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Homework Help: Cycloid Particle question

  1. Aug 23, 2010 #1
    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. jcsd
  3. Aug 23, 2010 #2

    ehild

    User Avatar
    Homework Helper


    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.

    You have found already that
    dy/dt = -Rw sin (wt)
    dx/dt= Rw cos (wt) + wR

    vx=dx/t and vydy/dt.
    At what time instants is y maximum or minimum? Find and plug in the values for t in the equations for vx and vy.

    ehild
     
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