Assignment 9

Problem 1:

A point particle with mass m is restricted to move on the inside surface of a horizontal ring.  The radius of the ring increases steadily with time as R = R0 + R't.
At t = 0 the speed of the particle is v0.
(a)  Are there any constants of motion?
(b)  Let E(t1) be the energy of the particle at the time the radius of the ring is 2R0 and E0 its energy at t = 0. 
Find the ration E(t1)/E0.

Problem 2:

A particle of mass m is constrained to move along the surface of a cone with half opening angle α under the influence of gravity.  The cone is oriented with its apex pointing down, and its symmetry axis along the direction of gravity, g.

image

(a)  Find the Lagrangian of the particle using (r,θ) as generalized coordinates, where r denotes the perpendicular distance from the cone axis to the particle, and θ is the azimuthal angle (see figure).
(b)  Calculate the generalized momenta pr and pθ that are conjugate to the coordinates r and θ, respectively.  Write down the Hamiltonian H(r, θ, pr, pθ) in terms of the coordinates and conjugate momenta.
(c)  Using the Hamiltonian from part (b), find the four corresponding Hamilton's equations of motion for dr/dt, dθ/dt, dpr/dt, and dpθ/dt.
(d)  Determine which, if any, of the following quantities are conserved: pr, pθ, and H.
Explain why the conservation follows from the form of the Hamiltonian, and provide a physical interpretation of any conserved quantities.

Problem 3:

A simple pendulum of mass m and length l is free to swing in the xy-plane.  It is fixed to a support of mass M that can move freely in the horizontal direction.  Using the coordinates x, the horizontal position of the support, and θ, the angle of the pendulum from vertical, write down the Lagrangian for this system.

Problem 4:

The Lagrangian for the system below can be written as

 image

where xi denotes the displacement of mass mi from its equilibrium position.

image

(a)  Find the matrices T and K.
(b)  Find the eigenvalues and eigenvectors for the oscillations of this system.
(c)  Write down the general solution for x1(t) and x2(t).
(d)  For the initial conditions

image
solve for the constants in your answer for (c) and determine for x1(t) and x2(t) for these initial values.

Problem 5: 

A particle of mass m moves under the influence of gravity on the inner surface of a paraboloid of revolution x2 + y2 = az which is assumed frictionless.
(a)  Introduce cylindrical coordinates ρ, φ, z.  Write a Lagrangian for the system employing ρ and φ as generalized coordinates.
(b)  Find two constants of motion.
(c)  Obtain the equations of motion.
(d)  Show that the particle will describe a horizontal circle in the plane z = h provided that it is given the proper angular velocity ω.  What is the magnitude of this velocity?