Question

Consider two identical masses m₁, and another massm₂ attached as seen below. The mass m₂ canslide up and down freely while each m₁ mass isconnected, through friction-free hinges and thin massless rods, tomass m₂ below it and, above it, to a fixed point on thethick shaft. The whole setup is rotating with a constant angularvelocity ω.

a. Find the Lagrangian, L, of thismotion

b. Find the Hamiltonian, H

c. Is H equal to the energy E = T +U? Is H conserved? Is E conserved?

d. Find the effective potential V(θ)and the effective kinetic energy K(θ) so that L = K - V and H = K+V

e.When ω is greater than a certainvalue ω_c then there is an equilibrium point at anon-zero θ_c. Find ω_c and θ_c.

f. Is θ = θ_c a stableequilibrium point? Explain.