Inspired by Curling

When the ring is pushed without rotation the equation of motion is simply $\dot u =-\mu g$ . The motion is a simple deceleration from the initial velocity to zero velocity.

For the rotating ring let us consider a small segment of the ring, of length $d\ell=r d\varphi$ at an angle $\varphi$ from the $x$ axis, as indicated in the Figure.

First we determine the direction of the friction force acting on this segment. The $x$ and $y$ components of the velocity are

$v_x=u-\omega r \sin \varphi$

$v_y=\omega r \cos \varphi$

and the components of the friction force are

$d F_x= - \frac{\mu mg}{2\pi} \frac{v_x}{|v|} d\varphi= - \frac{\mu mg }{2\pi} \frac{u-\omega r \sin \varphi}{|v|} d\varphi$

$d F_y= - \frac{\mu mg}{2\pi} \frac{v_y}{|v|} d\varphi= - \frac{\mu mg }{2\pi} \frac{\omega r \cos \varphi}{|v|} d\varphi$

where the absolute value of the velocity is

$|v|=\sqrt{v_x^2+v_y^2}=\sqrt{(u-\omega r \sin \varphi)^2+ \omega r \cos^2 \varphi}=\sqrt{u^2+(\omega r)^2-2u\omega r \sin \varphi}$

The total force is obtained by integration

$F_x= - \frac{\mu mg}{2\pi} \int_{0}^{2\pi}{ \frac{u-\omega r \sin \varphi}{\sqrt{u^2+(\omega r)^2-2u\omega r \sin \varphi}} d\varphi}$

$F_y= - \frac{\mu mg}{2\pi} \int_{0}^{2\pi}{ \frac{\omega r \cos \varphi}{\sqrt{u^2+(\omega r)^2-2u\omega r \sin \varphi}} d\varphi}$

One can divide the integral for $F_y$ into four quadrants and show that each quadrant has a pair where the integrand is the same, except for an opposite in sign. Therefore $F_y=0$ and the center of mass will move along the $x$ axis. The equation of motion for the center of mass is $m\dot{u}=F_x$ or

$\dot{u}= - \frac{\mu g}{2\pi} \int_{0}^{2\pi}{ \frac{\eta-\sin \varphi}{\sqrt{1+\eta^2-2\eta \sin \varphi}} d\varphi}$

where we introduced the dimensionless quantity $\eta=\frac{u}{\omega r}$ .

The torque around the center of mass is $d \tau = dF_y r \cos \varphi -dF_xr\sin \varphi$ or

$d \tau= - \frac{ r \mu mg}{2\pi}\left( \cos \varphi \frac{v_x}{|v|}-\sin \varphi \frac{v_y}{|v|}\right) d\varphi = - \frac{r \mu mg }{2\pi} \frac{\omega r cos^2\varphi - u \sin \varphi +\omega r \sin^2\varphi }{|v|} d\varphi = - \frac{r \mu mg }{2\pi} \frac{\omega r - u \sin \varphi}{|v|} d\varphi$

The total torque is

$\tau= - \frac{r \mu mg }{2\pi} \int_0^{2 \pi} \frac{1-\eta \sin \varphi}{\sqrt{1+\eta^2-2\eta \sin \varphi}} d\varphi$

The equation of motion for the rotation around the center is $I\dot\omega= \tau$ , where $I$ is the moment of inertia, $I=mr^2$ for a ring. We can express the equation of motion as

$\dot\omega r = - \frac{r^2 \mu mg }{2\pi I} \int_0^{2 \pi} \frac{1-\eta \sin \varphi}{\sqrt{1+\eta^2-2\eta \sin \varphi}} d\varphi = - \frac{\mu g }{2\pi } \int_0^{2 \pi} \frac{1-\eta \sin \varphi}{\sqrt{1+\eta^2-2\eta \sin \varphi}} d\varphi$ .

Notice that the integrand in the equation for the rotational motion can be easily transformed into the integrand in the equation for the translational motion using $\eta'=1/\eta$ , because

$\frac{1-\eta \sin \varphi}{\sqrt{1+\eta^2-2\eta \sin \varphi}}=\frac{1/\eta-\sin \varphi}{\sqrt{1+(1/\eta)^2-2(1/\eta) \sin \varphi}}$ .

Let us introduce the function

$f(\eta)= \int_0^{2 \pi} \frac{1-\eta \sin \varphi}{\sqrt{1+\eta^2-2\eta \sin \varphi}} d\varphi$

This function can be expressed in terms of elliptic integrals of second kind. For $\eta=1$ the integral can be done directly, and it yields $f(1)=2/\pi=0.636$ .

It is easy to see that the two equations of motions can be simplified to

$\dot u= -\mu g f(\eta)$

$\dot \omega r = -\mu g f(1/\eta)$

We started with the initial condition of $\eta=1$ . It is clear from the equations of motion that this condition is stable. Although $u$ and $\omega$ will change with time, but the rate of change is the same and the ratio $\eta=u/\omega r$ does not change. The motion is a simple deceleration with an "effective" gravitational acceleration of $g'=g f(\eta=1)= 0.636 g$ . Therefore the distance traveled is $1/0.636$ times the distance traveled in simple translational motion, $s'/s=1/0.636=1.57$ .

Note #1: If $\omega >0$ and $u>0$ but $\omega \ne u/r$ initially the motion is more complicated, but eventually it converges to $\omega =u/r$ before the ring stops. The translational and the rotational motion always stops at the same time.

Note #2: For an arbitrary object with axial symmetry we have $I=\alpha mr^2$ . A curling stone, for example, touches the ice like a ring, but its moment of inertia is different from the moment of inertia of a ring. If we use this more general formula for the moment of inertia, the equations of motion are

$\dot u= -\mu g f(\eta)$

$\dot \omega r = -\frac{\mu g}{\alpha} f(1/\eta)$

One may ask, is it possible to have a motion with constant $\eta$ ? To answer that question, let us look at the time derivative of $\eta$ and make it equal to zero:

$\dot \eta= \frac{d}{dt}\frac {u}{\omega r}= \frac{\dot u}{\omega r}-\frac{u\dot \omega}{\omega^2 r}=0$

This yields the condition $\frac{\dot u}{\dot \omega r}=\frac{u}{\omega r}$ . From the equations of motions we get:

$f(\eta)=\frac{\eta}{\alpha}f(1/\eta)$

First we note that for $\alpha=1$ we indeed have a solution with $\eta=1$ . This is what we had before. In the more general case we can solve the equation numerically, and we find that "stable" $\eta$ exists in the range of $0.5<\alpha<2$ . At the lower end $\eta$ approaches infinity as $\alpha \rightarrow 1/2$ . For $\alpha \rightarrow 2$ , $\eta \rightarrow 0$

Note #3, about curling: The main difference between this problem and the curling stone is that the friction between the ice and the stone has a much more complicated behavior. Also, the curling stone has a height that is comparable to the radius of the circle where it touches the ice. As a result, when the stone decelerates there will be higher pressure in the front relative to the back side of the stone and the friction forces pointing sideways will not cancel each other. This will cause the stone to go sideways ( $y$ direction in our problem). There is a discussion of this in the New Yorker magazine and in Scientific American . Mark Shegelski, a curler and a physicist at the University of Northern British Columbia, published several papers on the subject matter.