Lab M6: The Doppler Effect

Transcription

1 M6.1 Lab M6: The Dppler Effect Intrductin The purpse in this lab is t teach the basic prperties f waes (amplitude, frequency, waelength, and speed) using the Dppler effect. This effect causes the frequency f waes t be higher fr a surce that is appraching yu and lwer fr a surce that is ming further away. Sund is a pressure wae in air. When we hear a, we are sensing a small ariatin in the pressure f the air near ur ear. The speed f a wae in air,, is abut 340m/s; traels 1 mile in abut 5 secnds. This speed depends nly n the prperties f the air (temperature, cmpsitin, etc.), nt n the frequency r waelength f the wae and nt n the mtin f the surce r receier. Cnsider a sinusidal wae in air with frequency f and waelength λ. The speed is related t f and λ by (1) f λ. T see where this relatin cmes frm, think: speed change in distance change in time. The time it takes fr ne waelength f the t pass by is the perid T, s λ/t. But f 1 T s f λ. Nte that as f increases, λ ges dwn, but the speed stays the same. The frequency range f human hearing is abut 20 Hz t 20,000 Hz. (The upper end drps as we age; fr peple er 60, it is 8-13 khz, while dgs can hear up t abut 35 khz.) When the siren frm an ambulance passes by a listener, the listener reprts a change in the pitch f the siren. This is the Dppler effect fr waes. A surce f ming away frm the receier (listener) has a lwer apparent frequency (tne r pitch), while a surce ming tward the receier has a higher frequency. The frequency shift is apprximately prprtinal t the relatie speed f the surce and receier. T understand the Dppler effect, cnsider a statinary receier detecting the frm a ming surce, a surce which is emitting a wae with cnstant frequency f. Let us cnsider the case in which the surce is ming directly tward the statinary receier with a speed. In this cntext, statinary and ming mean at rest r nt with respect t the air which carries the. The diagram belw shws a series f successie waefrnts emitted when the surce was at psitins 1, 2, 3, 4, and 5.. The time between successie waefrnts emitted by the surce is the perid T distance x T 1. f 1, and during the time T f, the surce mes a

2 M6.2 wae 1 wae 2 wae 3 wae x statinary receier ming surce, speed, frequency f If the surce were nt ming, the spherical waefrnts wuld be cncentric, centered n the statinary surce, and the distance between successie waefrnts wuld be the waelength λ. The speed f, the waelength λ, the frequency f, and perid T are related by equatin (1) λ T λ f. Because the surce is ming, the waefrnts are crwded tgether n the side near the receier, and the distance between waefrnts, n the receier side, is (2) λ λ x λ T λ f λ λ λ 1. These waefrnts strike the receier at interals f T secnds with a frequency which is related t and λ by [equatin (1) again] f 1, T λ T λ f.

3 M6.3 Sling fr f, we hae (3) f λ f λ 1 1 (receier statinary, surce appraching). The frequency is shifted up and the receier hears a higher pitch. With sme algebra, ne can shw that the frequency shift f f f is gien by (4) f f 1, which, if <<, is apprximately (5) f f. Frm eq n (3), sling fr, we hae (6) f f f f f. Thus, ne can determine f the surce. frm measurements f the frequency shift and the speed If the surce is receding frm, rather than appraching the statinary receier, then the ( ) sign in eq n (4) is replaced with a (+) sign, and the frequency is shifted dwn. If the surce is statinary, and the receier is ming, then a new deriatin is required and ne btains a different frmula (nt shwn here). Radar guns use the Dppler effect, but with electrmagnetic waes, rather than waes. The pliceman s radar gun acts as bth a transmitter and receier. It emits a radar signal f knwn frequency, which is reflected frm a ming car, and the Dppler-shifted reflected signal is detected.

