A cycloid is a curve traced out by a point p on the rim of a wheel as it rolls along the ground. Using the coordinate system shown in Figure 8.27, show that the curve has the parametric representation

where θ is the angle through which the wheel has turned. Draw a sketch of the curve. Find the gradient of the curve at a general point (x, y). If the wheel rotates at a constant speed, with θ = ωt, where ω is constant and t is the time, show that the speed V of the point on the rim is given by

x = a(0 sin 0), y = a(1 - cos 0)