Sol.

After a laps of time t, the center C of the wheel will have traveled a distance as shown, and since it rolls without slip, the arc DA will also the length v 0 t. Thus the angle DCA will be v o t/r. Then from the geometry of the figure, we can express the coordinates x and y of the point A flows :

x = v 0 t – r sin

y = r – r cos ……(o)

with t as a parameter, these two equations define in rectangular coordinates path of point A which is called a cycloid.

Differentiating Eqs.(o) with respect to time gives the velocity-time equations as follows :

= v 0

……(p)

Using now the first of Eqs. for the magnitude of the resultant velocity, we find

v = v 0 = 2 v 0 sin ……(q)

From this expression we see that the maximum speed of point A is 2 v 0 when t = π r/ v 0 , that is, when point A is a the top A ′ of its path (see in figure)

Differentiating Eqs.(p) again with respect to time, we obtain the acceleration-time equations

……(r)

Then using the first of eq. (g) for the magnitude of the resultant acceleration

a = ……(s)

Thus the point A has acceleration of constant magnitude always directed toward the center C of the rolling wheel, as can be established from the last two of Eqs. (g)