Question

There are two wires, each produces a frequency of $500\;{\rm{Hz}}$. By what percentage tension in one wire should be increased so that $5$ beats per second can be heard?

Answer 1

Hint: To find the solution, we can use the relation connecting frequency and tension of the wire. Beats per second is the change in frequency.

Complete step by step answer:
Given, the frequency of two wires, $\upsilon = 500\;{\rm{Hz}}$
Beats per second, $\Delta \upsilon = 5\;{\rm{beats per second}}$
First, we express the relation connecting the frequency of the wave travelling through the wire and the tension of the wire. The relation is written as

$\upsilon = \dfrac{1}{{2l}}\sqrt {\dfrac{T}{m}} $

Here $\upsilon $ is the frequency of the wire, $T$ is the tension in the wire, $l$ is the length of the wire and $m$ is the mass of the wire.

Since the mass $m$ and the length $l$ of the wire are constants, the velocity of the wave is directly proportional to the tension in the wire.

$\upsilon \;\propto \;\sqrt T $

Or, we can write

$\begin{array}{l}
\upsilon \;\propto \;{T^{\dfrac{1}{2}}}\\
\upsilon = K{T^{\dfrac{1}{2}}}
\end{array}$

where $K$ is a constant of proportionality.

Now, we have to find the change in the tension.

For that, first let’s see how the change in a quantity raised to a power is calculated.

Let a quantity $z = {x^a}$. Here $a$ is the power.

The relative change is the quantity $z$ can be written as

$\dfrac{{\Delta z}}{z} = a\dfrac{{\Delta x}}{x}$

Using $\upsilon = K{T^{\dfrac{1}{2}}}$, the relative change in the velocity can be written as

$\dfrac{{\Delta \upsilon }}{\upsilon } = \dfrac{1}{2}\dfrac{{\Delta T}}{T}$

So, the relative change in the tension of the wire is

$\dfrac{{\Delta T}}{T} = 2\dfrac{{\Delta \upsilon }}{\upsilon }$

Substituting the values of $\upsilon $ and $\Delta \upsilon $ in the above equation, we get

$\begin{array}{c}
\dfrac{{\Delta T}}{T} = 2 \times \dfrac{5}{{500}}\\
 = \dfrac{2}{{100}}
\end{array}$

Hence, we can write the percentage change in the tension of the wire as

$\begin{array}{c}
\dfrac{{\Delta T}}{T} = \dfrac{2}{{100}} \times 100\\
 = 2\%
\end{array}$

Hence, a percentage change in the tension of the wire can produce $5\;{\rm{beats per second}}$.

Note:
In this question, we made the finding that a change in frequency is caused by a change in tension. It is because the quantities of mass and length are unchanged and the only factor which causes the change in frequency is the tension.