Moving on to 3D

• the bike turns in a circle of radius 20 feet in a counterclockwise direction;
• we let z=0 be the plane of the road;
• the reflector is 1 foot from the center of the wheel.

Question #5

• Let c(t) be the motion of the center of the wheel traveling in around the 20 foot circle. Parametrize c(t) as a three-dimensional curve.
• Find a function g(t) the describes the motion of the reflector relative to the wheel's center. Here are some hints:
  • At time t, the bicycle tire forms the plane spanned by the vertical direction and by a tangent vector to the 20 foot circle.
  • Determine the vertical component of the reflector by arguing that it has the form (0,0, Asin(wt)) where A and w are constants determined by the size and speed of the bicycle.
  • If T(t) is the unit tangent vector to the large circle at time t, then argue that the horizontal component of motion has the magnitude A cos(wt) and is in the direction of T(t).
• The motion of the reflector is therefore given parametrically by c(t)+g(t). Write down this expression and check your answer by using the ParamPlot3D or spacecurve command. An example of the syntax for ParamPlot3D is
  ParamPlot3D([2*cos(t), 2*sin(t), 5*sin(t)], t=0..2*Pi);

Question #6