Parametric equations of cycloid on a Ramp

Question

A small wheel of radius r is situated at the top of a ramp having an angle θ = π/3 rad as it appears in the figure below. At t = 0 the wheel is at rest and then it starts to rotate clockwise in the positive x direction with constant angular velocity ω.

Find the parametric equations of the x and y coordinates of the point p as a function of time, for t>0. Using Pythagoras’ theorem or otherwise verify your formulas at the points $x_p$(T) and $y_p$(T), where T = 2π/ω. the circle is standing on (0,0) origin and You can imagine as if the the circle is moving on the Ramo

I want to find parametric equations of the point P.

I know how to find the parametric of cycloid on x-axis when its touching the x-axis but really stuck with this one i have 2 problems with this its on a Ramp and the point P is not on the Ramp Completely lost please help as much as you can any hints tips appreciated and full answer would be awesome

If you are voting to close the question please leave comment why so with a reason I dont see why this question is off topic

Thanks

Answer 1

The parametric equation of a cycloid generated by rolling a wheel of radius $a$ at a constant rate by an angle $\theta$ is

$$x = a (\theta-\sin{\theta}) \quad y=a (1-\cos{\theta})$$

Rotating the $(x,y)$ coordinate system to a new coordinate system $(x',y')$ by an angle $\phi$ is accomplished by the tranformation

$$x'=x \cos{\phi}+y \sin{\phi} \quad y'=-x \sin{\phi}+y \cos{\phi}$$

so that the equation of the cycloid here is, after some simplification:

$$x'=a (\sin{\phi}+\theta \cos{\phi}) - a \sin{(\theta+\phi)} \quad y' = a(\cos{\phi}-\theta \sin{\phi}) - a \cos{(\theta+\phi)}$$

Note this assume that the point $P$ begins on the ramp. Note also that $\phi = \pi/2-\alpha$, the angle of incline.

Here is a plot for a constant rate at the specified angle of incline, $\alpha=\pi/3$.

Answer 2

Assume you can write cycloid equation $x=x(t),y=y(t)$, on the horizontal line with the origin as starting point. Then the first step is to rotate the line $\theta=\frac\pi3$ clockwisely using $$x'(t)=x(t)\cos\theta+y(t)\sin\theta\\ y'(t)=-x(t)\sin\theta+y(t)\cos\theta$$ The next step, the point $P$, in another view, seems like it has been rotated $\pi-\theta$ before it goes $(0,2r)$. So the parameterize equation is $$x''(t)=x'(t+(\pi-\theta))\\y''(t)=y'(t+(\pi-\theta))$$