Question

The frequency of the fundamental mode of vibration of a stretched string fixed at both ends is\[25Hz\] . If the string is made to vibrate with \[7\] nodes then what is the frequency of vibration? If the length of the string is \[3m\] then what is the frequency of the \[{4^{th}}\] harmonic.

Answer 1

Hint: A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency or constant pitch. If the length of the string is correctly adjusted, the sound produced is a musical tone. Vibrating strings are the basis of string instruments like guitars, cellos, and pianos.

Formula used:
\[l = \dfrac{{\left( {n - 1} \right)\lambda }}{2}\] and \[f' = 4\] (fundamental frequency)
Where, \[l = \]string length, \[n = \]node

Complete step-by-step solution:
Given:\[f = 25Hz\], \[l = 3m\].
As we know for vibration in n nodes, we have
\[l = \dfrac{{\left( {n - 1} \right)\lambda }}{2}\]
While given the fundamental frequency to be \[f = 25Hz\]
So, \[v = 2fl\]
\[ \Rightarrow v = 2 \times 25 \times 3\]
\[ \Rightarrow v = 150m/s\]
When it’s made to vibrate in \[7\] nodes
\[l = \dfrac{{\left( {7 - 1} \right)\lambda }}{2} = \dfrac{{6\lambda }}{2}\]
\[ \Rightarrow \lambda = \dfrac{{2l}}{6} = \dfrac{{2 \times 3}}{6} = 1m\]
Hence the frequency \[f = \dfrac{v}{\lambda } = \dfrac{{150}}{1} = 150Hz\]
Now, when the string is made to vibrate in the \[{4^{th}}\] harmonic
\[f' = 4(fundamental{\text{ }}frequency) = 4 \times 25 = 100Hz\].

Additional information:
The frequency is described as the number of waves that pass a fixed place in the given amount of time. It is denoted by the symbol, f and its unit Hertz can also be called cycles per second.

Note:Fundamental frequency is defined as the lowest resonant frequency of a vibrating object. The vibrating objects have more than one resonant frequency and those used in musical instruments typically vibrate at harmonics of the fundamental. An integer (whole number) multiple of the fundamental frequency is called harmonic. Conical air columns, open cylindrical air columns, and Vibrating strings will vibrate at all harmonics of the fundamental. Cylinders with one end closed will vibrate with only odd harmonics of the fundamental. Vibrations at harmonics produced by vibrating membranes, but also have some resonant frequencies which are not harmonics. It is for this class of vibrators that the term overtone becomes useful and they are said to have some non-harmonic overtones.