Introduction to Classical Mechanics · Problem 3.61

Oscillations — Morin

Physics — Mechanics · kinematics-cycloid

Problem

Rolling wheel * If you paint a dot on the rim of a rolling wheel, the coordinates of the dot may be written as

(x, y) = (R\theta + R\sin\theta,\, R + R\cos\theta). \tag{3.55}

The path of the dot is called a cycloid. Assume that the wheel is rolling at constant speed, which implies

\theta = \omega t

. ((a)) Find

\mathbf{v}(t)

and

\mathbf{a}(t)

of the dot. ((b)) At the instant the dot is at the top of the wheel, what is the radius of curvature of its path? The radius of curvature is defined to be the radius of the circle that matches up with the path locally at a given point. Hint: You know

v

and

a

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