Thread: Rolling Wheel - Parametric Paths

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?

Hello, monster!

I'll get you started . . .

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

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

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

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


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

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

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


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