Question

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 fundamental mode of vibration. The density of aluminium is 2600$\times {10^{10}}N/{m^2}$ and it's Young's modulus is 7.80$\times {10^{10}}N/{m^2}$. 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 t be 340m/s)

Answer 1

In this question, we will use different relations and equations to find the different physical properties like wavelength, speed, frequency . Further, we will also discuss the basics of sound for our better understanding.
Formula used:
$v = \left( {\dfrac{Y}{\rho }} \right)$
$\lambda = 2l$
$\nu = \dfrac{v}{l}$
$\lambda = \dfrac{v}{\nu }$

Complete step by step solution:
The speed of sound in aluminium is given by:
\eqalign{& v = \left( {\dfrac{Y}{\rho }} \right) = \sqrt {\left( {\dfrac{{7.80 \times {{10}^{10}}}}{{2600}}} \right)} \cr & \therefore v = 5480m/s \cr}
b) Here, as it is given that the rod is clamped at the middle, then the middle point is pressure antinode.The free ends of the rod are called nodes. When the rod vibrates in its fundamental mode, there are no other nodes or antinodes present at that time. The length of the rod is also the distance between the two successive nodes. Therefore, the length of the rod is equal to half of the wavelength. So, the wavelength of sound in the aluminium rod is given by:
$\lambda = 2l = 180cm$
c) The frequency of the sound produced which is also equal to the frequency of vibration of the rod is given by:
$\nu = \dfrac{v}{l} = \dfrac{{5480}}{{180}} = 3050Hz$
d) The wavelength of sound in air is given by:
\eqalign{& \lambda = \dfrac{v}{\nu } \cr}
\eqalign{& \Rightarrow \lambda = \dfrac{{340}}{{3050}} \cr}
\eqalign{& \therefore \lambda = 11.1cm \cr}
Therefore, we got all the required physical parameters in the above cases.

Additional information:
As we know from the basics of sound that it is produced when something vibrates.
Further, the matter that supports the sound is called the medium i.e., sound needs a medium to travel. Sound cannot travel through a vacuum.
A sound travels fastest in solids, slowest in air and water is in between. The reason behind this is that the particles in solids are closer together than the particles in liquids or gases. Also, the particles in liquids are closer together when compared to the particles in gases.
Since, the tighter particles are packed in a space, they collide more frequently. This allows sound, which is simply the combined collisions of particles, to travel fastest in solids. Therefore we can say that sound travels fastest in solids, then water, and slowest in the gases.

Note:
Here we should notice that the speed of sound is different in different mediums. Further, the speed that is more than the speed of sound is termed differently like- supersonic, hypersonic etc. we should also remember that the speed of sound cannot reach the speed of light.