Speed of Sound Physics Lab Report

Ankur Srivastava 11B

Physics Lab Report

Finding the velocity of Sound

Date of Experiment: 2nd March 2010 Aim: To find the velocity of sound using an air column. Research Question: What is the effect of the time taken for the sound waves emitted from the tuning fork to complete 1 oscillation on the height of the air column needed to achieve the maximum amplitude of this wave and what is the velocity of sound measured as a result? Background and Hypothesis: The cover sheet of this lab gives the speed of sound a value of 330 m/s. Therefore, the theoretical value that can be expected to be inferred from this lab can be 330 m/s. To find this value the relationship v = f x λ can be used. The theory of resonance can also be applied. Resonance is defined as the tendency of a system to oscillate at its maximum amplitude. A tube open at one end achieves resonance at ¼ of a wavelength. Using a glass tube with a vibrating tuning fork on top, the wavelength can be found by finding the height of the air column when the maximum sound is produced. As the frequency will already be known, the speed of sound can be calculated using this equation. From the equation it can also be predicted that the higher the frequency, the lesser the height of the air column as the speed of sound will always be constant (for our purposes). If we input the same relationship into the independent and dependent variables, it can also be concluded that the higher the value for the time taken to complete one oscillation, the higher the value for the length of the air column as the time taken to complete one oscillation is equal to the inverse of the frequency. Variables Independent: The time taken taken for the sound waves emitted from the tuning fork to complete 1 oscillation. Dependent: Height of the air column needed to achieve the maximum amplitude of the wave. Controlled: 1. Frequency of the tuning fork 2. Wavelength of the sound wave 3. Level of water in the plastic barrel 4. Angle that the glass tube makes to the surface of the water. 5. Edge of the glass tube. 6. Distance of the vibrating tuning fork above the edge of the glass tube. 7. Altitude of the location. 8. Temperature inside the room 9. Pressure inside the room

Procedure Apparatus required • 1 x large plastic barrel • 1 x glass tube • 8-12 tuning forks with varying frequencies.

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Metre ruler (± 0.0005m) 30 centimetre ruler (±0.0005m) A rubber bob which will be used to strike the tuning forks

Safety Precautions The tuning forks will be struck using the rubber bob so that they do not undergo damage. Method • Take a large plastic barrel and fill ¾ of it with water. • Place a stand next to the large plastic barrel. • Fasten the glass tube to the stand. • Loosen the clamp holding the glass tube and let one person (number 1) hold it. • The other (number 2) should strike the tuning fork and place it 1 cm above the edge of the glass tube and listen closely. • Number 1 should keep telling number 2 whether to raise or lower the glass tube in order to achieve the loudest sound. • When the loudest sound is achieved, number 2 should fasten the glass tube in place. • A meter stick should be placed next to the plastic barrel. The meter stick should have flat edges in order for it to be as perpendicular as possible. • A 30 centimeter ruler should be used to find the height of the water from the surface of the table by placing it along the straight lines along the meter stick in order for it to be perpendicular to the meter stick. • Use step 9 to find the height of the edge of the glass tube from the surface of the table. • Repeat steps 1-10 until sufficient data points (8-12) are found. Additional notes for controlling variables: 1. Frequency of the tuning fork: This will be controlled as the frequency will already be given and assumed to be accurate. 2. Wavelength of the sound wave: Sound travels at a constant speed at one location and therefore, the wavelength will be kept constant. 3. Level of water in the plastic barrel: The level of water will depend on the height of the stand, the height of the plastic barrel and the length of the glass tube. The water should be at a level which does not hinder any data collection and for the ease of calculation should be kept constant. 4. Angle that the glass tube makes to the surface of the water: This should be as perpendicular as possible, as the value for the height of the air column will be altered if the glass tube is slanted. 5. Edge of the glass tube: The edge of the glass tube should be straight and not protrude to remove any edge effect. 6. Distance of the vibrating tuning fork above the edge of the glass tube: The vibrating tuning fork should be kept close to the edge of the glass tube (around 1 centimetre), care should be taken to keep the distance constant as the glass tube moves upwards and downwards. 7. Altitude of the location: Performing the experiment at the same location should remove this source of error. 8. Temperature inside the room: The experiment should be done in one period so that the temperature is as constant as possible. Windows should be kept closed. 9. Pressure inside the room: Again, the experiment should be done in one stretch and the windows should be kept closed.

