STANDING WAVES AND RESONANCE

Transcription

1 Name: PHYSICS 119 SPRING 2008 S.N.: SECTION: Experiment 2 PARTNER: DATE: STANDING WAVES AND RESONANCE Objectives Observe standing waves in a stretched string Examine how an external oscillator can cause resonance Observe how changing the string length and tension affects the conditions for resonance Determine the frequency of oscillations in standing waves. Apparatus: Connect one end of a string to a vibrator while the other end passes over a pulley and supports a weight as shown in Figure 1. Make sure the blade of the vibrator is parallel to the string, so that the string vibrates into and out of the page (there are two possible frequencies of the string vibration induced by the vibrator, depending on the orientation of the vibrator with respect to the string). pulley string vibrator weighted stand table weight Figure 1. Apparatus for standing wave experiment. Both the length of the string and the tension can be varied. The weight pulls the string taut so the tension in the string is equal to the gravitational force on the hanging weight and its holder. 119LAB2-RESONANCE.DOC

2 In the string of Figure 1, a wave moving to the left is produced by the vibrator and is reflected at the pulley to make a wave moving to the right. Because there are two waves with the same amplitude and frequency moving in opposite directions through the same medium, we observe standing waves. Qualitative Procedures Set up the apparatus as shown in Figure 1 with a mass of 500g (plus the mass of the hanger which is 50g) and plug in the vibrator. Move the ring stand back and forth, observing the changes in the vibrating string. At certain locations of the ring stand, the amplitude of the standing wave pattern in the string is much larger than at the other locations. When the ring stand is at one of these locations, we are observing resonance in the vibrating string system. Resonance occurs when the vibrator is oscillating at a frequency equal to one of the natural frequencies of the system. In our setup, the frequency of the vibrator is fixed, but we can adjust the natural frequency of the system by changing the length of the string or the tension in it, and so make it the same as the oscillator. Find a position of the ring stand at which the amplitude of the oscillations is as large as possible and resonance is observed. 1. Are there points on the string where it is perfectly motionless? Are there points where the motion of the string is sufficiently small compared to the motion at other places that it is a good approximation to call them nodes? Discuss. 2. Is there a node, antinode, or neither at the vibrator? (Hint: Think about the approximation in question 1. The vibrator is moving, but how much is it moving compared to other places on the string?). Discuss. 3. Record the length of the string L between the knot at the vibrator and the pulley. How many nodes and antinodes are present in your standing wave (include the ends)? L = ± cm Is there a node, antinode, or neither at the pulley? number of nodes = number of antinodes = 4. Sketch the appearance of the string as viewed from above. Find the wavelength of your standing wave. The strobe lamp is very helpful to show what is actually going on. Show your work.! = ± m 119LAB2-RESONANCE.DOC

3 5. Predict the minimum distance you can move the ring stand so that resonance is observed again, and explain your prediction. 6. Move the ring stand to check your prediction. Sketch the resulting appearance of the string. How far did you move the ring stand? Was your prediction correct? If not, resolve the inconsistency. Changing the length of the string changes the number of nodes and antinodes at resonance. We can produce a similar effect by changing the tension in the string. Gently push down on the weights until resonance is observed again. Then gently lift up the weights to reach another resonance. The different oscillation patterns are called modes. 7. Describe in each case what changed when you changed the length of the string, and changed the tension in the string,. How many nodes are there, including the ends? Did the frequency change? Did the tension change? Did the wavelength change? You can make a little table to organize this. 8. If someone were to say that the natural frequencies had changed, would you agree or disagree, and why? Quantitative Procedures We can use the experimental setup to determine the frequency at which the vibrator is oscillating. We found above that it is possible to measure the wavelength of standing waves directly. If we knew the speed at which the waves were propagating, we could use the relation! = "f to determine the frequency. For small transverse oscillations on a stretched string, the wave speed is given by the relation! = F T / µ, where F T is the tension in the string and µ = m / L is the linear mass density (mass per unit length) of the string. 9. Using these equations, derive an expression for the frequency in terms of the wavelength, tension, and linear mass density. You will now use this expression to determine the frequency of the oscillations. 119LAB2-RESONANCE.DOC

