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In Melde’s experiment, in parallel position when mass \[{m_1}\] is kept in the pan, then the number of loops obtained is ${p_1}$ and when mass ${m_2}$ is kept the number of loops is ${p_2}$; then the mass of pan ${m_0}$ is
A. \[{m_0} = \dfrac{{p_1^2 - p_2^2}}{{{m_2}p_2^2 - {m_1}p_1^2}}\]
B. \[{m_0} = \dfrac{{{m_2}p_2^2 - {m_1}p_1^2}}{{p_1^2 - p_2^2}}\]
C. \[{m_0} = \dfrac{{{m_2}p_2^2 + {m_1}p_1^2}}{{p_1^2 - p_2^2}}\]
D. \[{m_0} = \dfrac{{{m_2}p_2^2 - {m_1}p_1^2}}{{p_1^2 + p_2^2}}\]

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Last updated date: 25th Jul 2024
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Answer
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Hint:Melde’s experiment consists of a light string which is tied to one of the prongs of a tuning fork that is mounted on a sounding board whereas the other end of the string is passed over a pulley which is horizontal and a pan which is lighter is suspended from the free end. By changing the weight placed in the pan the tension in the string can be adjusted while by moving the pulley towards or away from the fork the vibrating length can be altered.

Complete step by step answer:
In Melde’s experiment, the frequency of the tuning fork is given by
$N = \dfrac{p}{l}\sqrt {\dfrac{T}{M}} $ ………….. $\left( 1 \right)$
Where, $T = $ Tension, $M = $ Mass per unit length, $p = $ Number of loops and $N = $ Frequency.
Case 1:
$N = \dfrac{{{p_1}}}{l}\sqrt {\dfrac{{\left( {{m_0} + {m_1}} \right)g}}{M}} $ ……….. $\left( 2 \right)$
Case 2:
$N = \dfrac{{{p_2}}}{l}\sqrt {\dfrac{{\left( {{m_0} + {m_2}} \right)g}}{M}} $ ……….. $\left( 3 \right)\\$
Comparing equation $\left( 2 \right)$ and equation $\left( 3 \right)$
$\dfrac{{{p_1}}}{l}\sqrt {\dfrac{{\left( {{m_0} + {m_1}} \right)g}}{M}} = \dfrac{{{p_2}}}{l}\sqrt {\dfrac{{\left( {{m_0} + {m_2}} \right)g}}{M}} \\$
On simplifying and squaring on both sides above equation becomes
\[{\left( {{p_1}} \right)^2}\left( {{m_0} + {m_1}} \right) = {\left( {{p_2}} \right)^2}\left( {{m_0} + {m_2}} \right)\\\]
On simplifying above equation
\[{\left( {{p_1}} \right)^2}{m_0} + {\left( {{p_1}} \right)^2}{m_1} = {\left( {{p_2}} \right)^2}{m_0} + {\left( {{p_2}} \right)^2}{m_2}\\\]
Taking ${m_0}$ terms to L.H.S
\[\left( {p_1^2 - p_2^2} \right){m_0}_{} = {\left( {{p_2}} \right)^2}{m_2} - {\left( {{p_1}} \right)^2}{m_1}\\\]
\[\therefore {m_0} = \dfrac{{{{\left( {{p_2}} \right)}^2}{m_2} - {{\left( {{p_1}} \right)}^2}{m_1}}}{{\left( {p_1^2 - p_2^2} \right)}}\]

Hence, option B is correct.

Note:The fork is adjusted in such a way that its arms are transverse or perpendicular position to the length of the string. By gently hammering a prong the fork is set into vibration.The loops are formed due to the wave reflected back from the pulley and the wave starting from the fork and it will travel towards the pulley.