Q31 of 36 Page 110

Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops and 4 loops, the frequencies are in the ratio 1:2:3:4.

When a string is fixed at two ends then the equation for a wave on the sting is given by,


for a node y = 0, for which




For the string of length l, let one the position of one end be x = 0, then the position of other end will be x = L, therefore,



and the frequency



The frequencies for different values of n are called harmonics, for e.g. n =1 is called the first harmonic, n = 2 is the second harmonic and so on. Theses harmonics correspond to the number of loops for the vibrations. Thus the frequencies corresponding 1 loop, 2 loops, 3 loops and 4 loops are






respectively.


Hence these frequencies are in the ratio 1:2:3:4.


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29

The pattern of standing waves formed on a stretched string at two instants of time are shown in Fig. 15.3. The velocity of two waves superimposing to form stationary waves is 360 ms-1 and their frequencies are 256 Hz.


(a) Calculate the time at which the second curve is plotted.


(b) Mark nodes and antinodes on the curve.


(c) Calculate the distance between A′ and C′.


 30

A tuning fork vibrating with a frequency of 512Hz is kept close to the open end of a tube filled with water (Fig. 15.4). The water level in the tube is gradually lowered. When the water level is 17cm below the open end, maximum intensity of sound is heard. If the room temperature is 20° C, calculate

(a) speed of sound in air at room temperature


(b) speed of sound in air at 0° C


(c) if the water in the tube is replaced with mercury, will there be any difference in your observations?



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 33

If c is r.m.s. speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/v is constant and independent of temperature for all diatomic gases.