4 10. Find µ by measuring the mass and length of several meters of string (Use a different piece of string - don t dismantle your apparatus). There are scales at the back of the lab accurate to 0.1 g L = ± m m = ± kg µ = ± kg/m TA initials 11. Place a calibrated mass m s on the hanger and adjust the position of the ring stand until resonance is observed. In the table record the value of the calibrated mass. Estimate its uncertainty. Measure the distance L between two nodes (preferably with several antinodes between them) and your estimate of the uncertainty "L in L. Record the number of antinodes between the two nodes. Repeat this procedure five or six times, using a range of masses between 300 and 1000g. As you take your data enter it in the table. Consider a good spread of masses, such as 300, 400, 500, 600, 700 and 1000g (plus the weight of the hanger). Leave the "f column empty for Q. 12. Hints: To reduce errors when determining wavelengths, measure the length L of several half wavelengths (measure from node to node) and divide by the number of wavelengths (each antinode = one half wavelength). Show your work completely for the third entry in the table. Estimating "f (Hz) is discussed in more detail on the next page. Explain how you use the results of this example to calculate "f for the other trials. Be very careful with units! m s (g) Total mass (g) F T (Newtons) "F T (N) # antinodes L(cm) "L(cm)!(cm) "! f (Hz) "f (Hz) 119LAB2-RESONANCE.DOC

5 12. Compute the uncertainty "f(hz) of your third frequency measurement and enter the value in the table above. Remember to propagate the errors! There are 3 contributions to the uncertainty: the tension, the string mass per unit length, and the wavelength. Calculate the effect of these one at a time on the frequency. Then take the square root of the sum of the squares of the calculated frequency uncertainties to get the overall effect. Is there one contribution that is much more important or much less important in the final uncertainty? Since the calculation is quite lengthy, just do it for the third frequency, and estimate the rest based on that. (The argument is that by selecting a measurement in the middle, we will be reasonably close in our estimate for all of them.) You can organize this calculation by making a small table. 13. Compute the average of your frequency measurements. f ave = Hz 14. Compute the uncertainty in the average. If the uncertainty in the average is dominated by random errors, it is smaller by #N than the uncertainty in a single measurement); if the uncertainty is dominated by systematic errors, the uncertainty of the average is the same as the uncertainty of each measurement. Which is appropriate here? This is a difficult question you need to think about where the uncertainties in your table above are coming from. Ignore the uncertainty in the tension, because it is small. What would your answer be if a. you had no uncertainty in the string mass per unit length, or b. you had no uncertainty in the wavelength? There is a way to handle the uncertainty in the average correctly when you have both. Keep a separate account for each frequency of the random part and the systematic part of the uncertainty. When you form the average, the random part becomes #N smaller and the systematic part stays the same. Then, as the last step, combine the random and systematic contributions by the square root of the sum of the squares to get the overall uncertainty in the average. f ave = ± Hz 119LAB2-RESONANCE.DOC

6 15. The vibrators are designed to oscillate at a frequency that is twice the frequency of the AC power in the building, which is 60.00±0.02Hz. Compare the frequency you determined to the vibrator frequency. If they do not agree, explain why you think this is so. 16. Is the standing wave you have measured transverse or longitudinal? Explain. Consider both the motion of the vibrator blade and the motion of the string you actually observed. 17. Set up your apparatus so that you have an even number of antinodes. Keeping the length of the string as constant as you can, rotate the ring stand 90 so that the blade of the vibrator is perpendicular to the string. Describe briefly, what happens. Is the new standing wave transverse or longitudinal? Write a short explanation. 18. Determine the frequency of the new standing wave without making any additional measurements. Explain your reasoning. Use the strobe lamp, if available, to check your result. 19. Predict what you would have observed if you had started with an odd number of antinodes before rotating the vibrator and describe your prediction. 119LAB2-RESONANCE.DOC

7 20. Test your prediction. Describe the results of your test and explain any inconsistencies. 21. Explain why the frequency changes when you rotate the vibrator. [Hint: Consider the motion of the string as the vibrator alternately moves toward and away from the pulley. The diagram may help.] Vibrator deflection string 119LAB2-RESONANCE.DOC