
269Coupled Oscillations and Normal Coordinates
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Gravity acts vertically downward. Find the frequencies of the system and the normal coor-
dinates, assuming that the motion is only in the vertical direction.
10. A bead of mass m slides on a smooth uniform circular ring of mass M and radius R. If the
ring oscillates vertically under gravity about a xed point on its circumference as shown in
Figure 8.13, nd the frequencies and the normal coordinates.
11. Show that the equations of motion for longitudinal vibrations of a loaded string are of
exactly the same form as the equations for transverse motion.
12. ...