know how use the large of the rope, this never change and maybe use

$\frac{d\alpha}{dt}=w$

Thx anyway

• What have you considered so far? It appears that you have access to triangular sections of your diagram. What analysis can you apply to these sections? – abiessu Aug 29 '13 at 3:35

## 1 Answer

I am not sure if your question can be assimilated to the usual piston problem in a clockwise rotation of course: read piston motion equations.
If so, after choosing a natural frame of reference with coordinate $x$, the law of cosines gives $$x^2+r^2-2rx \cos \omega t=l^2$$Differentiating with respect to $t$, one obtains $$2x \dot x-2r(\dot x \cos \omega t- \omega x \sin \omega t)=0$$so that $$\dot x=\frac {r \omega x \sin \alpha}{r \cos \alpha-x}$$Then for $\alpha=\frac \pi 2$, the speed is $\,-r\omega$ .