Question

A bead of mass m slides frictionlessly on a circle of wire with radius R. The circle stands up in a vertical plane and rotates about the z-axis with constant angular velocity $\omega$. Write down the Lagrangian. Find the equations of motion. For an angular velocity greater than some critical angular velocity $\omega_{c}$, the bead will experience small oscillations about some stable equilibrium point $\theta_{0}$. Find $\omega_{c}$ and $\theta_{0}$($\omega$).

Answer 1

A level of mass m slides frictionlessly on a circle of wire with radius R. Circle stands in vertical plane and rotates about z- axis with constant angular velocity w. (as shown) in the figure. rightarrow Angle between the diameter the is ow and the diameter hoop is spinning around is (theta) therefore In spherical coordinates the bond�s velocity is: - r = rr + r theta theta + r sin theta phivariant phivariant in the case r = R(a costant) so r = 0 and phivariant = w(hoop�s angular velocity) on substitution: - r = R theta theta + R sin theta w phivariant Hence Kinetic energy T = 1/2 m(R^2 theta^2 + R^2w^2 sin^2theta) The potential energy: - on taking line passing through the centre of the hoop in horizontal plane as reference for potential energy. Then potential energy = -mg(R cos theta) = -mg R cos theta Hence Langragins is L = T - V