2.3.33∗. The body slides along a flat surface that smoothly passes into another flat surface located at an angle $\alpha$ to the first one. Coefficient of friction $\mu$. Determine the kinetic energy at the end of the interface section if it is equal to $K_0$ at the beginning.
Solution
For problem $2.3.33$
Assume that the speed $v$ of the body is much larger than $\sqrt{gr}$, where $r$ the radius of curvature of the bend, so that the gravitational field $g$ has negligible effect on centripetal acceleration. Consequently, the normal force
$$N=\frac{mv^2}{r}=\frac{2K}{r},$$
where $K$ is the kinetic energy of the body. The change in kinetic energy is due to the work done by friction, i.e.
$$dK=-\mu Nr\,d\theta=-2\mu K\,d\theta,$$
where $d\theta$ is a small angle subtended by a small arc of the bend. Integrating, we have $K(\theta)=K_0\cdot e^{-2\mu\theta}$. In the figure, the angle subtended by the bend is $\alpha$, so $K=K_0\cdot e^{-2\mu\alpha}$ at the end of the bend.
