E X P E R I M E N T 13

Transcription

1 E X P E R I M E N T 13 Stadig Waves o a Strig Produced by the Physics Staff at Colli College Copyright Colli College Physics Departmet. All Rights Reserved. Uiversity Physics, Exp 13: Stadig Waves o a Strig Page 1

2 Purpose I this experimet, you will explore the relatioship betwee strig legth, wavelegth, frequecy, liear desity, ad strig tesio i a stadig wave, thus gaiig a empirical uderstadig of the ormal modes of vibratio i a stretched strig. Equipmet 1 Power Amplifier 1 Super Pulley 1 Mass Set w/ Hager 1 Mechaical Wave Driver 1 C Clamp 1 Support Rod 1 Meter Stick 1 Roll of White Strig 2 Baaa Plug Patch Cords 1 Uiversal Table Clamp Itroductio A wave movig withi ay material is evidece that eergy is beig trasported as the result of a disturbace. There are two distict categories of waves: mechaical ad electromagetic. Mechaical waves require some kid of material to travel i, but electromagetic waves, icludig light, do ot. The speed of both categories of waves depeds o two properties of the material they are movig through. For mechaical waves they are a iertial property ad a elastic property. For electromagetic waves they are the permittivity ad permeability of the material. For a mechaical wave i a stretched strig, the iertial property is its liear desity (its mass per uit legth), ad the elastic property is the tesio force i it. A wave will propagate alog the strig if you disturb its equilibrium state at ay positio. Whe the wave reaches either ed, it will reflect ad propagate back toward the disturbace. If you make the disturbace repetitive by usig, say, a electric vibrator at oe ed, the waves propagatig away from the vibrator iterfere with those that are reflected back from the other ed. If the legth of the strig is a itegral multiple of the wavelegth of the iterferig waves, the iterferece patter will be statioary i the strig. Such a statioary wave patter is called a stadig wave. I this experimet, you will create stadig waves i a stretched strig ad the measure their wavelegth. You will explore the relatioship betwee strig legth, wavelegth, frequecy, liear desity, ad strig tesio i a stadig wave, thus gaiig a empirical uderstadig of the ormal modes of vibratio i a stretched strig. You will compare your measuremets of stadig waves to the theory that relates these properties. Whe you are fiished, you will be able to 1. Explai how stadig waves are created. 2. Idetify the odes ad atiodes ad the umber of segmets i a stadig wave. 3. Discuss the factors that determie the atural frequecies of a vibratig strig. Uiversity Physics, Exp 13: Stadig Waves o a Strig Page 2

3 Theory The properties that characterize a wave are its wavelegth λ, its frequecy of oscillatio f (measured i hertz, or 1/s = s 1 ), ad its speed v. These properties are related by the equatio: λ f = v Mechaical waves propagate through a medium i either a logitudial or a trasverse mode. I a logitudial wave, each particle i the medium oscillates i the same directio as the wave propagatio. Waves i a vibratig sliky sprig ad soud waves i ay material travel i this maer. I trasverse waves, each particle oscillates perpedicular to the directio of wave propagatio. The waves i a stretched strig vibrate i a traverse mode. As each particle oscillates, its maximum displacemet up ad dow is called the wave s amplitude, desigated as +A or A. Figure 13.1 is a plot of displacemet vs. either positio or time. The eergy beig carried by the wave is related to its amplitude. The period of oscillatio is iversely related to the frequecy T = 1/f. Two waves meetig each other will iterfere. The combied wave they produce is a simple superpositio of the two waves. If two waves movig i opposite directios have the same amplitude ad frequecy, their iterferece produces a stadig wave as show i Fig The positios of miimum displacemet (destructive iterferece) are called odes, ad the positios of maximum displacemet (costructive iterferece) are called atiodes. The legth of oe segmet of the stadig wave is equal to oe-half its wavelegth. Whe a strig is vibrated at oe ed, waves travelig from the vibrator iterfere with waves reflected from the opposite fixed ed. This iterferece produces a stadig wave i the strig at specific frequecies that deped o the strig s desity, tesio, ad legth. If the strig is vibrated at multiples of this frequecy, stadig waves with multiple segmets will appear. The higher frequecies are kow as harmoics (see Figure 13.3). Uiversity Physics, Exp 13: Stadig Waves o a Strig Page 3

