At a temperature of $27^{\circ} C$, two identical organ pipes produce notes of frequency $140 \,Hz$. If the temperature of one pipe is raised to $57.75^{\circ} C$, then the number of beats produced per second is

An organ pipe $P_1$, closed at one end and containing a gas of density $\rho_1$ is vibrating in its first harmonic. Another organ pipe $P_2$, open at both ends and containing a gas of density $\rho_2$ is vibrating in its third harmonic. Both the pipes are in resonance with a given tuning fork. If the compressibility of gases is equal in both pipes, the ratio of the lengths of $P_1$ and $P_2$ is (assume the given gases to be monoatomic)

For air at room temperature, the atmospheric pressure is $1.0 \times 10^{5} \text{ Nm}^{-2}$ and the density of air is $1.2 \text{ kgm}^{-3}$. For a tube of length $1.0 \text{ m}$, closed at one end, the lowest frequency generated is $84 \text{ Hz}$. The value of $\gamma$ (ratio of two specific heats) for air is:

An air column,closed at one end and open at the other,resonates with a tuning fork when the smallest length of the column is $50\, cm$. The next larger length of the column resonating with the same tuning fork is .... $cm$.

$A$ glass tube is open at both ends. $A$ tuning fork of frequency $f$ resonates with the air column inside the tube. Now,the tube is placed vertically inside water so that half the length of the tube is filled with water. Now,the air column inside the tube is in unison with another fork of frequency $f^{\prime}$. Then,