Lab 11. Vibrating Strings Goals To experimentally determine relationships between fundamental resonant of a vibrating string and its length, its mass per unit length, and tension in string. To introduce a useful graphical method for testing wher quantities x and y are related by a simple power function of form y = ax n. If so, constants a and n can be determined from graph. To experimentally determine relationship between resonant frequencies and higher order mode numbers. To develop one general relationship/equation that relates resonant of a string to four parameters: length, mass per unit length, tension, and mode number. Introduction Vibrating strings are part of our common experience. Musical instruments from all around world employ vibrating strings to make musical sounds. Anyone who plays such an instrument knows that changing tension in string changes resonant of vibration. Similarly, changing thickness (and thus mass) of string also affects its. String length must also have some effect, since a bass violin is much bigger than a normal violin. The interplay between se factors is explored in this laboratory experiment. Water waves, sound waves, waves on strings, and even electromagnetic waves (light, radio, TV, microwaves, etc.) have similar behaviors when y encounter boundaries from one medium to anor. In general all waves reflect part of energy and transmit some into new medium. In some cases amount of energy transmitted is very small. For example a water wave set up in your bathtub moves down length of tub and hits end. Very little energy is transmitted into material of tub itself and you can observe a wave of essentially same size as incident wave being reflected. The clamps at ends of a string provide similar boundaries for string waves such that virtually all energy of wave is reflected back and wave travels from one end to or. The wave bounces back and forth. If waves are sent down a string of some length at a constant, n re will be certain frequencies reflected waves and waves being generated on string interfere constructively. That is, peaks of incident waves and peaks of reflected waves coincide spatially and thus add toger. When 53
54 CHAPTER 11. VIBRATING STRINGS this occurs, composite wave no longer travels along string but appears to stand still in space and oscillate transversely. This is called a standing wave for obvious reasons. A marching band that is marching in place but not moving is a fair analogy. You can easily demonstrate this phenomenon with a stretched rubber band. These standing waves occur only at particular frequencies, known as resonant frequencies, when all necessary conditions are satisfied. These necessary conditions depend on factors mentioned above, such as wher string is clamped tightly at ends or not (i.e., boundary conditions), length of string, its mass per unit length, and tension applied to string. With this in mind, we will systematically explore how resonant depends on three of four factors listed above. In all cases our strings are clamped or held tightly at both ends; we consistently use same boundary conditions. Finally, we will search for a single equation that describes effect of length, tension, and mass per unit length on resonant. Equipment set up A schematic diagram of set up is shown in Figure 11.1. Connect speaker unit to output terminal (marked with a wave symbol) and ground terminal (marked with ground symbol) of ScienceWorkshop interface unit. The interface unit can be configured to produce a voltage that varies sinusoidally at a known. In Experimental Setup window, click on image of output terminal (marked with a wave symbol). In window that appears, make sure that waveform pull down menu is set to Sine Wave. Use and voltage windows to set and output voltages, respectively. Keep output voltage below 4.5 V. Click Auto button (which toggles Auto function off), n click On button to start voltage generator. This voltage drives an audio speaker mechanism that lacks diaphragm that normally produces sound. You will neverless hear some sound from speaker drive mechanism. This sound can be irritating, so use minimum voltage required to make a good measurement. This speaker drive oscillates in synchrony with drive voltage and is connected to string via an alligator clip. Caution: Do not apply loads greater than 10 kg to end of string! Effect of string length on resonant Start with 1.3 g/m string (see tag attached to end of string) and hang a total mass of 5 kg, including mass of mass hanger, on end of string. Determine fundamental resonant for five or six different string lengths. Plucking string with your finger near middle point excites a vibration of string primarily in its fundamental resonant mode (also called first harmonic). Pluck string and note how string vibrates. The vibration of string stops a short time after you pluck it because of energy losses due to air friction. The speaker drive allows you to pump energy into vibrating system at same rate that it is lost, so that vibration can be maintained for as long you wish. The string will vibrate strongly only at certain well-defined frequencies. By adjusting of speaker drive slowly while watching
