Question

A wheel of radius R rolls to the right with an angular velocity w.

a) write the parametric equations that express the movement of a spot on the wheel that is a distance r<R from the center. Assume the point is at its maximum height at time t=0.

b) suppose the wheel is pushed up a ramp that makes an angle X with the horizontal. Because it is being pushed it maintains its angular velocity w. Write the parametric equations tha express the movement of a spot on the wheel that is a distance r from the center

Answer 1

a) Radius =R

Velocity =W

Distance from the centre = r<R

Any point on the curve has to combine these two motions - the linear motion of the center moving forward and the rotational motion of the point around the center.

The rotational motion around the center is given by (−2sint2,−2cost2) compared to the center of the circle during this time. Note the signs are such as to give the correct starting position and direction of rotation around the center.

Putting these together gives position, p¯¯¯(t) as:

p¯¯¯(t)=(2t,4)+(−2sint2,−2cost2)

p¯¯¯(t)=(2t−2sint2,4−2cost2

b) Distance from centre = r

X = R(t-sint) =. Distance travelled by center (arc length) - adjustment to spot on edge

= R - Rsint.

y = R(1-cost) = R- Rcost

X (t =π) = R π - Rsinπ = Rπ

y (t = π) = R - Rcosπ = R-(-R) = 2R