An aluminium rod having a length of 90.0 cm is clamped at its middle point and is set into longitudinal vibrations by stroking it with a rosined cloth. Assume that the rod vibrates in its fundmental mode of vibration. The density of aluminium is `2600 kgm^-3` and its Young's modulus is `7.80xx10^10 Nm^-2`. Find a. The speed of sound in aluminium, b. The wavelength of sound waves produced in the rod. c. The frequency of the sound produced and d. The wavelength of the sound produced in air. Take the speed of sound in air to be `340 ms^-1`.

Explanation

a. The speed of sound in aluminium is
`v=rho(Y/rho)=sqrt((7.80xx10^10Nm^-2)/(2600kgm^-3))`
`=5480ms^-1`
b. since the rod is clamped at the middle, theh middle point is pressure antinode.The fre ends of the rod are the nodes. As the rod vibrates in it fundamental mode, there are no other nodes or antinodes.The length of the rod which is also the distance between the succissive nodes, is there fore, equal to hal fthe wavelength. Thus, the wavelength of sound in the aluminium rod is
`lamda=2l=180cm`
c. THe frequency of the sound produced which is also equal to the frequency of vibrationof the rod is
`v=v/lamd(5480ms^-1)/(180cm)=3050Hz.`
d. The wavelength of sound in air is
`lamda=v/v=(340ms^-1)/(3050Hz)=11.1cm`.