Question

The speed of longitudinal waves in a steel bar is (\[Y = 2.0{\text{ }} \times {\text{ }}{10^{11}}Pa\] and Density\[ = 7800kg{m^{ - 3}}\] )

Answer 1

Hint: In this question, we need to evaluate the speed of the longitudinal waves in the steel bar such that the stiffness coefficient is \[Y = 2.0{\text{ }} \times {\text{ }}{10^{11}}Pa\] and the density of the steel is \[7800kg{m^{ - 3}}\]. For this, we will use the relation between the speed, stiffness coefficient and the density of the material (here, steel). Substitute the given values and then, calculate the final answer.

Formula used:
The speed of sound in any substance is given by $v = \sqrt {\dfrac{Y}{\rho }} $ where $\rho $ is the density of the substance and $Y$ is the stiffness coefficient (or, the young’s modulus).

Complete step by step answer:
Sound travels through means of compression and rarefaction. It cannot travel in vacuum. Speed of sound is not universal. It is highly dependent on the medium, more specifically the physical state of the medium. Speed of the sound is most in solids and least in gases. This is because in solids the molecules are tightly packed and thus the vibrations can be transferred easily. This lags in case of gases.
In solids, we use the Young’s Modulus or $Y$ as the stiffness coefficient. The Young’s Modulus is defined by the ratio of longitudinal stress and longitudinal strain. It has the unit of Pascal.
Hence the formula for sound in solids becomes $v = \sqrt {\dfrac{Y}{\rho }} $ where, Y is the stiffness coefficient and $\rho $ is the density of the material.
Given that,
Young’s Modulus, \[Y = 2.0{\text{ }} \times {\text{ }}{10^{11}}Pa\]
And
Density, \[\rho = 7800kg{m^{ - 3}}\]
Putting the values in the equation of speed of sound we get that
\[
  v = \sqrt {\dfrac{{2.0{\text{ }} \times {\text{ }}{{10}^{11}}}}{{7800}}} m{s^{ - 1}} \\
   = \sqrt {\dfrac{{2.0{\text{ }} \times {\text{ }}{{10}^9}}}{{78}}} m{s^{ - 1}} \\
   = \sqrt {\dfrac{{2.0{\text{ }} \times {\text{ }}10 \times {\text{ }}{{10}^8}}}{{78}}} m{s^{ - 1}} \\
   = \sqrt {\dfrac{{20{\text{ }}}}{{78}}} \times {10^4}m{s^{ - 1}} \\
   = 0.5063 \times {10^4}m{s^{ - 1}} \\
   = 5063m{s^{ - 1}} \\
 \]
Thus the velocity of sound is \[v = 5063m{s^{ - 1}}\].

Note:
The stiffness coefficient is different for different states of matter. For gases we use the Bulk’s Modulus. Modulus in the Elasticity chapter of physics is found out by dividing stress by strain. Also students need to keep in mind the different types of modulus, stress and strains. Avoid making calculations mistakes. Use standard units for calculation.