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).

Similar questions

Can you show me how to solve this?
Engineering Dynamics need help from 4,5,6,7 thank you A ball of mass m is moving along a vertical semi-cylinder of radius R as it is guided by the arm OA. The arm moves in a clockwise direction with a constant angular velocity ω. Assume 0° ≤ Φ ≤ 90°. Neglect any friction. Neglect also the size of the ball and the thickness of the arm. Find the relationship between r, R and θ where r is the distance between O and the ball. Draw a free body diagram of the ball assuming that it is in contact with the cylinder and the arm OA. Write the equations of motion in the (r, θ) coordinate system. Find the normal force acting on the ball by the cylinder for Φ = Φ0. Find the normal force acting on the ball by the bar for Φ = Φ0. Determine the angle Φ at which the ball loses contact with the cylinder. Take m = 1 kg, R = 1.4 m, ω = 0.5 rad/s, and Φ = 60°
Let B be the solid bounded by the surfaces z = x^2+2y^2, x^2+y^2=16, and the xy-plane(distances in cm). If B has a constant mass density of 5g/cm^3, find the moment of inertia of B about the axis through(4,3,2) that is perpendicular to the yz-plane
A particle of mass m is located at x = 1, y = 0,2 = 2. Find the tensor of inertia for the particle relative to the origin. The particle rotates about the z axis through a small angle a <<1 as shown below. Show that the moments of inertia are unchanged to second order in a but the products of inertia can change linearly with a.