Rolling as Translation and Rotation Combined — Question 6

Question 6

A marked tire traces a cycloid rather than a circle

A radius-$R$ wheel rolls right without slip. A mark starts at the ground origin. After a clockwise angle $\phi$, its coordinates are $x=R(\phi-\sin\phi)$ and $y=R(1-\cos\phi)$. The center speed is constant $V$.

1. Find the mark velocity and evaluate it at $\phi=\pi$.

2. Locate the mark after a full revolution and explain the cusp at $\phi=0$.

Question 6 - Solution

A marked tire traces a cycloid rather than a circle

(a) Derivation and calculation

The clockwise angle grows at $\dot\phi=V/R$. Differentiating the given cycloid, $$v_x=R(1-\cos\phi)\dot\phi=\boxed{V(1-\cos\phi)},$$ $$v_y=R\sin\phi\dot\phi=\boxed{V\sin\phi}.$$ At $\phi=\pi$, $\boxed{\vec v=2V\hat x}$: the mark is at the top.

(b) Result and interpretation

After one complete turn, $$\boxed{(x,y)=(2\pi R,0)}.$$ Near contact, the Taylor expansions give $$x\simeq R\phi^3/6,\qquad y\simeq R\phi^2/2.$$ Both velocity components vanish at $\phi=0$. Since $x$ changes cubically while $y$ rises quadratically on either side, the trajectory has a cusp with a vertical limiting tangent.

Physical check

One turn advances the center and successive contact points by the circumference $2\pi R$.