Rolling Wheel - Parametric Paths

monster · #1 $Old$ March 24th, 2009, 01:19 AM

Need help with this probelm not with the maths involved but the theory,

Wheel radius R rolls to the right along straight line with speed v, the path of a point on the rim is given by;

c(t)= ( vt - Rsin((vt)/R) , R-Rcos((vt)/R) )

When is the velocity vector of this point horizontal and what is the speed at this point?

Trying To work out when velocity is horizontal?

Soroban · #2 $Old$ March 24th, 2009, 04:01 AM

Hello, monster!

I'll get you started . . .

Quote:

Wheel radius $R$ rolls to the right along straight line with speed $v$ .

The path of a point on the rim is given by: . $c(t) \:=\:\begin{Bmatrix} x &=& vt - R\sin\left(\frac{v}{R}t\right) \\ \\[-4mm] y &=& R-R\cos\left(\frac{v}{r}t\right) \end{Bmatrix}$

When is the velocity vector of this point horizontal and what is the speed at this point?

We have: . $\begin{array}{ccc}\dfrac{dx}{dt} &=& v - v\cos\left(\frac{v}{R}t\right) \\ \\[-3mm] \dfrac{dy}{dt} &=& v\sin\left(\frac{v}{R}t\right) \end{array}\quad\hdots\quad \text{and the velocity is: }\:\frac{dy}{dx} \:=\:\frac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$

. . Hence: . $v(t) \;=\;\frac{\sin\left(\frac{v}{R}t\right)} {1-\cos\left(\frac{v}{R}t\right)}$

If the velocity is horizontal, then: . $v(t) \,=\,0 \quad\Rightarrow\quad \sin\left(\tfrac{v}{R}t\right) \:=\:0$

Then: . $\frac{v}{R}\,t \:=\:\pi n \:\text{ for }n \in I$

. . Hence: . $t \:=\:\frac{R\pi}{v}\,n$

You'll have to fine-tune this result.
Some values of $t$ produce an undefined form for $v(t).$