Determination of an unknown frequency (beats)
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1 Teacher's/Lecturer's Sheet Determination of an unknown frequency (beats) (Item No.: P ) Curricular Relevance Area of Expertise: Physics Education Level: Age Topic: Acoustics Subtopic: Wave properties and oscillations Experiment: Determination of an unknown frequency (beats) Difficulty Preparation Time Execution Time Recommended Group Size Easy 10 Minutes 10 Minutes 1 Student Additional Requirements: Experiment Variations: Microphone Loudspeaker Pair of scissors PC Keywords: Task and equipment Information for teachers Additional information In this experiment, the students determine the frequency of an off-tune tuning fork by the measurement of the beat frequency with the aid of a second tuning fork. Prior to the execution of the experiment, the students should be familiar with the concept of beats as well as with the operation of the measure Acoustics software. Notes concerning the execution of the experiment The experiment should be performed by two students. Note An additional comparison measurement of the tuned tuning fork with the PC (i.e. the execution of part 2 also with the tuned tuning fork) can be helpful in the event of unclarities, or if the students work is carried out particularly quickly in order to clarify the relative consideration.
2 Student's Sheet Determination of an unknown frequency (beats) (Item No.: P ) Task and equipment Task How can an unknown frequency be determined with the aid of beats? Beats occur when two tones have nearly the same frequency. If one frequency is known, then the other frequency can be determined by analysing the beat. Determine the frequency of an off-tune 440 Hz tuning fork. 1. Measure the beat frequency and calculate the possible frequencies of the off-tune tuning fork. 2. Determine whether the frequency of the off-tune tuning fork is higher or lower than the frequency of the tuned tuning fork in order to determine the frequency of the off-tune tuning fork. Equipment Position No. Material Order No. Quantity 1 Tuning fork 440 Hz Striking hammer Frame drum, d = 20 cm Silicone tubing, inner diameter 3 mm Software "Measure Acoustics", single user license Digital stop watch, 24 h, 1/100 s & 1 s Additional material Microphone 1 Loudspeaker 1 PC 1 Pair of scissors 1
3 Student's Sheet Set-up and procedure Part 1: Measurement of the beat frequency Cut a piece of 3 mm off the tubing and push it onto one of the prongs of a tuning fork so that the upper edge of the piece of tubing is flush with the upper edge of the prong (Fig. 1). Fig. 1: Tuning fork with a piece of tubing Fig. 2: Frame drum with the two tuning forks Student 1: Take the stop watch and make it ready to start. Student 2: Strike both of the tuning forks and hold their lower ends on the frame drum (Fig. 2). Student 2: Count 30 beat maximums. Student 1: Take the time that student 2 needs for counting 30 beat maximums. Note down your result in the report (under "Result - Observations 1"). Part 2: Measurement of the unknown frequency with the PC Now, measure whether the frequency of the off-tune tuning fork is higher or lower than the frequency of the tuned tuning fork. The value of the frequency of the off-tune tuning fork can then be determined based on the results from part 1. Connect the microphone correctly to the PC. Start the software "measure Acoustics". Open the experiment "3.5 Determination of an unknown frequency (beats)". Help 1: Open the experiment overview (Menu "File" "Open experiment" or select "Open experiment" on the menu bar). Open the folder "3 Applications in the field of medicine, music, and everyday life" and select the experiment "3.5 Determination of an unknown frequency (beats)". Student 1: Strike the tuning fork that has been put out of tune with the piece of silicone tubing and hold it in front of the microphone (Fig. 3).
4 Student's Sheet Fig. 3: The tuning fork with the piece of tubing held in front of the microphone Student 2: Freeze the diagram. Help 2: Select "Activate/freeze diagram" in the window "Spectrum of the signal at the audio input (microphone)". Zoom the area of the maximum amplitude and use the crosshair to read the frequency of the off-tune tuning fork. Note down your result in the report (under "Result - Observations 2"). Help 3: Select "Zoom" on the menu bar in the window "Spectrum... (microphone)". Proceed as follows in order to zoom in on a suitable diagram section: Keep the mouse button pressed and drag a rectangle from the upper left-hand corner to the lower right-hand corner. Then, select the crosshair with "Mark". Hold the crosshair onto the position of the amplitude maximum and read the x-value (frequency in Hz) in the lower left-hand corner of the diagram window. Readjust the standard diagram section and activate the diagram. Help 4: Select "Standard diagram section" and "Activate/freeze diagram" in the window "Spectrum... (microphone)". Repeat the experiment of part 2 at least 10 times. Each time, note down the frequency value.
5 Teacher's/Lecturer's Sheet Report: Determination of an unknown frequency (beats) Result - Observations 1 (10 points) Part 1: Measurement of the beat frequency Time for 30 beats: With a piece of tubing of 3 mm, 30 beats take approximately 10 seconds. Result - Observations 2 (10 points) Part 2: Measurement of the unknown frequency with the PC Measured values of the unknown frequency: The frequency that can be read in the frequency spectrum is approximately 437 Hz. Result - Observations 3 (10 points) Part 2: Measurement of the unknown frequency with the PC The frequency of the tuned tuning fork is 440 Hz. Is the frequency of the off-tune tuning fork on average higher or lower than the frequency of the tuned tuning fork? On average, the frequency of the off-tune tuning fork is lower than that of the tuned tuning fork. Evaluation - Question 1 (10 points) Calculate the duration of one beat based on the measured time for 30 beat maximums. The beat period is approximately 1/3 s.
6 Teacher's/Lecturer's Sheet Evaluation - Question 2 (10 points) Use the result of point 1 in order to calculate the beat frequency. The beat frequency is approximately 3 Hz Evaluation - Question 3 (10 points) Explain the effect that the piece of tubing has on the oscillation and frequency of the tuning fork. When a ballast is put on the prong of the tuning fork, then it is hard to cause this prong to oscillate. This, in turn, reduces the oscillation frequency and, thereby, also the frequency of the tone. Evaluation - Question 4 (10 points) If the sound volume maximums are measured, the beat frequency equals the difference of the two frequencies of the individual tones. Calculate the two possible frequencies of the off-tune tuning fork and - based on your observations during part 2 of the experiment - state which of the frequencies makes sense from your point of view. The first frequency of 440 Hz leads to two possible second frequencies: 437 Hz or 443 Hz. Based on the observation (point 3) of part 2 as well as on the evaluation (point 3), the frequency must be lower so that 437 Hz is the only possible result. Evaluation - Question 5 (10 points) For technical and hardware-related reasons, the resolution of the measurement with the PC is limited to approximately 2.6 Hz. State why the frequency of the off-tune tuning fork cannot be measured exactly with the PC for this reason. The PC measurement should result in a spectrum with a maximum at a frequency of exactly 437 Hz. Since, however, the resolution of the experiment set-up is limited to approximately 2.6 Hz, the frequency can only be read with a level of precision of 2.6 Hz so that an exact measurement is not possible. In addition due to the technical FFT implementation (Fast Fourier Transformation) a leakage effect may occur. If the measurement range is unfavourable, a double peak may occur instead of the real, single peak at 437 Hz and/or the spectrum may show a lobed amplitude curve around the amplitude maximum. The relative consideration during this experiment, however, enables a qualitative comparison on average. Note: An additional comparison measurement of the tuned tuning fork with the PC (i.e. the execution of part 2 also with the tuned tuning fork) can be helpful in the event of unclarities, or if the students work is carried out particularly quickly in order to clarify the relative consideration
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