4.II.11C

A puck of mass $m$ located at $\mathbf{r}=(x, y)$ slides without friction under the influence of gravity on a surface of height $z=h(x, y)$. Show that the equations of motion can be approximated by

$\ddot{\mathbf{r}}=-g \nabla h,$

where $g$ is the gravitational acceleration and the small slope approximation $\sin \phi \approx \tan \phi$ is used.

Determine the motion of the puck when $h(x, y)=\alpha x^{2}$.

Sketch the surface

$h(x, y)=h(r)=\frac{1}{r^{2}}-\frac{1}{r}$

as a function of $r$, where $r^{2}=x^{2}+y^{2}$. Write down the equations of motion of the puck on this surface in polar coordinates $\mathbf{r}=(r, \theta)$ under the assumption that the small slope approximation can be used. Show that $L$, the angular momentum per unit mass about the origin, is conserved. Show also that the initial kinetic energy per unit mass of the puck is $E_{0}=\frac{1}{2} L^{2} / r_{0}^{2}$ if the puck is released at radius $r_{0}$ with negligible radial velocity. Determine and sketch $\dot{r}^{2}$ as a function of $r$ for this release condition. What condition relating $L, r_{0}$ and $g$ must be satisfied for the orbit to be bounded?

*Typos? Please submit corrections to this page on GitHub.*