Central Forces 2022
(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 bowl-shaped 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 \(r = a\).
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Sketch the potential energy for this system (just the potential energy, not the effective potential). Set the zero of
potential energy at the top of the sides of the bowl.
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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?
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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?
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In situation 2, what is the angular momentum of the puck immediately after it is struck?
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Draw a sketch of the effective potential for situation 2.
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In situation 2, for what minimum value of \(\vec v_0\) does the puck
just escape the rink?