55 pulley alligator clip string Mechanical vibrator waveform display Digital function generator (model may vary) range mass hanger amplitude amplitude a adjust adjust Figure1. 11.1. Apparatus Apparatus for forvibrating vibrating string string experiment. experiment. Make sure that string lengths that you test are approximately uniformly spaced between 0.4 m and string approximately you should 1.7 be m. able (The tomaximum find string length that makes is limited by string vibrate length of in its fundamental table.) resonant mode. You can recognize fundamental resonant easily because whole By middle graphical portion means of determine string oscillates a mamatical up and down function likefor a jump fundamental rope; fundamental resonant, resonance can f, as bea thought function of of as L, jump L is rope length mode. of For best vibrating resultsstring you must as determined continue adjusting by placement speaker of drive alligator until you clip. have Do found you get a middle linear graph of if you resonance, plot f on y-axis amplitude and L on of x-axis? vibration is Instead, try plotting f on y-axis and 1/L on x-axis. What important property of wave on maximized. string can be determined from this graphical analysis? (Feel free to refer to your text or web.) Note The that units distance of slope from of this alligator graph (assuming clip to it top is linear) of pulley provide information string on is what held this tightly quantity might be. Explain your reasoning! determines length of vibrating string. The alligator clip does vibrate slightly but string behaves very nearly as if clip defines a clamped end. (The motion of alligator clip cannot 3. be Effect ignored of string for verymass-per-unit-length heavy strings. For se, on youresonant may have to visually locate point near alligator clip which appears to be clamped and doesn t vibrate.) For this set of experiments, use maximum string length employed in above and hang a total of 5 Make kg on sure that end of string string. lengths Test that you four test are strings approximately in box, noting uniformly mass spacedper between unit 0.4 m length and (μ) approximately indicated on 1.7 m. attached (The maximum cards. Find string fundamental length is limited oscillation by length of for each table.) of By graphical strings at means your station. determine Remember a mamatical that you function already took for one fundamental data point while resonant observing, f, as effect of string length. Determine graphically wher relationship between fundamental, a function f, and of L, mass L per is unit length length, of μ, that vibrating is f(μ), string is a simple as determined power function. by placement If so, find of alligator clip. Do you get a linear graph if you plot f on y-axis and L on x-axis? Instead, try 61
56 CHAPTER 11. VIBRATING STRINGS plotting f on y-axis and 1/L on x-axis. What important property of wave on string can be determined from this graphical analysis? The units of slope of this graph (assuming it is linear) provide information on what this quantity might be. Explain your reasoning! Effect of string mass-per-unit-length on resonant For this set of experiments, use maximum string length employed in above and hang a total of 5 kg on end of string. Test four strings in box, noting mass per unit length (µ) indicated on attached cards. Find fundamental oscillation for each of strings at your station. Remember that you already took one data point while observing effect of string length. Determine graphically wher relationship between fundamental, f, and mass per unit length, µ, that is f (µ), is a simple power function. If so, find equation for as a function of mass/unit length. Refer to Uncertainty-Graphical Analysis Supplement in this lab manual for details. Effect of string tension on resonant For se experiments, use a string with a mass/length between 1.0 and 6.0 g/m and a length of at least 1.5 m. Determine fundamental resonant of string as total mass on end of string is increased from 1.0 to 10.0 kg. The weight of hanging mass will equal to tension in string, T. Graphically determine wher relationship between fundamental resonant, f, and string tension, T, is a simple power function. Again refer to Uncertainty-Graphical Analysis Supplement in lab manual. Effect of harmonic mode number on resonant Using 1.3 g/m string and 3 kg hanging mass, set length of string to at least 1.5 m. So far you have looked at fundamental or first harmonic of string vibration. The second harmonic (mode number n = 2) will have a jump rope mode on each half of string but y will oscillate in opposite directions. Increase driver until you find this resonance and record it. The third harmonic will have three jump rope modes on string, etc. At very least you should collect data for n = 1, 2, 3, and 4. If time allows, determine frequencies for even higher n values. Determine relationship f (n) between resonant, f, and mode number, n, by graphical means. Summary Summarize your findings clearly and succinctly. Can you write a single mamatical function that encapsulates all relationships that you have discovered? That is f (T, µ,l,n). Note that taking sum of four relations you determined above will not work. Compare your experimental results with those oretically predicted in your textbook. (This is sometimes included in a section
on musical instruments.) Show that textbook formula is dimensionally correct. Be quantitative in your comparisons. 57 Before you leave lab please: Turn off power to all equipment. Leave only 1 kg mass hanger on end of 1.3 g/m string. Straighten up your lab station. Report any problems or suggest improvements to your TA.