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 it sfundmental mode of vibration. The density of aluminiumis `2600 kgm^-3` and it s 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. THefrequency of the sound produced and d.the wavelength of the sound produced in air. Take the speed of sound in air t be `340 ms^-1`.

Options

A. `5477 ms^(-1)` and 30.42 Hz
B. `5000 ms^(-1)` and 20.24 Hz
C. `3438 ms^(-1)` and 60.34 Hz
D. None of the above

Answer

A

Explanation

Given, Young's modulus, `Y=7.80xx10^(10) Nm^(-2)`
density of aluminium, `rho=2.6" gcm"^(-3)=(2.6xx(100)^(3))/(1000)"kgm"^(-3)`
or, `rho=2600 kgm^(-3)`
wavelength, `lambda=180 m`
Speed of sound in aluminium, `v=sqrt(Y/rho)=sqrt((7.80xx10^(10) Nm^(-2))/(2600 kgm^(-3)))`
or, `v=5477 ms^(-1)`
As, `lambda=v/c implies v=v/lambda =(5477 ms)/(180 m)=30.42 Hz`