Question

A weight of the 5 kg required to produce the fundamental frequency of a sonometer wire. What is required to produce its octave?
(A) 20
(B) 32
(C) 26
(D) None of the above

Answer 1

Hint: Sonometer is a device that can be used to produce transverse standing waves and demonstrate the relationship between the frequency of the sound produced by plucking of string and the tension, length and mass per unit length. For production of octaves on the sonometer- wire the necessary condition will be- \[f':f = 2:1\] where $f’$ and $f$ are the octave and fundamental frequency.

Complete step by step solution:
Given
Mass of the load = 5 kg
Formula used: \[f = \dfrac{1}{{2L}}\sqrt {\dfrac{T}{\mu }} \]
Where, f = frequency of the sound wave.
L= length of the wire
T = Tension in the wire and
μ = Mass per unit length of the wire.
For octave: \[f':f = 2:1\]
Since, \[f \propto \sqrt T \] as the wire is the same.
\[\Rightarrow f':f = \sqrt {\dfrac{T’}{T}} = 2:1\]
\[\Rightarrow \dfrac{{T'}}{T} = \dfrac{{m'}}{m} = 4\]
\[ \Rightarrow m' = 4m\]
\[\therefore m' = (4 \times 5)kg = 20kg\]

Hence option (A) is the correct answer.

Note: Velocity of a transverse wave travelling on stretched string of sonometer is given by: \[v = \dfrac{T}{\mu }\] Where T is the tension in the string and μ is the mass per unit length. When the frequency of the applied force is equal to the natural frequency of the sonometer the body vibrates with a very large amplitude with intensity of sound will be maximum. This phenomenon is known as resonance.