Three resonant frequencies of a string are 90, 150 and 210 Hz. (a) Find the highest possible fundamental frequency of vibration of this string. (b) Which harmonics of the fundamental are the given frequencies ? (c) Which overtones are these frequencies ? (d) If the length of the string is 80 cm, what would be the speed of a transverse wave on this string ?

Answer

A, B, C, D

Explanation

The resoN/Ant frequencies of a string are
`f_1=90Hz`
`f_2=150Hz`
`f_3=210Hz`
a. The highest possible fundamental frequency of the string is
`f=30Hz`
[because `f_1,f_2 and f_3 ` are integral multiples of `30 Hz`]
b. The frequencies are
`f_1=3f`
`f_2=5f`
`f_3=7f`
so, `f_1, f_2` and `f_3` are 3rd harmonic, `5`th harmonic and `7`th harmonic respectively.
c. The frequecy in the string are `f, 2f, 3f, 4f, 5f,`.........
So, `3f=2`nd overtone and `3rd harmonic`
`5f=4`th overtone and `5th harmonic` l `7th=6th` overteone and 7th harmonic.
d. Length of the string is
`L=80cm`
`rarr f_1=(3/2)v`
(v=velocity of the wave)
`rarr 90={3/(2xx80)}xxK`
`=30xx2xx80`
`=4800(cm)/sec`
`=48m/s`