**Coupled Harmonic Oscillators**

This animation shows two equal masses
attached to springs. The outside springs (k) are the same, and the middle
spring (k_{p}) can be different. The initial positions and velocities of each
mass can be specified.

a)
To excite the Symmetric Normal Mode, use X_{01}=X_{02} and V_{01}=V_{02}.

b)
To excite the Anti-Symmetric Normal Mode, use X_{01}= -X_{02} and V_{01}= -V_{02}.

Try using a very small middle spring
constant (0.4) and a very large outside spring constant (k>20) - and
start only one mass initially ... can you see the motion of that moving
mass transferred to the other mass? That shows the weak coupling. Now try
a very strong inner spring, and a weak outer spring - and move only one
mass. What happens here is an internal vibration that slowly moves from
side to side.

The "X" that is plotted below
the springs, and the black line in the top graph, is the Center of Mass of
the system. Note that since this is a conservative system, the center of
mass motion is uniform (regardless of how the two masses individually
move, the center of mass of the system remains regular in its motion).

On the top right, there is a "configuration
space" diagram. This plots X_{1} on the horizontal against X_{2} on the
vertical. This is usually a "space-filling" type curve unless
you use specific starting values :

a) Try the "normal modes"
above, and see what the configuration space shows.

b) Or, to get a "lissajous"
type figure, try using these values : m=1, k=25, k_{p}=5.5, X_{20}=3,
X_{10}=V_{10}=V_{20}=0 (this will create W_{1}=5 and W_{2}=6). Or, try k=9, k_{p}=3.5.

c) What about m=0.1, k=10, k_{p}=15? And
then there is m=0.1, k=10, k_{p}=40?

What conditions create the "simple" lissajous
figures, and what create the more complicated graphs?

*Note: This physlet seems to do a great deal of
processing .. and occasionally "freezes" - just press the play
button to start it again .. after a few restarts it seems to be fine - I
can't figure out why at the moment.*