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Chapter 16: Problem 13

At what temperature will the speed of sound be double of its value at \(0^{\circ} \mathrm{C}\) ?

Short Answer

Expert verified
The speed of sound doubles at approximately 552.17掳C.

Step by step solution

01

Understand the speed of sound at 0掳C

At \(0^{\circ} \mathrm{C}\), the speed of sound is approximately 331.3 m/s. This value is crucial for finding when the speed will double.
02

Calculate the doubled speed of sound

To find the speed of sound when it is doubled, calculate: \( 2 \times 331.3 \) m/s = 662.6 m/s.
03

Use the speed of sound formula

The speed of sound in air is given by the formula: \( v = 331.3 + 0.6T \), where \( T \) is the temperature in degrees Celsius. We need this formula to find when sound speed reaches 662.6 m/s.
04

Set up the equation

Set the formula for speed equal to the doubled speed and solve for \( T \): \( 331.3 + 0.6T = 662.6 \).
05

Solve for T

Isolate \( T \) by subtracting 331.3 from both sides: \( 0.6T = 662.6 - 331.3 \). Calculate: \( 0.6T = 331.3 \).
06

Final Calculation

Divide both sides of the equation by 0.6 to solve for \( T \): \( T = \frac{331.3}{0.6} = 552.1667 \). Therefore, the temperature is approximately \(552.17^{\circ} \mathrm{C}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature and Sound Speed
The speed of sound is not constant; it changes with the temperature of the medium it's traveling through, particularly in the air. At a standard room temperature of 0掳C, the speed of sound is around 331.3 meters per second (m/s).
This is because sound is a type of mechanical wave that travels through the movement of molecules. As the temperature increases, molecules move faster. They collide more often and transmit sound waves more quickly.
Therefore, when the temperature rises, sound waves can travel faster, increasing the speed of sound in the air.
  • At higher temperatures, the kinetic energy of air molecules increases.
  • This accelerates the passage of sound waves through the air.
  • The faster the molecules, the quicker they interact and move the wave along.
Consequently, the speed of sound is inherently linked to the ambient temperature of the environment.
Sound Velocity Equation
Knowing how to calculate the speed of sound can be quite useful, especially in various scientific and engineering applications.
The equation to determine the speed of sound in air is given by:

\( v = 331.3 + 0.6T \)

In this formula, \(v\) represents the speed of sound in meters per second, and \(T\) denotes the temperature in degrees Celsius. The equation shows how the speed of sound varies with temperature.
  • At \(0^{\circ} C\), the equation simplifies to 331.3 m/s.
  • For every 1-degree increase in Celsius, the speed of sound increases by 0.6 m/s.
  • This linear relationship helps us predict sound speeds at varying temperatures.
Understanding this equation allows us to determine the speed at any temperature. For example, if you wanted to find out at what temperature the speed of sound would double from its value at 0掳C, you would set up the equation accordingly.
Thermal Effects on Sound Speed
When considering how thermal changes affect sound speed, it's essential to recognize the impact of temperature on the air's properties. Heat energy affects the motion of particles in the medium.
The warmer the air, the more energetic its molecules, leading to increased movement rates. This rise in temperature lowers the density of the air, making it less resistant to sound waves.
  • Molecules collide and interact more frequently when heated.
  • This frequent interaction allows sound waves to pass through more readily.
  • The result is a direct increase in sound speed in warmer conditions.
Understanding the thermal effects on sound speed is crucial not only for physics but also for applications in meteorology, aviation, and audio engineering.
Each of these fields benefits from predicting and adapting to the way sound velocities change with temperature variations.

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Most popular questions from this chapter

The noise level in a classroom in absence of the teacher is \(50 \mathrm{~dB}\) when 50 students are present. Assuming that on the average each student outputs same sound energy per second, what will be the noise level if the number of students is increased to \(100 ?\)

In a resonance column experiment, a tuning fork of frequency \(400 \mathrm{~Hz}\) is used. The first resonance is observed when the air column has a length of \(20 \cdot 0 \mathrm{~cm}\) and the second resonance is observed when the air column has a length of \(62 \cdot 0 \mathrm{~cm} .\) (a) Find the speed of sound in air. (b) How much distance above the open end does the pressure node form ?

A car moving at \(108 \mathrm{~km} \mathrm{~h}^{-1}\) finds another car in front of it going in the same direction at \(72 \mathrm{~km} \mathrm{~h}^{-1}\). The first car sounds a horn that has a dominant frequency of \(800 \mathrm{~Hz}\). What will be the apparent frequency heard by the driver in the front car ? Speed of sound in air \(=330 \mathrm{~m} \mathrm{~s}^{-1}\).

A small source of sound oscillates in simple harmonic motion with an amplitude of \(17 \mathrm{~cm}\). A detector is placed along the line of motion of the source. The source emits a sound of frequency \(800 \mathrm{~Hz}\) which travels at a speed of \(340 \mathrm{~m} \mathrm{~s}^{-1}\). If the width of the frequency band detected by the detector is \(8 \mathrm{~Hz}\), find the time period of the source.

Find the change in the volume of \(1 \cdot 0\) litre kerosene when it is subjected to an extra pressure of \(2 \cdot 0 \times 10^{5} \mathrm{~N} \mathrm{~m}^{-2}\) from the following data. Density of kerosene \(=800 \mathrm{~kg} \mathrm{~m}^{-3}\) and speed of sound in kerosene \(=1330 \mathrm{~m} \mathrm{~s}^{-1}\).
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