Sound. Measure the speed of sound in air by means of resonance in a tube; Measure the speed of sound in a metal rod using Kundt s tube.

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1 Sound Purpose Problem 1 Problem 2 Measure the speed of sound in air by means of resonance in a tube; Measure the speed of sound in a metal rod using Kundt s tube. Introduction and Theory When a vibrating tuning fork is held at the top of a glass tube which is partially immersed in water, the air column inside the tube starts vibrating along with the tuning fork (Fig. 1). If the length of the air column is an odd number of quarter-wavelengths, like ¼λ or ¾λ, the sound emitted by the tuning fork can set up a standing wave of large amplitude inside the tube, creating a loud sound. This condition is known as resonance. The air molecules cannot move up and down at the surface of the water, so that is a displacement node of the wave. The biggest vibration happens near the top of the tube, making this is an antinode. tuning fork air column large cylinder surface of water glass tube Figure 1 We can determine the wavelength of the sound by measuring the length of the air column at resonance. If we also know the frequency of the sound wave, we can calculate the speed of the sound from v=fλ. Experimentally, we increase the length of the air column by raising the glass tube. The first resonance happens when the length of the air column L 1 equals one quarter of a wavelength, as shown in Fig. 2. The next two resonances happen when the length is three quarters and five quarters of the wavelength. That is, n L = λ, where n = 1, 4 By plotting a graph of L vs. the wavelength fraction, we can find the wavelength of the sound in the air. 3, Sound - 1

2 L 3 L 2 L1 1 st resonance 2 nd resonance 3 rd resonance Figure 2: The heavy dashed lines represent the magnitudes of the vibrations of the air molecules. The true vibrations of the molecules are parallel to the sides of the glass tube. To calculate the theoretical speed of sound in air, we use and the following values: γ : v = γrt M the ratio of the specific heat capacity of at constant pressure to the specific heat capacity at constant volume of the air: γ = R : the molar gas constant: R = J/mol K. M : T : the molecular mass of the gas: M = kg/mol. the temperature of the gas in Kelvin Sound - 2

3 In Problem 2, we determine the speed of sound in a metal rod using a Kundt s tube device, which consists of a metal (Aluminum) rod and a glass tube with one end closed (see Fig. 3). A sound wave is created in the metal rod by rubbing it. We assume that the fundamental mode is excited. The midpoint of the rod is clamped, so it is a displacement node for the sound wave. The two ends of the rod are free to vibrate longitudinally and are displacement anti-nodes. Therefore we can find the wavelength of the sound in the metal rod from the length of the rod. Kundt s tube L a Cork dust piles n spaces (here n=7) Figure 3 The piston on the end of the rod excites a sound wave in the air column in the glass tube. When the length of the tube is appropriate, a standing wave of large harmonic number is created in the air column. Cork dust is put inside the glass tube to mark the locations of the nodes. When the air molecules vibrate, the cork dust is swept away from the regions where the vibration is large (antinodes) and piles up where the vibration is least (at the nodes). The positions of the cork dust piles give the wavelength of sound in the air. Knowing the wavelengths of the sound in the metal and in air, the speed of sound in the metal can be found from the fact that the frequencies of the sound waves in the 2 media are the same: vaλm v m = (1) λa where v m and v a are the speeds of sound in the metal and in air, λ m and λ a are the wavelengths in the metal and in air. The theoretical speed of sound in the rod is given by: E v= ρ where E is the Elastic or Young s modulus of the solid and ρ its density Sound - 3

4 Sound Prelab Print this page and finish the questions before coming to the lab. Derive Eq. (1) in Problem Sound - 4

5 Problem 1 The Speed of Sound in Air Your task is to find the speed of sound in air. Start with the Purpose and Apparatus sections in your report. Data The frequency in cycles/second or Hertz (Hz) is marked on each tuning fork. Assume the uncertainty to be ± 2 Hz. Fill the large plastic cylinder with water to within about 3 cm of the top. Insert a short glass tube into the water so that it is almost submerged. Strike the short tuning fork (1024 C) with the hammer, and then place it just above the top of the tube, as shown in Fig. 1. Be very careful when handling the glass tube! When striking the tuning fork, keep the fork away from the glass tube, or you may break the tube! Starting from the lowest position of the glass tube, slowly raise the tube and the tuning fork together until a loud sound is heard. This is the position of the first resonance, where the length of the air column equals 1/4 of the sound wavelength. Use an elastic band to mark the position of the water surface and measure the length of the air column L 1. Repeat the procedure until you are confident of your result. Repeating this also helps you to decide the uncertainty in L 1. Continue raising the glass tube to find the 2nd and 3rd resonance positions. Get longer glass tubes as needed. Similarly, measure L 1 to L 3 for the long tuning fork (512 C). Record the temperature of the air in the room, with its uncertainty. The thermometer is to the left of the blackboard. You will use this to calculate the theoretical speed of sound in the air. Empty the water from the large cylinder and leave it in the sink area. Calculations Graphically determine the wavelengths of each tuning fork and their uncertainties. Do this with only one sheet of graph paper, using a single set of axes. Calculate the speed of sound in air for each fork. You should have 2 similar results average them. Uncertainty Analysis / Conclusions / Discussion Calculate the uncertainty in your final speed of sound, for both the experimental and theoretical results. When calculating the uncertainty of the theoretical speed of sound, you may neglect the uncertainties of γ, R and M, because they are more than 10 times smaller than the uncertainty of T. State both results in proper format. Does your experimental value agree with the theoretical value within the uncertainty? Discuss Sound - 5

6 Problem 2: The Speed of Sound in a Metal Start your report with the Purpose and Apparatus sections. Draw the setup of the apparatus and clearly mark the quantities you measure. Data The lab demonstrator will show you the setup of the apparatus. Note that the rod is clamped at its midpoint and the two ends are free to move. The lab demonstrator will force the rod to vibrate longitudinally by stroking the rod with the small piece of chamois, which should have had rosin put on it. This creates a screeching sound. Watch the cork dust inside the glass tube while the lab demonstrator creates the sound. The sound wave, which passes from the rod into the closed air column, causes the cork dust to be collected into piles at the nodes of the sound wave inside the air column. To find the wavelength of the sound in the metal and in the air λ m and λ a, you need to know the length of the metal rod L m and the spacing between the cork dust piles, which is given by L a and n. Make n as large as possible to reduce the fractional uncertainty. Calculations From the standing wave patterns, derive the relations between the wavelengths of the sound wave and the lengths that you measured. Find the wavelengths λ m and λ a. Calculate the speed of sound in the metal rod, using the speed of sound in the air that you measured in Problem 1. Complete as usual with the Uncertainty Analysis, Conclusions, and Discussion sections Calculate the uncertainty in your speed of sound in the metal. State the final result in proper format. Calculate the theoretical value of the speed of sound in the metal using the properties of Aluminum alloy #6, which can be found in a data sheet in the lab. You may ignore the uncertainty in the theoretical value because it is much smaller than the uncertainty in the experimental value. If you want to calculate it, assume that the properties of Aluminum have an uncertainty of ± 1 in the last significant digit. Discuss the agreement Sound - 6