If a wheel rolls along a horizontal surface at constant speed, the coordinates of a certain point on the rim of the wheel are x(t) = R[(2wt/T) - sin(2wt/T)] and y(t) = R[l- cos(2wt/T)], where R and T are constants. (a) Sketch the trajectory of the point from t = 0 to t = 2T. A curve with this shape is called a cycloid.
(b) What are the meanings of the constants R and T?
(c) Find the x- and y-components of the velocity and of the acceleration of the point at any time t.
(d) Find the times at which the point is instantaneously at rest. What are the x- and y-components of the acceleration at these times?
(e) Find the magnitude of the acceleration of the point. Does it depend on time? Compare to the magnitude of the acceleration of a particle in uniform circular motion, and = 4'1T2R/T'. Explain your result for the magnitude of the acceleration of the point on the rolling wheel, using the idea that rolling is a combination of rotational and translational motion.