Stage: 5 Challenge Level: 
This problem consists of a sequence of ideas concerning the configuration of hoop and pole as shown in the diagram below.
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A rigid hoop of radius $r$ is attached to a vertical pole. A light ideal spring of length $L$ with spring constant $k$ is attached at one end at the point of contact of the hoop to the pole. A bead of mass $m$ is attached to the other end of the spring and threaded onto the hoop, so that it can slide around the hoop. The hoop contains a shallow groove so that as the bead moves, the spring stretches around the circumference of the hoop, as in the diagram. |
If the hoop configuration is frictionless, what will be the
possible range of positions of the bead for a stationary
configuration?
What constants in the problem will affect the locations (as
measured by the angular displacement from the horizontal) of
these stationary positions? In particular, will they depend
on the value of $r$? Test your intuition by finding these
values.
How will these results change if there is a coefficient of
friction $\mu$ between the bead and the hoop?
Can values of the constants be found so that the bead can
remain stationary at any point on the hoop? Back up your
conjecture with a mathematical analysis.
Imagine that the bead is stretched to a certain position and
released. How many qualitatively different types of motion
might result?
biology. Equilibrium of a particle. Fluids and aerodynamics. engineering. Rigid Bodies. Resultants and components of forces. Friction. physics. chemistry. Short problems. Forces.
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