A hoop of mass $M$ and radius $R$ rolls without slipping along a straight line on a horizontal surface. A point mass $m$ slides without friction along its inner surface and performs small oscillations about its mean position. Find the number of degrees of freedom of the system as an integer.
One coordinate specifies the hoop’s horizontal displacement. Its rotation is then fixed by the no-slip constraint $x=R\phi$, while its centre remains at height $R$. A second independent coordinate specifies the point mass’s angular position on the hoop. The frictionless point is not constrained to rotate with the hoop. Thus two independent generalized coordinates, and hence two degrees of freedom, are required.
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