Central Forces 2023 (3 years)
(Synthesis Problem: Brings together several different concepts from this unit.) Use effective potential diagrams for other than \(1/r^2\) forces.
Consider the frictionless motion of a hockey puck of mass \(m\) on a perfectly
circular bowlshaped ice rink with radius \(a\). The central region of the bowl
(\(r < 0.8a\)) is perfectly flat and the sides of the ice bowl smoothly rise to a
height \(h\) at \(r = a\).

Draw a sketch of the potential energy for this system. Set the zero of
potential energy at the top of the sides of the bowl.

Situation 1: the puck is initially moving radially outward from the
exact center of the rink. What minimum velocity does the puck need
to escape the rink?

Situation 2: a stationary puck, at a distance \(\frac{a}{2}\) from the
center of the rink, is hit in such a way that it's initial velocity
\(\vec v_0\) is perpendicular to its position vector as measured from
the center of the rink. What is the total energy of the puck
immediately after it is struck?

In situation 2, what is the angular momentum of the puck immediately after it is struck?

Draw a sketch of the effective potential for situation 2.

In situation 2, for what minimum value of \(\vec v_0\) does the puck
just escape the rink?