Question

a) A uniform wire of length ` and mass per unit length, µ, is stretched between two fixed points ×. A point close to one end is plucked and a pulse travels along the wire at speed v, as shown in Fig. 1, to reflect off the right-hand fixed end.

(i) Sketch a diagram of the reflected pulse.
(ii) After a second reflection from the left-hand end, the pulse would be passing the initial point and traveling in the same direction. At that moment the wire could be plucked again to reinforce the pulse. What is the period of plucking, t(`, v) which would accomplish this?
(iii) Calculate the corresponding frequency of plucking the wire, f1, in terms of v and `.
(iv) If a mechanical vibrator is attached to the wire to produce oscillations of frequency fn = nf1 (which are the harmonics or multiples of the 1st harmonic, f1) what resonant frequencies of the wire will be sustained, in terms of integer n, v and `.
(v) The wire is now plucked near one end whilst a finger is lightly touching the wire at a point 40% of the length from one end. Sketch the resultant amplitude of vibration along the wire, and determine the two lowest frequencies heard in terms of f1.
(vi) The speed of transverse waves along a stretched wire wave is given by v =
qTµ,  with T the tension in the wire. What would be the tension required to sustain the fifth harmonic, f5, of frequency 162 Hz in the wire if its length is 2.3 m and mass is 42 g?

b) In this example we hang the cable vertically and use its own weight to provide the tension. A massive, uniform cable of length L and mass M hangs with its upper end fixed to support and the lower end free. The extension is negligible.

(i) What is the tension at a distance x below the support?
(ii) The cable is shaken slightly at the bottom end. Using the information in part (a) (vi), calculate the ratio of the speeds of the pulse on the cable at points 1/4 of its length from the top end and 1/4 of its length from the bottom end.
(iii) The cable is now plucked at the top end as in Fig. 2 and the pulse travels down the cable. Write an expression for the speed of a pulse traveling down the cable in terms of x, L, and g.
(iv) Now obtain an expression for the time taken, stop for a pulse to travel a distance x down the cable from the support at the top end.
(v) At the same moment that the cable is plucked at the top end, a ball is dropped from rest from the same starting height. At what value of x in terms of L will the ball overtake the pulse traveling down?