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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. (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).

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. (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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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).
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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