In the early seventeenth century Mersenne (1588–1648) conducted experiments with long lengths of rope and so obtained the law for the frequency of transverse vibrations of strings. Assuming that the frequency depends on products of powers of T, the tension in the rope, l, the length of the rope, and m, the mass per unit length of the rope,
i) find, by dimensional analysis, the form of the relationship. A rope of length 24 m and mass 0.5 kg m-1 under tension of 72 N is found to vibrate with a frequency of of a cycle per second.
ii) State the exact relationship between the frequency, T, l and m.
iii) Find the frequency of vibration of a string of length 20 cm and mass 0.005 g cm-1 under a tension of 8 105 dynes. (The dyne is the cgs unit of force: 1 dyne is the force required to give a mass of 1 g an acceleration of 1 cm s-2 .)