Question

The frequency of vibration of sonometer wire is directly proportional to the linear density of the wire:
a)True
b)False

Answer 1

The sonometer is nothing but a wooden box with a long string that is used to suspend the load. The frequency of vibration of the string depends on the length, tension in the string and the linear density of the string as well. Hence knowing the exact relation between the frequency of the sonometer wire and the linear density of the wire will enable us to select the correct alternative.
Formula used:
$f=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu }}$

Complete step-by-step solution:
A sonometer is a device that is used to verify the relationship between the wavelength and the frequency of the sound waves. The sonometer works on the principle of transverse vibrations of a string when plucked.
Let us say we fix the length of the sonometer wire as ‘L’, the tension in the string is ‘T’ and the mass per unit length of the wire be $\mu$. Then the frequency (f) of vibration of the wire is given by,
$f=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu }}$
If we observe the above expression, the frequency of vibration of the wire is directly proportional to the tension in the string and inversely proportional to the mass per unit length and the total length of the wire.
Therefore the above statement is false and hence the correct answer of the above question is option b.

Note: Mass per unit length of a wire is an intrinsic property. Hence different materials will have different
$\mu$. When the frequency of the applied source is equal to the natural frequency of the sonometer wire for given conditions, the wire vibrates with maximum frequency due to resonance. Hence when conducting the experiment for a particular source frequency, the amplitude of vibration is maximum.