e moment of inertia
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Transcribed Image Text:Problem 5
A homogeneous ring of mass m and radius R is able roll without slipping on a horizontal plane. The centre
of the ring G is connected to a fixed point O on the plane by a spring with an elastic constant k and a rest
length of zero. See the figure below for a representation of the system.
X
(a) Write down the moment of inertia of the ring about an axis orthogonal to it and passing through G.
(b) Write down the Lagrangian of the system and the Euler-Lagrange equation(s).
(c) Find the frequency of the small oscillations around the equilibrium point.
Hint: Taylor-expand the Lagrangian up to second order around the equilibrium point.
(d) At time t = 0 the centre of the ring moves with velocity equal to v, and the spring length is equal to 1.
Determine the position of the centre of mass G as a function of time, i.e. determine x(t).
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