# What causes angular and linear deceleration in a sliding and rotating ring?

I was solving an AP Physics problem involving a ring sliding and rotating over a frictional surface. When I started to think about why the ring eventually comes to a stop I started to become confused.

The linear velocity of the ring causes to it experience a kinetic frictional force which opposes the relative motion between the hoop and the ground. This frictional force will act horizontally and hence a torque is applied to the ring. I do not understand how the torque decreases the ring's linear velocity, instead I think it will only decrease the angular velocity. If this was the case however the ring would not come to a stop and I know this can not be the case because of what I have observed in the world!

The definition of the scenario was: A ring of mass $M$, radius $R$, and rotational inertia $MR^2$ is initially sliding on a frictionless surface at a constant velocity $v_0$ to the right. At time $t = 0$ it encounters a surface with coefficient of friction $\mu$ and begins sliding and rotating. After traveling a distance $L$, the ring begins rolling without sliding.

In attempting the first question which asks for the linear velocity $v$ of the ring as a function of time $t$ I did the following.

$$F_{kf} = \begin{bmatrix} -\mu * F_n \\ 0 \end{bmatrix}$$

This force acts as a torque on the ring.

$$\tau = I \alpha$$ $$R * (-\mu * F_n) = (MR^2) \alpha$$ $$\frac{-\mu * g}{R} = \alpha$$

At this point I thought, perhaps the angular deceleration in turn causes the linear velocity to decrease. I decided that I don't think this is the case because say you pushed the ring forward in space then applied a torque, the linear velocity would be not be effected. At this point I started thinking, what force could contribute to the linear deceleration of the ring? In the end I was unable to figure out what was causing the ring to decelerate.

Any help understanding the physics behind the rings motion would be a great help!

## 1 Answer

The force due to sliding friction causes the center of mass to decelerate - since it is the only external force acting on the ring, and we know the center of mass moves as though all forces act there. You might find this earlier answer of mine and the links therein helpful.

As for the "in the real world the ring will stop completely" part of your question - this is due to rolling friction which is a fairly subtle phenomenon usually beyond the scope of AP physics. But you can think of it as the ring digging itself a very small hole in the surface it rests on, and always trying to roll 'uphill' to get out of the hole. I address some of that in this answer

• Thanks @Floris I wasn't aware of rolling friction, I suppose then the question can't be answered unless you know the coefficient of sliding friction as well as a laster question asks for a formula for the angular velocity over time). – HennyH Apr 17 '15 at 15:54
• I suspect that this question is ignoring the existence of rolling friction, and assumes that once the ring rolls without slipping, its (angular) velocity will be constant. That is the usual simplification for problems like this, which makes them solvable. – Floris Apr 17 '15 at 15:56