** Next:** Transverse Standing Waves
** Up:** Coupled Oscillations
** Previous:** Three Spring-Coupled Masses

- A particle of mass
is attached to a rigid support by means of a spring of
spring constant
. At equilibrium, the spring hangs vertically
downward. An identical oscillator is added to this system, the
spring of the former being attached to the mass of the latter.
Calculate the normal frequencies for one-dimensional vertical
oscillations about the equilibrium state, and describe the associated normal modes.

- Consider a mass-spring system of the general form shown
in Figure 15 in which the two masses are of mass
, the two outer springs have spring constant
, and the middle spring has spring constant
. Find the normal
frequencies and normal modes in terms of
and
.

- Consider a mass-spring system of the general form shown
in Figure 15 in which the two masses are of mass
, the two leftmost springs have spring constant
, and the rightmost spring is absent. Find the normal
frequencies and normal modes in terms of
.

- Consider a mass-spring system of the general form shown
in Figure 15 in which the springs all have spring constant
, and the
left and right masses are of mass
and
, respectively. Find the normal
frequencies and normal modes in terms of
and
.

- Find the normal frequencies and normal modes of the coupled LC circuit
shown in the preceding figure terms of
and
.

- Consider two simple pendula with the same
length,
, but different bob masses,
and
. Suppose
that the pendula are connected by a spring of spring constant
.
Let the spring be unextended when the two bobs are in their equilibrium
positions. Demonstrate that the equations of motion of the system (for small
amplitude oscillations) are

where
and
are the angular displacements of the respective
pendula from their equilibrium positions. Show that the
normal coordinates are
and
. Find the normal frequencies and normal
modes. Find a superposition of the two modes such that at
the
two pendula are stationary, with
, and
.

- A linear triatomic molecule (e.g., carbon
dioxide) consists
of a central atom of mass
flanked by two identical atoms of
mass
. The atomic bonds are represented as springs of spring constant
.
Find the molecule's normal frequencies and modes of linear oscillation.

** Next:** Transverse Standing Waves
** Up:** Coupled Oscillations
** Previous:** Three Spring-Coupled Masses
Richard Fitzpatrick
2013-04-08