4 Note that each segmet is equal to oe-half of a wavelegth. Thus, for a give harmoic, the wave-legth becomes: 2L λ = where L is the strig legth ad is the umber of segmets. You ca therefore express the velocity of a wave i a stretched strig as: v = 2Lf You ca also fid the velocity of a wave i a stretched strig from the relatioship: v = μ where the tesio force is the elastic property i the strig ad the liear desity μ is the iertial property. You ca fid the value of μ by weighig a kow legth of strig. μ = mass legth You ca solve for the tesio force by elimiatig v betwee the two equatios above: = 2 2 4L f μ 2 If you keep the legth ad frequecy costat but allow the tesio to vary, a graph of versus 1/ 2 yields a straight lie whose slope is the umerator of this equatio: 4L 2 f 2 μ. Kowig the legth ad frequecy, you ca the fid the value of μ from the slope of this graph. You ca also solve the last equatio for the frequecy: f = 2L μ A graph of f vs. will yield a straight lie whose slope is the coefficiet of : 1 2L μ. Uiversity Physics, Exp 13: Stadig Waves o a Strig Page 4

5 Procedure You will apply tesio to a legth of strig by hagig mass from it over a pulley, as show i Figure You will the create waves i the strig with a computer-drive vibrator ad adjust the tesio (i Part A) ad the frequecy (i Part B) to create stadig waves havig from 1 to 7 segmets. 1. Cut a piece of strig about 5 m log. Stretch it out o the table (doublig it if ecessary) ad measure its legth l s (to 3 SD). Wad the strig up ad carefully measure its mass m s (to 3 SD) o the electroic balace i the lab prep area. Record both values i Data Table Calculate the strig s liear desity μ ad record its value i Table Cut about a 2-m piece of your strig ad tie a loop i each ed. Slip oe loop over a vertical support rod that is clamped to the table. Pass the strig over a pulley that is clamped to the ed of the lab table about 1.5 m away ad hook a mass hager i the other loop. 4. Place the wave driver uder the strig ear the vertical support rod. Slide the loop dow the support rod util the strig rests i the slot o the top of the wave driver. Use baaa-plug patch cords to coect the wave driver to the output of the power amplifier. With the power switched off, plug the Power Amplifier ito Aalog Chael A. A. Variable Tesio, Costat Frequecy ad Legth 1. Measure the legth of strig L betwee the vibrator ad the top of the pulley. Record this legth (to 3 sig. dig.) uder Table Switch o the computer system ad the power amplifier (the switch is o the back). Ope Data Studio ad select Create Experimet. Double-click the Power Amplifier ico i the Sesors list. Click the O butto i the Sigal Geerator widow. Doubleclick the Output ico uder Sigal Output. Set the sie wave frequecy f to Hz ad the Amplitude to V. Note: Vary the amplitude ad frequecy by clickig o the left/right arrowheads to set the step icremet, the click o the + or butto to step the value up or dow by that icremet. 3. Pull dow hard eough o the hager to make the strig vibrate i its fudametal mode (oe segmet). Now put eough mass o the hager to maitai that strig Uiversity Physics, Exp 13: Stadig Waves o a Strig Page 5

6 tesio. Adjust the mass to maximize the cetral atiode ad make odes at the two eds that are very dark ad clea (ot vibratig). Record the hagig mass m h (to 3 SD), icludig the mass of the hager, i Table Calculate ad record the tesio force i the strig. 4. Now remove eough mass from the hager to create maximum-amplitude stadig waves at each of its higher harmoics (2 to 7 segmets). Record each mass ad tesio (to 3 SD) i Table Plot a graph of the tesio vs. 1/ 2 ad draw a best-fit straight lie through the data poits. Calculate the slope of the lie (to 3 SD), ad record its value i Table Usig this measured value of the slope, calculate the liear desity of the strig. Record it (to 3 SD) i Table Calculate ad record the percet differece betwee your two values of liear desity. B. Variable Frequecy, Costat Tesio ad Legth 1. Put 1000 g o the mass hager. Calculate ad record the resultig tesio (to 3 SD) i Table Vary the output frequecy f of the Sigal Geerator util you fid the fudametal frequecy (a stadig wave havig oe segmet). Record this frequecy (to 3 SD) i Table Fid the frequecies required for the higher harmoic stadig waves ( = 2 through = 7), ad record these (to 3 SD) i Table Plot a graph of f vs. ad draw a best-fit straight lie through the data poits. 5. Calculate the slope of the lie ad record it (to 3 SD) i Table Usig the value of the slope, calculate the liear desity of the strig. Record it (to 3 SD) i Table Calculate ad record the percet differece betwee this value of μ ad your directly measured value. 8. Switch off the computer ad power amplifier. Coil the coectig cable ad secure it with a twisty-tie. Retur all your equipmet to the lab cart ad clea up your area o the lab table. Uiversity Physics, Exp 13: Stadig Waves o a Strig Page 6