Question 270: Waves

Question

Two uniform wires of same material are vibrating under the same tension. If the $1^{	ext {st }}$ overtone of $1^{	ext {st }}$ wire is equal to the $2^{	ext {nd }}$ overtone of $2^{	ext {nd }}$ wire and radius of $1^{	ext {st }}$ wire is twice the radius of $2^{	ext {nd }}$ wire, the ratio of length of $1^{	ext {st }}$ wire to that $2^{	ext {nd }}$ wire is

Accepted Answer

Let the lengths of the first and second wires be $L_1$ and $L_2$.
Let their radii be $r_1$ and $r_2$.
Given:
$r_1 = 2r_2$
Both wires are of
same material
and are under
same tension
.
Frequency of $n^	ext{th}$ overtone
(i.e., $(n+1)^	ext{th}$ harmonic) of a stretched string:
$$ f_n = (n+1)\frac{1}{2L}\sqrt{\frac{T}{\mu}} $$
where
$L$ = length of wire
$T$ = tension (same for both)
$\mu =$ mass per unit length $= \pi r^2 ho$ (as $ho$ is density and $r$ is radius)
Given:
$1^{	ext{st}}$ overtone of $1^{	ext{st}}$ wire = $2^{	ext{nd}}$ overtone of $2^{	ext{nd}}$ wire
So,
$1^{	ext{st}}$ overtone means $n=1$; frequency $= 2\frac{1}{2L_1}\sqrt{\frac{T}{\mu_1}} = \frac{1}{L_1}\sqrt{\frac{T}{\mu_1}}$
$2^{	ext{nd}}$ overtone means $n=2$; frequency $= 3\frac{1}{2L_2}\sqrt{\frac{T}{\mu_2}} = \frac{3}{2L_2}\sqrt{\frac{T}{\mu_2}}$
Set them equal:
$$ \frac{1}{L_1}\sqrt{\frac{T}{\mu_1}} = \frac{3}{2L_2}\sqrt{\frac{T}{\mu_2}} $$
Tension $T$ is same, so cancel it:
$$ \frac{1}{L_1}\frac{1}{\sqrt{\mu_1}} = \frac{3}{2L_2}\frac{1}{\sqrt{\mu_2}} $$
Rewriting:
$$ \frac{1}{L_1} \cdot \frac{\sqrt{\mu_2}}{\sqrt{\mu_1}} = \frac{3}{2L_2} $$
So,
$$ \frac{L_1}{L_2} = \frac{2}{3} \cdot \frac{\sqrt{\mu_2}}{\sqrt{\mu_1}} $$
We know:
$\mu = \pi r^2 ho$
So,
$$ \frac{\sqrt{\mu_2}}{\sqrt{\mu_1}} = \frac{\sqrt{\pi r_2^2 ho}}{\sqrt{\pi r_1^2 ho}} = \frac{r_2}{r_1} $$
Given: $r_1 = 2r_2 \implies r_2 = \frac{r_1}{2}$
Therefore,
$$ \frac{r_2}{r_1} = \frac{\frac{r_1}{2}}{r_1} = \frac{1}{2} $$
Plug this back:
$$ \frac{L_1}{L_2} = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3} $$
So,
the ratio of lengths
$L_1 : L_2 = 1:3$
Answer:
Option A: $1:3$

Answer

$1: 3$

Answer

$3: 1$

Answer

$2: 3$

Answer

$3: 5$

Waves

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