Question One: For a system of n particles and starting from pre-Lagrange equation (the relation between the time derivative of the kinetic energy and the generalized force): Prove that the time derivative of it total angular momentum equals the total external torque (moment of the external force). az Question Two: The energy function h is known as h = 45,-L where L is the Lagrangian. For non conservative system in the presence of frictional force with the Rayleigh dissipation function F + 2 + 2F = 0. prove that at Question Three: Obtain the Lagrange equations of motion for a conical pendulum shown in the Figure using the suitable generalized coordinates and write these equations in a tidy simple form. b
Question Four: For the simple pendulum Shown: If the string is an elastic one with elastic (force constant) C and original (relaxed) length Lo. Set up the Lagrangian Find the equations of motion. Question Five: A solid sphere of mass m and radius a is free to roll (without slipping) inside a hollow cylinder of radius R as shown in the Figure. Set up the Lagrangian. Set up the equations of constraints. Use Lagrange multipliers to: Derive the equations of motion. Find the Forces of constraints. R