(Lateral vibration of hanging rope) Consider a flexible rope or chain that hangs from the...

(Lateral vibration of hanging rope) Consider a flexible rope or chain that hangs from the ceiling under the sole action of gravity (see the accompanying sketch). If we pull

the rope to one side and let go, it will oscillate from side to side in a complicated pattern which amounts to a superposition of many different modes, each having a specific shape Y (x) and temporal frequency w. It can be shown (from Newton’s of second law of motion ) that each shape Y(x) is governed by the differential equation

where p is the mass per unit length and g is the acceleration of gravity.

Use computer software to obtain the zeros quoted above (2.405, 5.520, 8.654), and to obtain computer plots of the three mode shapes. (Set A = 1, say.)