Data Collection

Table 1: Collection of raw data: The frequency of the tuning fork that was held over the air column and the length of the air column for the maximum amplitude measured as a result. Trial no.

Frequency of the tuning forks (Hz) 1 2 3 4 5 6 7 8 9

271.2 304.4 320.0 341.3 406.4 426.6 456.1 480.0 512.0

Height of the water level Height of the edge of the glass above the surface of the tube above the surface of the table (m) (±0.005) table (m) (±0.005) 0.390 0.694 0.390 0.661 0.390 0.648 0.390 0.632 0.390 0.593 0.390 0.583 0.390 0.577 0.390 0.562 0.390 0.551

Note: The frequency values on the turning forks were taken as 100% accurate. Data Processing

Step 1: Deriving the independent and dependent variables. 1. Starting with the given equation v = f x λ (where v is the speed of the wave, f is the frequency, λ is the wavelength). 2. Substituting f with 1/T and λ with 4h (where T is the time taken to complete one oscillation and h is the height of the air column), we get → v = 4h/T →T=4h/v

Step 2: Calculating the independent and dependent variables. Part 1: The dependent variable was the height of the air column in the glass. The following set of calculations find the value of this height for each of the 9 data points. Table 2: The values for the heights of the water level and the edge of the glass above the surface of the table and using them to calculate the height of the glass tube Trial no. Height of the water level above the surface of the table (m) (±0.005)

Height of the edge of the glass tube above the surface of the table (m) (±0.005)

Height of the air column in the glass tube (m) (±0.01)

1

0.390

0.694

0.304

2

0.390

0.661

0.271

3

0.390

0.648

0.258

4

0.390

0.632

0.242

5

0.390

0.593

0.203

6

0.390

0.583

0.193

7

0.390

0.577

0.187

8

0.390

0.562

0.172

9

0.390

0.551

0.161

Sample Calculations for Table 2 Theory In order to find the height of the air column of the glass tube, the height of the water level above the surface of the table has to be subtracted from the height of the edge of the glass tube above the surface of the table. Therefore, Height of air column of the glass tube = Height of edge of glass tube above the surface of the table – Height of the water level above the surface of the table. Calculation Value For trial 1, Height of air column = 0.694m – 0.390m = 0.304m Uncertainty For adding or subtracting values with uncertainties, the uncertainties are simply added. Therefore, uncertainty of final value is = 0.005m + 0.005m = ±0.01m Final value 0.304m ±0.01 Part 2: In order to find the independent variable (the time taken for the sound wave to complete one oscillation) the following calculations can be used. Since f(frequency) = 1/T, The values for the independent variable can be found using the frequency values of the tuning forks. Table 3: Finding the time taken for the sound waves generating by the tuning fork to complete one oscillation. Trial no. 1 2 3 4 5 6 7

Frequency values for the tuning forks (Hz) 271.2 304.4 320.0 341.3 406.4 426.6 456.1

The time taken for these waves to complete one oscillation (s) 0.003687 0.003285 0.003125 0.002930 0.002461 0.002344 0.002193

8 9

480.0 512.0

0.002083 0.001953

Note: The tuning forks did not have an uncertainty as they were assumed to be 100% accurate. Sample Calculations for Table 3 Theory As we had seen in the theory section above, we need to find the inverse of the frequency value in order to find the time taken for one oscillation. Calculation Using trial number 1 we get =1/271.2 =0.003687s (4 significant figures) Note: No uncertainty values could be calculated because the frequency values were assumed to be perfect

Step 3: Graphing this relationship Referring back to step 1, the equation being used for calculating the speed of sound is T=4h/v The speed of sound can be calculated if T is plotted as the y axis, h as the x axis and then the gradient will be found to be 4/v. Table 4: The time taken for the sound waves to complete one oscillation and the corresponding heights of the glass tube upon achieving the maximum amplitude. Trial no. 1 2 3 4 5 6 7 8 9

Height of the air column in the glass tube (m) (±0.01) 0.304 0.271 0.258 0.242 0.203 0.193 0.187 0.172 0.161

The time taken for these waves to complete one oscillation (s) 0.003687 0.003285 0.003125 0.002930 0.002461 0.002344 0.002193 0.002083 0.001953

Step 4: Finding the value for the speed of sound using the lines of best, maximum and minimum fit. •

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Line of best fit: Gradient = (y2-y1)/((x2-x1) = (0.002924-0)/(0.24516-0) = 0.0119769049 (GDC) = 0.01198 (4sf) Finding the value for speed of sound = 4/0.01198 = 333.9 m/s (4sf) Line of maximum fit

