Physics IA Final

INTERNATIONAL BACCALAUREATE II. GIMNAZIJA MARIBOR

INTERNAL ASSESSMENT Subject: Physics Title: Correlation between speed of sound and temperature

CANDIDATE NAME: Benjamin Knez SCHOOL CODE: 000563 CANDIDATE PERSONAL CODE: 0014 MENTOR/SUPERVISOR: Gorazd Žiberna EXAMINATION SESSION: MAY 2016

Introduction and theoretical background Sound is something that is always present in our lives, but we never stop and think about where it comes from and how long has it already been travelling. As a musician, I always worry about the sound that comes from my guitar amplifier, the sound that my drum heads emit and the sound that my clarinet produces. One thing I never usually think about is how these sounds travel and are affected by temperature, such as playing in summer or winter. What might the delay be, before the sound reaches the audience? Questions like these started intriguing me and encouraged me to start finding answers. Therefore, when I had to choose the experiment for my internal assessment, I knew I was going to explore the very important relationship between physics and music. After discussing with my professor, I decided to design an experiment where I would measure the velocity of sound depending on temperature using an oscilloscope and a frequency generator. The purpose of this investigation is to find the relationship between temperature and the speed of sound using resonance. The velocity of sound is calculated by the formula: v =f × λ

Where

v

equals velocity of sound in meters per second (

sound in hertz ( Hz ) and

λ

m s ),

f

the frequency of

the wavelength in meters ( m ). (“The Speed of Sound”,

n.d.) [1] Speed of sound in a certain material depends on two main concepts: the density of the material and its rigidity (elasticity). The effect of rigidity is observed in different states This is also observed in solids, which have the ability to transmit the sound the fastest. Then there are liquids and at the end gases, that transmit sound the slowest. Another important factor that influences the speed of sound is the air humidity. As humidity increases, the number of water molecules in the air increases. The water molecules have a 1

smaller molecular mass than oxygen, carbon dioxide or nitrogen. When there are more water molecules there is a lower mass to volume ratio. This leads to an overall smaller density of the air. The denser the air, the slower the sound propagates through it. The effect of humidity is a little greater at lower air pressures. At high altitudes (6000 m), the speed was calculated to be around 0.7 percent larger in 100% humid air than in 0% humid air, keeping the room temperature constant. When increasing the temperature, the effect of humidity was found to be modestly bigger. (Brennan, J., n.d.) [2] It was also experimentally determined that the speed of sound is proportional to the square root of absolute temperature and the ratio of specific heats. (NASA, 2015) [3]

Exploration Research question: How does the speed of sound in air depend on increasing air temperature? Dependent variable: 

Wave frequency

Independent variable: 

Air temperature

Controlled variables: 

Wave length (was always constant for all measurements, equal to two heights of the

   

jar in which the experiment is carried out) Thermometer (was equally calibrated for all measurements) Oscilloscope (measuring range was kept constant through all the measurements) Frequency generator output (output was always sinusoidal wave) Receiver and speaker type (both were the same simple sound generators that could act both as a receiver or a speaker)

Experiment Hypothesis: The speed of sound will increase as the temperature increases because the density of air decreases with higher temperatures, due to gas laws. Equipment used in the experiment:

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       

Frequency generator Oscilloscope Speaker Receiver Thermometer Jar Pot with water Stove

Experiment design and procedure I designed my experiment following this schema Receiver Jar

Oscillosco pe

Frequency generator Speaker

Thermom eter

First, I took a jar of a known height (16.8 cm) and two sound emitters from sound postcards. I glued one at the top, which I used as a sound receiver, or a microphone. The other one, I glued at the bottom and used it as a speaker. I also needed a thermometer in order to measure the temperature inside the jar. Next I connected the speaker to the frequency generator and the receiver to the oscilloscope to be able to analyse the longitudinal wave that occurs. By incrementally increasing the frequency from small values, I was searching for the maximum amplitude seen on the oscilloscope. This point is known as the resonance point, as this is where the first standing wave occurs, known as the fundamental 1 st harmonic that has the wavelength equal to 2 lengths of the jar. If we then further increase the frequency, we eventually reach 2 nd harmonic, 3rd harmonic and so on.

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Picture 1: Standing waves1

In order to increase the temperature, I put the jar in a pot full of water on the stove. Then I slowly started heating the pot and the jar on low fire. The heat from the water was transferred from the water to the air inside of the jar. I started recording the temperatures shown on my thermometer, and for every change of exactly 2 Kelvins, I changed the frequency in order to form a standing wave (resonance) in the jar and then recorded that frequency. I repeated the experiment 3 times, and calculated the average value of the frequency for each temperature.

