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Cycloids and related curves questions

  1. Sep 17, 2008 #1
    I have three questions that I'm going to roll into one. I'm going insane trying to figure these out.

    1. Find the unit tangent vector T to the cycloid. Also find the speed at theta=0 and theta=Pi, if the wheel turns at dtheta/dt=1.

    that dtheta/dt is the speed, right? I'm a little baffled as to how I'm supposed to find a tangent vector when I don't even know what the cycloid is. I think it has something to do with v/|v|, but that doesn't change the fact I don't know how to answer the question. I'm sorry for not having gotten further with this, I need a nudge.

    2. The slope of the clycloid is infinite at theta=0: dy/dx= sin(theta)/(1-cos(theta)). Estimate the slope at theta=(1/10) and theta=(-1/10).

    I know I need to use L'Hopital's rule, but the answer my key is giving me is confusing me a bit. here's my main question: how does (sin(theta))/(1 - cos(theta))=approx (theta)/(((theta)^2)/2)? after that, I think I understand.

    3. For a cardioid the radius C - 1 of the fixed circle equals the radius 1 of the circle rolling outside (epicycloid with C=2). (a) The coordinates of P are x = -1+2cos(theta)- cos(2theta), y=____________. (b) the double-angle formulas yield x= 2 cos(theta)(1-cos(theta)), y=_______________. (c) x^2 + y^2 =_______________ so its square root is r=________________.

    this is the one I'm really struggling with. I've been sitting here trying to make sense of it for about an hour and I just DO NOT GET IT AT ALL. I'm getting really frustrated and I really really really need someone to guide me through it. I'm sorry I don't have any work to show for my efforts, but I know nothing I've got is right. I have an answer key for this one, but it's not proving very useful.

    please, I really need help!
  2. jcsd
  3. Aug 24, 2009 #2
    Read the short history of Mersenne the French mathematician and his associates.Your problem took years of work before they got an answer.Some failed completely.Mersenne studied cycloids for over 20 years.If you are impatient after a couple of hours where does that leave you?For really difficult problems ,when you explain the result nobody will understand you at all.
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