Gradient =( y2-y1)/((x2-x1) =(0.002924-0)/(0.24516-0.0339) = 0.012094139 (GDC) = 0.01209 (4sf) Finding the value for the speed of sound =4/0.01209 = 330.9 (4sf) •

Line of minimum fit Gradient = ( y2-y1)/((x2-x1) =(0.002924-0.0003684)/(0.24516-0) =0.0104653061 (GDC) = 0.01050 (4sf) Finding the value for the speed of sound = 4/0.0105 =380.9 (4sf)

Conclusion Looking at the graph, it can be observed that as the time taken to complete one oscillation increased, the height required to achieve the maximum amplitude also increased as predicted in the hypothesis. After doing the calculations, the maximum value for the speed of sound was 380.9 m/s, the minimum value was 330.9 m/s and the value obtained by the line of best fit was 333.9 m/s. These values give us is a positive error of 13.8% and a negative error of 0.009% and the speed of light to be 333.9 m/s. The final result is 1.2% off the theoretical value of 330 m/s. Evaluation If the final calculated value and the theoretical value are compared, they are found to be quite close to each other. This shows us that the method used for the calculation of the speed of sound was quite accurate. However an outlier was found, the positive uncertainty was found to be quite large, and the final value was still 3.9 m/s off. Perhaps some sources of error can be found that outline why this is so and suggestions to the improvement are made to the method in order to achieve more accuracy. Sources of error Random •

Error: Angle that the glass tube made to the surface of the water This could have been affected by the loosening of the clamp and also because of the fact that 90 degrees was not actually measured. Solution Measuring this value using a protractor could reduce this error.

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Refraction of light Error: While measuring the height of the air column, refraction of light through the glass medium could have given values other that what they really were. This could have affected the calculations as the height recorded for the air column would have been different. Solution: A solution for this error would be measuring this value at eye level because no refraction occurs when the incident ray enters at the normal

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Damage to the tuning fork Error: Perhaps the existence of the one outlier can be explained by assuming that the one tuning fork was damaged. Solution: A solution for this error would have been measuring the frequencies of the tuning forks beforehand using a data logger so that the actual values are used for best results.

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Interference from the other lab partners around you Error: Sometimes it could be hard to judge the height at which the maximum amplitude was reached because sounds were coming from other experiments that were set up in the same room. This could impede judgment. Solution: A solution for this error could be performing this experiment in a separate room

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Error:Angle that the 30 centimeter ruler made to the set square This could have affected the data because in the experiment, the ruler might not have been completely perpendicular to the metre ruler because the divisions along which the 30 centimeter ruler was placed were quite short. Solution: A solution for this could be using a large set square instead of the 30 centimeter ruler so that more accurate values are measured.

Systematic •

Range was limited Error: While measuring the gradient of the graph it can be seen that the range was quite limited as the values did not go further downwards as the highest frequency used was 512 Hz. Solution: If more data points with higher frequencies (and lower oscillation times as a result) were used, the lines of maximum, minimum and best fit could have been a lot tighter and the positive uncertainty could have been significantly reduced

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Error: The uncertainty of both rulers Solution: If the error bars had been smaller, perhaps the final values for the speed of sound would have been more precise. This could have been done using other equipment such as stretching a non-elastic string from the water level to the edge of the glass tube and then measuring the length. This would have halved the error.

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Damping effect of the tuning fork Error: Due to the fact that the experiment was done in a non-ideal environment, the tuning fork would have undergone damping effects. Therefore, it could have been difficult to judge precisely the height of the glass tube where it achieves the maximum amplitude Solution:A solution for this error could be hitting the tuning fork on the bob every 5 seconds

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Temperature of the location After some research it was found that temperature had a large role to play in the speed of sound. Perhaps the calculated values would have been closer to the theoretical value if the temperature was taken into account while obtaining the theoretical value for the speed of sound. (Source: Speed of Sound http://www.sciencedaily.com/articles/s/speed_of_sound.htm)

Suggestions for improvement to the method • Using more accurate equipment for measuring the change in height such as the string or even laser equipment.. • Using a larger range of frequencies. If tuning forks with higher frequencies were used, the graph would have been more accurate. • Taking temperature into account while finding the theoretical value. • Taking care to measure all values at eye level. • Using a set square instead of a 30 centimetre ruler in order to the random error of the ruler not being perpendicular.