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10 Nov 2019
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 Ï.
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 Ï is greater than a certainvalue Ïc then there is an equilibrium point at anon-zero θc. Find Ïc and θc.
f. Is θ = θc a stableequilibrium point? Explain.
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 Ï.
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 Ï is greater than a certainvalue Ïc then there is an equilibrium point at anon-zero θc. Find Ïc and θc.
f. Is θ = θc a stableequilibrium point? Explain.