Question

What is the percentage change in the tension necessary in a sonometer of fixed length to produce a note octave lower (lower half of original frequency) than before?
A) 25%
B) 50%
C) 67%
D) 75%

Answer 1

Hint: Velocity v is directly proportional to the square root of the tension produced in a sonometer.
$v$ ∝ $\sqrt T $ (where v is the velocity and T is the tension.)
We also have the relation:
 f ∝ v (frequency is directly proportional to the velocity of the sonometer.)
Using the above two relations we will calculate the percentage change in the tension of the sonometer.

Complete step by step solution:
Let’s define the sonometer and note octave first.
A sonometer is a device which shows the relationship between frequency of a sound produced when we use it to plug the string, and the tension, length and mass per unit length of the string. It is an ancient laboratory instrument.

Note: Any musical sound which is produced by the simple harmonics oscillation of the source is called note:
Octave: The tone of the musical sound having double the frequency of the fundamental frequency is called octave.
Now comes to the calculation part:
We have the relation:
$v$ ∝$\sqrt T $............ (1)
Let’s take $v_1$ be the velocity one and $v_2$ be the velocity two, similarly tension as $T_1$ and $T_2$.
We also have the relation as:
f ∝ v
Therefore, from the given question we bring out the relation because the note produced in octave lower than before.
${v_1}$ = ${2v_2}$ ............... (2)
using equation 1 and 2 we have
$
   \Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{{{T_1}}}{{{T_2}}}} \\
   \Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = 2 \\
 $ ($v_1 = 2 v_2$)
$
   \Rightarrow \sqrt {\dfrac{{{T_1}}}{{{T_2}}}} = 2 \\
   \Rightarrow \dfrac{{{T_1}}}{{{T_2}}} = 4 \\
   \Rightarrow {T_2} = \dfrac{{{T_1}}}{4} \\
 $ (removing the square root from both sides and writing $T_2$ terms of $T_1$)
Percentage change:
$
   \Rightarrow \dfrac{{{T_1} - {T_2}}}{{{T_1}}} \times 100 \\
   \Rightarrow \dfrac{{{T_1} - \dfrac{{{T_1}}}{4}}}{{{T_1}}} \times 100 \\
   \Rightarrow \dfrac{{\dfrac{{3{T_1}}}{4}}}{{{T_1}}} \times 100 \\
   \Rightarrow \dfrac{3}{4} \times 100 \\
   \Rightarrow 75\% \\
 $ (putting the value of $T_1$ and cancel $T_1$ both numerator and denominator)
Percentage change is 75%.

Hence, Option (D) is correct.

Note: Sonometer is a diagnostic instrument used for measuring frequency, tension, and density of the vibrations which are generated when sound is produced. Sonometer is also used in the medical field for tests of both hearing and bone density. It is also used in green chemistry.