# 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!