Dynamics | Part IA, 2005

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

r¨=gh,\ddot{\mathbf{r}}=-g \nabla h,

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

Determine the motion of the puck when h(x,y)=αx2h(x, y)=\alpha x^{2}.

Sketch the surface

h(x,y)=h(r)=1r21rh(x, y)=h(r)=\frac{1}{r^{2}}-\frac{1}{r}

as a function of rr, where r2=x2+y2r^{2}=x^{2}+y^{2}. Write down the equations of motion of the puck on this surface in polar coordinates r=(r,θ)\mathbf{r}=(r, \theta) under the assumption that the small slope approximation can be used. Show that LL, 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 E0=12L2/r02E_{0}=\frac{1}{2} L^{2} / r_{0}^{2} if the puck is released at radius r0r_{0} with negligible radial velocity. Determine and sketch r˙2\dot{r}^{2} as a function of rr for this release condition. What condition relating L,r0L, r_{0} and gg must be satisfied for the orbit to be bounded?

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