# Homework Help: Cycloid Particle question

1. Aug 23, 2010

### shyta

1. The problem statement, all variables and given/known data

A particle moves in a plane according to

x= Rsin(wt) = wRt
y= Rcos(wt) + R

where w and R are constants. This curve, called a cycloid, is the path traced out by a point on the rim of a wheel which rolls without slipping along the x-axis.

Question:
(a) Sketch the path.

(b) Calculate the instantaneous velocity and acceleration when the particle is at its maximum and minimum value of y.

2. Relevant equations

x= Rsin(wt) = wRt
y= Rcos(wt) + R

3. The attempt at a solution

I drew up the path of the particle, as a cycloid of course. No problem with that.
I am having problems understand part b, with relation to the curve.

I differentiated the given equations wrt to x.

i.e.
dy/dt = -Rw sin (wt)
dx/dt= Rw cos (wt) + wR

then i proceed to find dy/dx = [Rw sin (wt)]/[Rw cos (wt) + wR]

Ymaximum should be 2R (diameter of the rim of the wheel)

I also went ahead to find values of x which corresponds to the maximum and minimum values of y.
Values of X for maximum Y = 0, 2pie, ...
Values of X for minimum Y = pie, 3pie, ...

Instantaneous velocity is tangent of the curve of a position time graph, which is it the same as the graph i drew out?

Then back to the original question, the instantaneous velocity at maximum y is therefore 0? and instantaneous velocity at minimum y is infinity?

I know I have a conceptual error somewhere but I just can't figure it out.

2. Aug 23, 2010

### ehild

You mean x=Rsin(wt) + wRt, don't you?

Your graph shows y in terms of x. Both the velocity and the acceleration are vectors. vx=dx/dt, vy=dy/dt. The tangent of this graph shows the direction of velocity at the given point. The x and y components of the acceleration are the time-derivatives of vx and vy, respectively.