Question

23.In a meldes experiment when the tension is 100gm and the tuning fork vibrates at right angles to the direction of the string, the later is thrown into four segments.If now the tuning fork is set to vibrate along the string find what additional weight which will make the string vibrate in one segment.

Answer 1

In Melde's Experiment, the frequency of the tuning fork when the wires are perpendicular to the vibrations will be equal to the vibration of the strings.
Hence
$$\begin{gathered} f=\sqrt{\frac{gM}{4\mu l^2}} \\ Where \\ M=Massofthepanandmasspresentinit \\ l=halfofthedis\tan cebetweentwoconsecutivenodes \\ \mu =massperunitlengthofthestring \end{gathered}$$
When the string is set to vibrate along the tuning fork
$f=\sqrt{\frac{gM}{\mu l^2}}$
Hence
$$\begin{gathered} \sqrt{\frac{g{M}_2}{\mu {l_2}^2}}=\sqrt{\frac{g{M}_1}{4\mu {l_1}^2}} \\ \frac{M_2}{{l_2}^2}=\frac{M_1}{4{l_1}^2} \\ {M}_2=\frac{M_1}{4}{\left(\frac{l_2}{l_1}\right)}^2 \\ {M}_2=\frac{100}{4}{\left(\frac{1}{4}\right)}^2 \\ {M}_2=\frac{100}{64} \\ {M}_2=1.5625gm \end{gathered}$$
[This question is answered with an understanding that the string is cut into four parts and one part is made to vibrate using the same tuning fork longitudinally.]