- A horizontal rod rotates with constant angular velocity about
its mid-point . A particle is attached to it by equal-length strings , .
If is the inclination of the plane to the vertical, prove that
- A double pendulum consists of two simple pendula, with one pendulum
suspended from the bob of the other. Suppose that the two pendula have equal lengths, ,
and bobs of equal mass, , and are confined to move in the
same vertical plane.
Let and —the angles that the upper and
lower pendula make with the downward vertical (respectively)—be the
generalized coordinates. Demonstrate that Lagrange's equations of motion for the system
are
- Consider an elastic pendulum consisting of a bob
of mass attached to a light elastic string of stiffness and unstreatched
length . Let be the extension of the string, and the angle that the
string makes with the downward vertical. Assume that any motion is confined to a vertical plane.
Demonstrate that Lagrange's equations of motion for the system are
- 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. Let be the displacement of the center of mass of the disk down the slope, and let
be the angle subtended between the pendulum and the downward vertical.
Demonstrate that Lagrange's equations of motion for the system are
- 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. Let the generalized coordinate be the angle shown in the diagram. Here, is a horizontal Cartesian coordinate,
a vertical Cartesian coordinate, and the center of the hoop. Demonstrate that the equation
of motion of the system is
- The kinetic energy of a rotating rigid object with an axis of symmetry
can be written
- Demonstrate that the components of acceleration in the spherical coordinate system are
(From Lamb 1923.)
- A particle is constrained to move on a smooth spherical surface of radius . Suppose that the particle is
projected with velocity along the horizontal great circle.
Demonstrate that the particle subsequently falls a vertical height
,
where
- 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
*Rayleigh dissipation function*.