- A horizontal rod rotates with constant angular velocity about
its midpoint . A particle is attached to it by equal strings , .
If is the inclination of the plane to the vertical, prove that

where . Deduce the condition that the vertical position of should be stable. - A double pendulum consists of two simple pendula, with one pendulum
suspended from the bob of the other. If the two pendula have equal lengths, ,
and have bobs of equal mass, , and if both pendula are confined to move in the
same vertical plane, find Lagrange's equations of motion for the system.
Use and --the angles the upper and
lower pendulums make with the downward vertical (respectively)--as the
generalized coordinates. Do
not assume small angles.
- The surface of the Diskworld is a disk of radius which
rotates uniformly about a perpendicular axis passing through
its center with angular velocity . Diskworld gravitational acceleration is of magnitude
, and is everywhere directed normal to the disk.
Find the Lagrangian
of a projectile of mass using co-rotating cylindrical polar coordinates as the generalized
coordinates. What are the momenta conjugate to each coordinate? Are
any of these momenta conserved? Find Lagrange's equations of motion for
the projectile.
- Find Lagrange's equations of motion for an elastic pendulum consisting of a particle
of mass attached to an elastic string of stiffness and unstretched
length . Assume that the motion takes place in a vertical plane.
- A disk of mass and radius rolls without slipping down a plane inclined at an angle to the horizontal.
The disk has a short weightless axle of negligible radius. From this axle is suspended a simple pendulum
of length whose bob is of mass . Assume that the motion of the pendulum takes place in the
plane of the disk. Find Lagrange's equations of motion of the system.
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Chapter09/fig9.04.eps - A vertical circular hoop of radius is rotated in a vertical plane about a point on its
circumference at the constant angular velocity . A bead of
mass slides without friction on the hoop. Find the kinetic energy, the potential energy, the Lagrangian, and Largrange's equation
of motion of the bead, respectively, in terms of the angular coordinate
shown in the above diagram. Here, is a horizontal Cartesian coordinate,
a vertical Cartesian coordinate, and the center of the hoop.
Show that the beam oscillates like a pendulum about the point on the
rim diagrammatically opposite the point about which the hoop
rotates. What is the effective length of the pendulum?
- Consider a spherical pendulum of length . Suppose that the string is initially horizontal, and the bob is rotating horizontally with tangental velocity . Demonstrate that, at its lowest subsequent point, the bob will have fallen a vertical height
,
where

Show that if is large compared to then this result becomes approximately . - The kinetic energy of a rotating rigid object with an axis of symmetry
can be written

where is the moment of inertia about the symmetry axis, is the moment of inertia about an axis perpendicular to the symmetry axis, and , , are the three Euler angles. Suppose that the object is rotating freely. Find the momenta conjugate to the Euler angles. Which of these momenta are conserved? Find Lagrange's equations of motion for the system. Demonstrate that if the system is precessing steadily (which implies that , , and are constants) then

- Consider a nonconservative system in which the
dissipative forces take the form
, where the
are Cartesian coordinates, and the are all positive. Demonstrate that
the dissipative forces can be incorporated into the Lagrangian formalism
provided that Lagrange's equations of motion are modified to read

where

is termed the Rayleigh Dissipation Function.