Lagrange's Equation of Motion Q.3 A double pendulum consists of two particles supended by massless rods. Assuming that all motion is in a vertical plane, find Lagrange's equations of motion. Linearize these equations, assuming small (angle approximations) motions. Q.4 Consider a particle of mass m attached to a fixed point by a massless string or rod of length l. The particle is free to swing in any direction under the action of gravity. Since the particle is constrained to move on the inner surface of a sphere, this system is called a spherical pendulum. Find its differential equations of motion. Q.5 A pendulum consists of a mass m and a massless stick of lenght l. The pendulum support oscillates horizontally with a position given by x(t) = Acos(wt). What is the general solution for the angle of the pendulum as a function of time? Q.6 Consider a frictionless horizontal hoop of radius r in which a bead can slide freely. The center of the hoop further travels in a horizontal circle of radius R, about a point with constant angular frequency w. Find the equation of motion for the bead. Also find out the frequency of small oscillations about the equilibrium point.