Answer
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Hint:We know the frequency produced by a vibrating string is inversely proportional to the length of the string. $f \propto \dfrac{1}{L}$
By using this relation we can find the length of the string for any frequency.
Complete step-by-step answer:
First we find the relation between length of the string and wavelength of the wave produced.
Let us assume the length of the string is $L$ which tight between two point ${S_1}$, ${S_2}$ as shown in figure then the fundamental tone produced by the string have wavelength $\lambda $ and frequency $f$ velocity of wave in string is $v$. Then
We can clearly see from figure the length of the string equal to the $\dfrac{\lambda }{2}$
$ \Rightarrow L = \dfrac{\lambda }{2}$
Wavelength of fundamental wave produced
$ \Rightarrow \lambda = 2L$ ........... (1)
We know the relation between velocity, wavelength and frequency of wave is
$ \Rightarrow v = f \times \lambda $
So the frequency of wave
$ \Rightarrow f = \dfrac{v}{\lambda }$
Put value of $\lambda $ from eq (1)
$ \Rightarrow f = \dfrac{v}{{2L}}$
So in his step we get the relation between frequency and length of the string. Velocity of the wave is constant for a string. So we can write.
$\therefore f\propto \dfrac{1}{L}$ ................ (2)
Step 2
From equation (2) we can write.
${f_1}\propto \dfrac{1}{{{L_1}}}$ ....... (3)
${f_2}\propto \dfrac{1}{{{L_2}}}$ ...... (4)
Divide (3) by (4)
$ \Rightarrow \dfrac{{{f_1}}}{{{f_2}}} = \dfrac{{{L_2}}}{{{L_1}}}$
$ \Rightarrow {L_2} = \dfrac{{{f_1}}}{{{f_2}}}\left( {{L_1}} \right)$
Now we take value which given in question
${f_1} = 800Hz$ ${L_1} = 50cm$
${f_2} = 1000Hz$
From these values we can calculate the length of string $\left( {{L_2}} \right)$for 1000Hz.
$ \Rightarrow {L_2} = \dfrac{{{f_1}}}{{{f_2}}}\left( {{L_1}} \right)$
$ \Rightarrow {L_2} = \dfrac{{800}}{{1000}}\left( {50cm} \right)$
Further solving it.
$ \Rightarrow {L_2} = 40cm$
So now we get the length of the string which can produce $1000Hz$ frequency.
$\therefore {L_2} = 40cm$
Hence in this question option B is correct.
Note:
We use in above question the velocity of wave in string is constant how it is constant the velocity wave in string is given by $v = \sqrt {\dfrac{T}{m}} $
Where $T \Rightarrow $ tension in the string
$m \Rightarrow $ Mass per unit length of string.
We did not change the string in the above question so we take the velocity of the wave as constant.
By using this relation we can find the length of the string for any frequency.
Complete step-by-step answer:
First we find the relation between length of the string and wavelength of the wave produced.
Let us assume the length of the string is $L$ which tight between two point ${S_1}$, ${S_2}$ as shown in figure then the fundamental tone produced by the string have wavelength $\lambda $ and frequency $f$ velocity of wave in string is $v$. Then
We can clearly see from figure the length of the string equal to the $\dfrac{\lambda }{2}$
$ \Rightarrow L = \dfrac{\lambda }{2}$
Wavelength of fundamental wave produced
$ \Rightarrow \lambda = 2L$ ........... (1)
We know the relation between velocity, wavelength and frequency of wave is
$ \Rightarrow v = f \times \lambda $
So the frequency of wave
$ \Rightarrow f = \dfrac{v}{\lambda }$
Put value of $\lambda $ from eq (1)
$ \Rightarrow f = \dfrac{v}{{2L}}$
So in his step we get the relation between frequency and length of the string. Velocity of the wave is constant for a string. So we can write.
$\therefore f\propto \dfrac{1}{L}$ ................ (2)
Step 2
From equation (2) we can write.
${f_1}\propto \dfrac{1}{{{L_1}}}$ ....... (3)
${f_2}\propto \dfrac{1}{{{L_2}}}$ ...... (4)
Divide (3) by (4)
$ \Rightarrow \dfrac{{{f_1}}}{{{f_2}}} = \dfrac{{{L_2}}}{{{L_1}}}$
$ \Rightarrow {L_2} = \dfrac{{{f_1}}}{{{f_2}}}\left( {{L_1}} \right)$
Now we take value which given in question
${f_1} = 800Hz$ ${L_1} = 50cm$
${f_2} = 1000Hz$
From these values we can calculate the length of string $\left( {{L_2}} \right)$for 1000Hz.
$ \Rightarrow {L_2} = \dfrac{{{f_1}}}{{{f_2}}}\left( {{L_1}} \right)$
$ \Rightarrow {L_2} = \dfrac{{800}}{{1000}}\left( {50cm} \right)$
Further solving it.
$ \Rightarrow {L_2} = 40cm$
So now we get the length of the string which can produce $1000Hz$ frequency.
$\therefore {L_2} = 40cm$
Hence in this question option B is correct.
Note:
We use in above question the velocity of wave in string is constant how it is constant the velocity wave in string is given by $v = \sqrt {\dfrac{T}{m}} $
Where $T \Rightarrow $ tension in the string
$m \Rightarrow $ Mass per unit length of string.
We did not change the string in the above question so we take the velocity of the wave as constant.
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