A simple apparatus for demonstrating resonance in an air column is depicted in figure. A vertical pipe open at both ends is partially submerge in water, and a tuning fork vibration at an unknown frequency is placed near the top of the pipe. The length $L$ of the air column can be adjusted by moving the pipe vertically. The sound waves generated by the fork are reinforced of the pipe. For a certain pipe, the smallest value of $L$ for which a peak occurs in the sound intensity is $9.00 \mathrm{cm}.$

$\left(\mathrm{a}\right)$ What is the frequency of the tuning fork?
$\left(\mathrm{b}\right)$ What are the values of $L$ for the next two resonance conditions?

An air column in a glass tube is open at one end and closed at the other by a movable piston. The air in the tube is warmed above room temperature, and a $384 \mathrm{Hz}$ tuning fork is held at the open end. Resonance is heard when the piston is $22.8 \mathrm{cm}$ from the open end and again when it is $68.3 \mathrm{cm}$ from the open end.

What speed of sound is implied by these data?

An air column in a glass tube is open at one end and closed at the other by a movable piston. The air in the tube is warmed above room temperature, and a $384 \mathrm{Hz}$ tuning fork is held at the open end. Resonance is heard when the piston is $22.8 \mathrm{cm}$ from the open end and again when it is $68.3 \mathrm{cm}$ from the open end.

How far from the open end will the piston be when the next resonance is heard?

An aluminium $rod1.60 \mathrm{m}$ long is held at its centre. It is stroked with a rosin-coated cloth to set up a longitudinal vibration. The speed of sound in thin rod of aluminium is $5100 \mathrm{m}/\mathrm{s}.$

What is the fundamental frequency of the waves established in the rod?

An aluminium $rod1.60 \mathrm{m}$ long is held at its centre. It is stroked with a rosin-coated cloth to set up a longitudinal vibration. The speed of sound in thin rod of aluminium is $5100 \mathrm{m}/\mathrm{s}.$

What harmonics are set up in the rod held in this manner?

An aluminium $rod1.60 \mathrm{m}$ long is held at its centre. It is stroked with a rosin-coated cloth to set up a longitudinal vibration. The speed of sound in thin rod of aluminium is $5100 \mathrm{m}/\mathrm{s}.$

What would be the fundamental frequency if the rod were copper, in which the speed of sound is $3560 \mathrm{m}/\mathrm{s} ?$

A metal rod of length $l=100 \mathrm{cm}$ is clamped at two points $A$ and $B$ as shown in figure. Distance of each clamp from nearer end is $a=30 \mathrm{cm}.$

If density and Young's modulus of elasticity of rod material are $\rho =9000 \mathrm{kg}/{\mathrm{m}}^3$ and $Y=144\mathrm{GPa},$ respectively, calculate minimum and next higher frequency of natural longitudinal oscillations of the rod.