Raw data Table 1: Raw data of frequencies T (°C)

f (Hz)

±0.05 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0

±0.005 1026 1028 1032 1036 1038 1042 1046 1048 1052 1056

42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0

1060 1062 1066 1070 1074 1078 1080 1084 1088 1092

62.0 64.0 66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0

1096 1098 1102 1106 1110 1114 1118 1122 1126 1130

1 Image can be found at: https://cnx.org/resources/aa67fb155f2f0a86a459138ef2af6da96c5b86d0/StandingWaveHarmonics.png 4

Figure 1: Graph of frequencies with temperature

Figure 2: Linearized graph of frequencies with square root of absolute temperature

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Analysis Velocities were calculated by the formula v =f × λ . Absolute uncertainties: Frequency: ± 2 Hz Wave length: ± 0.0005 m Relative uncertainty of wavelength: 0.3% Absolute uncertainty of the slope is calculated as half of the difference between maximum and minimum slopes in the linearized graph of frequency with square root of absolute

temperature:

± 2.5. Slope of the best fit line is 64.6

Hz √K

Relative uncertainty is then

2.5 ×100=3.9 64.6 Relative uncertainty of speed of sound is 3.9 +0.3 =4.2 Calculated velocities of sound according to different temperatures are then calculated and shown in Table 2.

Table 2: Calculated velocities of sound T (°C) ±0.05

22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0

V (

m ) s

±4.2% 345 346 347 348 349 350 352 352

42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0

356 357 358 359 361 362 363 364 6

62.0 64.0 66.0 68.0 70.0 72.0 74.0 76.0

368 369 370 372 373 374 376 377

38.0 40.0

354 355

58.0 60.0

366 367

78.0 80.0

378 380

Figure 3: Graph of speed of sound with square root of absolute temperature

Figure 4: Graph of velocities of sound with temperature As we can see, the velocity of sound increases with an increase in temperature. By calculating the slope of the graph at the beginning for the first four measurements, we can see it measures 7

at approximately 0.5

m s ℃

and towards the end for the last four measurements, around 0.75

m s .

Evaluation Research question was: How does the speed of sound in air depend on increasing air temperature? It was found that the results speak in favour of my hypothesis as the speed of sound did actually increase with increasingly higher temperature. This is most probably because of the density change of air at higher temperature. Due to gas laws, a higher temperature results in lower density, which in turn leads to a higher speed of sound. It was also found that the rate of increase in speed of sound started increasing at higher temperatures. This is because the speed of sound is linearly dependent on the square root of absolute temperature. Also, the fact that the jar was not completely sealed could have a small effect. In order to install the wires and the thermometer, the jar was slightly open and as the temperatures increased also the amount of water vapour produced from the water in the pot increased. Some of the vapour went into the jar, which resulted in a graph with exponential growth because of the effect of humidity. Main source of systematic errors was the inability to perfectly determine when resonance in the jar was reached. The display on the oscilloscope allowed me to see when resonance was reached, but it couldn’t help me to determine the point of maxima on the displayed curve. Therefore, there were probably some mistakes present in determining the wave frequency. Another important error was controlling the density of air as it is not an ideal gas, so I assumed its composition was constant throughout the experiment with the exception of water vapour, as previously discussed. Another important error was the way the thermometer measured the temperature. Sometimes, it inconsistently jumped through larger value increments. This error was minimised by carefully observing its measurements throughout the experiment in order to get an accurate representation. It would be better to use a newer and more advanced oscilloscope as it was hard to detect maximum amplitude. More accuracy would be needed for the measurements to be completely legitimate. I have also only measured the frequencies every two degrees Celsius. Combined with the uncertainty of the oscilloscope, the readings could be quite inaccurate, resulting in an 8

uneven graph. It would be better to use smaller temperature increments, combined with a precise and reliable thermometer and also a more modern oscilloscope that could also help me analyse the points on the graph. It would also be interesting to measure the speed of sound depending on temperature in other types of mediums and see what kind of graph would occur in them. This could in the end help us to also analyse and develop new techniques for recording sounds that instruments emit, and develop a more advanced and modern approach to recording in studios.

References [1]The Speed of Sound. (n.d.). Retrieved March 23, 2016, from http://www.physicsclassroom.com/class/sound/Lesson-2/The-Speed-of-Sound [2]

Brennan, J. (n.d.). How Does Humidity Affect Speed of Sound? Retrieved March 23,

2016, from http://science.opposingviews.com/humidity-affect-speed-sound-22777.html [3]

Hall, N. (2015, May 5). Speed of Sound. Retrieved March 23, 2016, from

https://www.grc.nasa.gov/www/k-12/airplane/sound.html [4]

Giancoli, D. (2005). Physics. Harlow: Pearson Education Limited.

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