A tuning fork of frequency $392 Hz,$ resonates with $50 cm $ length of a string under tension ($T$). If length of the string is decreased by $2\%$, keeping the tension constant, the number of beats heard when the string and the tuning fork made to vibrate simultaneously is

The frequency of a sonometer wire is $f$, but when the weights producing the tensions are completely immersed in water the frequency becomes $f/2$ and on immersing the weights in a certain liquid the frequency becomes $f/3$. The specific gravity of the liquid is: 

In Melde's experiment, the string vibrates in $4$ loops when a $50 \,gram$ weight is placed in the pan of weight $15\, gram.$ To make the string to vibrates in $6$ loops the weight that has to be removed from the pan is

To increase the frequency from $100 Hz$ to $400 Hz$ the tension in the string has to be changed by ..... $times$

Two perfectly identical wires are in unison. When the tension in one wire is increased by $1\%$, then on sounding them together $3$ beats are heard in $2 \,sec$. The initial frequency of each wire is .... ${\sec ^{ - 1}}$

A string is stretched between fixed points separated by $75.0\ cm$. It is observed to have resonant frequencies of $420\ Hz$ and $315\ Hz.$ There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is  .... $Hz$