An organ pipe $P_1$ closed at one end has a vibrating air column in its first overtone,and another pipe $P_2$ open at both ends has a vibrating air column in the third overtone. Both are in resonance with a given tuning fork. The ratio of the length of pipe $P_1$ to that of $P_2$ is

$A$ resonance air column of length $20 \ cm$ resonates with a tuning fork of frequency $250 \ Hz$. The speed of sound in air is .... $m/s$.

$A$ resonance tube is old and has a jagged end. It is still used in the laboratory to determine the velocity of sound in air. $A$ tuning fork of frequency $512\,Hz$ produces the first resonance when the tube is filled with water to a mark $11\,cm$ below a reference mark,near the open end of the tube. The experiment is repeated with another fork of frequency $256\,Hz$,which produces the first resonance when water reaches a mark $27\,cm$ below the reference mark. The velocity of sound in air,obtained in the experiment,is close to .... $ms^{-1}$

$A$ tuning fork is used to produce resonance in a glass tube. The length of the air column in this tube can be adjusted by a variable piston. At room temperature of $27\,^{\circ}C$,two successive resonances are produced at $20\,cm$ and $73\,cm$ of column length. If the frequency of the tuning fork is $320\,Hz$,the velocity of sound in air at $27\,^{\circ}C$ is .... $m/s$.

In a resonance tube experiment,when the tube is filled with water up to a height of $17.0 \, cm$ from the bottom,it resonates with a given tuning fork. When the water level is raised,the next resonance with the same tuning fork occurs at a height of $24.5 \, cm$. If the velocity of sound in air is $330 \, m/s$,the tuning fork frequency is $...... \, Hz$.

For sound waves, if the number of nodes for the $5^{th}$ harmonic of an open-ended pipe is $n$ and that for the $9^{th}$ harmonic of the same pipe with one of its ends closed is $m$, the ratio $\frac{n}{m}$ is :