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Chapter 2: Q6E (page 539)

Question: (a) In a liquid with density \({\bf{1300}}\;{\bf{kg/}}{{\bf{m}}^{\bf{3}}}\), longitudinal waves with frequency 400 Hz are found to have wavelength 8.00 m. Calculate the bulk modulus of the liquid. (b) A metal bar with a length of 1.50 m has density \({\bf{6400}}\;{\bf{kg/}}{{\bf{m}}^{\bf{3}}}\). Longitudinal sound waves take \({\bf{3}}{\bf{.90}} \times {\bf{1}}{{\bf{0}}^{ - {\bf{4}}}}\;{\bf{s}}\) to travel from one end of the bar to the other. What is Young鈥檚 modulus for this metal?

Short Answer

Expert verified

(a) The bulk modulus of liquid is \(1.331 \times {10^{10}}\;{\rm{Pa}}\) .

Step by step solution

01

Given data

The data given in the question can be listed below as;

  • The density of the liquid is,\(\rho = 1300\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\) .
  • Frequency is,\(f = 400\;{\rm{Hz}}\).
  • Wavelength is,\(\lambda = 8.00\;{\rm{m}}\)
02

Concept

Longitudinal sound waves are a type of sound waves in which the vibration/oscillation of the particles is in the direction of the propagation of the moving wave in the medium they are traveling.

03

(a) Calculation of the bulk modulus of the liquid

The formula that we can use to calculate the bulk modulus of the liquid is given as,

\(B = {\lambda ^2}{f^2}\rho \)

Substitute the values in the above expression, and we get,

\(\begin{array}{c}B = {\left( {8.00\;{\rm{m}}} \right)^2}{\left( {400\;{\rm{Hz}}} \right)^2} \times 1300\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\\ = 1.331 \times {10^{10}} \cdot \left( {1\;{{\rm{m}}^2} \times 1\;{\rm{H}}{{\rm{z}}^2} \times 1\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}} \times \frac{{1\;{\rm{Pa}}}}{{1\;{\rm{kg}} \cdot {{\rm{s}}^{\rm{2}}}{\rm{/m}}}} \times \frac{{1\;{{\rm{s}}^{\rm{2}}}}}{{1\;{\rm{H}}{{\rm{z}}^2}}}} \right)\\ = 1.331 \times {10^{10}} \cdot \left( {1\;{\rm{Pa}}} \right)\\ = 1.331 \times {10^{10}}\;{\rm{Pa}}\end{array}\)

Thus, the bulk modulus of liquid is \(1.331 \times {10^{10}}\;{\rm{Pa}}\) .

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