Question

The fundamental frequency of string stretched with a weight of \[4{\rm{ kg}}\] is \[256{\rm{ Hz}}\]. The weight required to produce its octave is
(1) \[4{\rm{ kg wt}}\]
(2) \[12{\rm{ kg wt}}\]
(3) \[16{\rm{ kg wt}}\]
(4) \[24{\rm{ kg wt}}\]

Answer 1

Hint:The meaning of octave frequency is that the final fundamental frequency is twice its initial value. In other words, we can say that the ratio of final and initial frequency is \[2\]. We know that the given string's fundamental frequency is directly proportional to the square root of tension in that string.

Complete step by step answer:
The initial mass attached to the string is \[{m_1} = 4{\rm{ kg}}\].
The fundamental frequency of string when it is loaded with mass \[{m_1}\] is \[256{\rm{ Hz}}\].
Assume:
The mass required to produce octave is \[{m_2}\].
The fundamental frequency of string when it is loaded with mass \[{m_2}\] is \[{f_2}\].
It is given that the ratio of final to the initial fundamental frequency is:
\[\dfrac{{{f_2}}}{{{f_1}}} = 2\]
From the concept of equilibrium, we can write:
\[{T_1} = {m_1}\]
Here \[{T_1}\] is the tension in the string when it is loaded with mass \[{m_1}\].
Substitute \[4{\rm{ kg}}\] for \[{m_1}\] in the above expression.
\[{T_1} = 4{\rm{ kg}}\]
We can write the relation between initial fundamental frequency and respective tension of the string as below:
\[{f_1} \propto \sqrt {{T_1}} \]
Removing the sign of proportionality, we can write:
 \[{f_1} = k\sqrt {{T_1}} \]……(1)
Here, k is the constant of proportionality.
The relation between final fundamental frequency and the respective tension of the string is given as:
\[{f_2} \propto \sqrt {{T_2}} \]
Removing the sign of proportionality, we can write:
 \[{f_2} = k\sqrt {{T_2}} \]……(2)
Here, k is the constant of proportionality.
On dividing equation (1) and equation (2), we get:
\[\dfrac{{{f_2}}}{{{f_1}}} = \sqrt {\dfrac{{{T_2}}}{{{T_1}}}} \]
On substituting \[4{\rm{ kg}}\] for \[{T_1}\] and \[2\] for \[\dfrac{{{f_2}}}{{{f_1}}}\] in the above expression, we get:
\[2 = \sqrt {\dfrac{{{T_2}}}{{4{\rm{ kg}}}}} \]
On squaring both sides, we have,
\[\begin{array}{l}
{2^2} = {\left( {\sqrt {\dfrac{{{T_2}}}{{4{\rm{ kg}}}}} } \right)^2}\\
{T_2} = 16{\rm{ kg}}
\end{array}\]
We know that the string's tension is equal to the mass applied to it, and the value of mass in kilogram-weight is \[16{\rm{ kg wt}}\].
Therefore, the weight required to produce the octave is \[16{\rm{ kg wt}}\], and option (3) is correct.

Note: Do not get confused with the unit of options given in the questions because mass can also be expressed in kilogram-weight. Take extra care while dividing equation (1) and equation (2) to find the relationship between fundamental frequencies and tensions of the string.