Homework 8    

Homework 8
1) A particle of mass m can slide freely along a wire AB whose perpendicular distance to the origin
O is h. The line OC rotates about the origin at constant angular velocity    . The position of
the particle can be described in terms of the angle  and distance q to the point C. If the particle
is subject to a gravitational force, and if the initial conditions are
 (0)  0,
q(0)  0,
q(0)  0
show that the time dependence of the coordinate q is
q(t ) 
g
2 2
(cosh(t )  cos(t ))
Sketch this result. Compute the Hamiltonian for the system, and compare with the total energy. Is
the total energy conserved?
2) A particle is constrained to move (without friction) on a circular wire rotating with constant
angular speed  about a vertical diameter. Find the equilibrium position of the particle, and
calculate the frequency of small oscillation around this position. Find and interpret physically a
critical angular velocity   c that divides the particle’s motion into two distinct types.
Construct phase diagram for the two cases   c and   c .
3) A particle of mass m moves under the influence of gravity along the helix z  k , r  const. ,
where k is a constant and z is vertical. Obtain the Hamiltonian equations of motion.
4) (a) Formulate the canonical equations including friction.
A hanger of mass m for a car with a spring-damping system is moving along the x-direction with
constant velocity v. It performs sinusoidal oscillations in y according to: ys  a cos(
masses of both tires can be ignored.
2
x) . The
l
(b) Formulate the equation of motion for the y-coordinate of the hanger using the canonical
equations.
(c) How does the spring constant D and the friction coefficient c have to be dimensioned to ensure
that the hanger is moving as quietly as possible?
5)
Proof the Jacobi identity:
[f, [g, h]] + [g, [h, f]] + [h, [f, g]] = 0