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Found problems: 1

1952 Putnam, B4
A homogeneous solid body is made by joining a base of a circular cylinder of height $h$ and radius $r,$ and the base of a hemisphere of radius $r.$ This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if $r$ is large as compared to $h$, the equilibrium will be stable; but if $r$ is small compared to $h$, the equilibrium will be unstable. What is the critical value of the ratio $r\slash h$ which enables the body to rest in neutral equilibrium in any position?
hemispheres ratio cylinder algebra coordinate method