4 M6.4 Experiment The surce is an ultrasnic speaker, which emits a tne at abut 40 khz well abe the range f human hearing. The surce rides n an air track glider, whse speed is determined with a pair f phtgates and a timer. An ultrasnic receier detects the frm the surce and the frequency is displayed n a high-precisin frequency cunter. Bth the transmitter and the receier signals are displayed n the scillscpe. The experiment is t measure the frequency shift as a functin f the speed f the surce, and, using these data, t cmpute the speed f. A schematic f the experimental apparatus is shwn belw. Begin by leeling the air table. After turning n the air supply, place the glider midway between the phtgates, and adjust the height f the leeling ft until the glider remains statinary. Make sure that the blcking card attached t the glider interrupts the phtgates. The timer starts when the card interrupts gate 1, stps when the card interrupts gate 2, and displays the time interal t. The reslutin f the timer can be set t either 1 r 0.1 msec reslutin, but with 0.1msec reslutin, the timer erflws at nly 2 sec; use 1 msec reslutin t aid erflw. By measuring the distance x between the tw phtgates, the speed f the x glider can be cmputed frm. Make sure that nthing interferes with thin wires t leading t the surce, as the glider traels back and frth n the air track. On the scillscpe, bsere bth the signal t the surce (n channel 2) and the signal frm the receier (n channel 1). The ultrasnic transducers used fr the surce and receier resnate at a fixed frequency f arund 40 khz; they d nt functin well at frequencies far frm 40 khz. With the surce statinary, turn the fine adjust knb n the sine-wae generatr t tune the frequency f the surce fr maximum signal n the receier. After allwing the sine-wae generatr seeral minutes t warm up and stabilize, recrd the frequency f f the surce with the frequency meter. With the glider statinary, there is n Dppler shift, and the frequency f the surce and receier are identical. Hld the glider statinary and read f frm the receier frequency meter, which has a 1 Hz reslutin. Check f frequently; if it is drifting yu will hae t recrd it each time yu take data. Make sure that the receier is directly in frnt f the transmitter, at the same height, etc. s that the surce mes directly tward the receier. The TTL utput f the sine-wae generatr is used t externally trigger the scillscpe. This ensures that there is always a stable display n the scillscpe.

5 M6.5 Thin flppy wire t surce f adjust sine-wae generatr TTL f ut timer Ch.1 ext trig x scillscpe Ch.2 gate 1 gate 2 card t trip phtgates glider ultrasnic receier amplifier air track transmitter delay kHz f leeling ft HOLD RESET frequency meter Prcedure Part 1. Speed f frm the measured Dppler shift Yur task is t measure the frequency f at the receier and hence the frequency shift f f f, as a functin f the speed f the surce. Frm these data, yu will cmpute the speed f, as shwn belw. Reset the timer t zer and push the glider tward the receier s that it passes thrugh gates 1 and 2 at a unifrm speed. Recrd the frequency f n the frequency meter and the phtgate time interal t. Repeat this prcedure fr seeral (at least 10) trials, arying the speed f the glider er as large a range as pssible. Yu cannt let the speed f the glider g much faster than 1 m/sec (which is pretty fast!) r the frequency cunter wn t hae time t recrd the frequency. The frequency meter wrks by cunting cycles fr ne secnd (frequency in Hz number f cycles in ne secnd). There is a little red light which is n while the meter is cunting. T prepare the frequency meter t take a reading, make sure the FREQ buttn is depressed, turn the DELAY knb t the HOLD psitin, and then press the HOLD buttn. When yu are ready t recrd the frequency, press the RESET buttn. As sn as yu press RESET, the meter will cunt cycles (the red light will cme n) fr 1 sec and then the frequency will be displayed (in kilhertz, khz). When taking measurements at faster speeds, yu shuld press the RESET buttn immediately after pushing the glider s that the 1 secnd cunt time is up befre the glider reaches the end f the track. On the table near the track there is sme -absrbing fam material which seres the imprtant functin f preenting reflected (eches) frm reaching the receier. Make sure the fam cers the side f the track. If the absrbers were nt present, then the receier wuld receie nt nly the direct-path frm the

6 M6.6 transmitter but als arius reflected-path eches. The eches can interfere destructiely with the direct-path causing a lss f signal at the receier. transmitter direct path receier air track Table tp ech If the receier signal drps belw abut 0.2V peak-t-peak, then the frequency cunter cannt cunt crrectly and the displayed frequency will be t lw. Check that the receier is getting adequate signal by bsering the scillscpe utput as yu slide the transmitter alng the whle length f the track. Stp the transmitter at any pints alng the track where the receier signal is unusually small, and check that the frequency cunter is still displaying f. Plt yur measured frequency shift f f f s. the speed f the surce. Yu shuld get a ery straight line, as predicted by eq n (5). Frm the appearance f the graph, decide whether any f the data pints shuld be thrwn ut. (If ne pint deiates substantially frm the thers, it is usually an indicatin that a number was recrded incrrectly.) Fr each f the gd data pints, cmpute the speed f frm eq n (6). Using yur seeral alues f, cmpute the mean, the standard deiatin, and the standard deiatin f the mean. (T aid subscript chas, we write s fr belw.) s 1 si σ N i ( si s) i N 1 2 σ mean σ N Part 2. Cmparisn with the knwn speed f The speed f in air depends n the temperature accrding t (7) m ( / s) T where the temperature T is in degrees Celsius. This expressin is nly accurate fr temperatures near rm temperature. (Yu can read the temperature frm the big dial thermmeter n the wall in the lab). Cmpare this theretical thery cmputed frm eqn(7) with yur measured meas frm Part 1. Part 3. Calculatin f the waelength f the ultrasnic wae

7 M6.7 Frm the knwn frequency f and yur measured frm Part 1, cmpute the waelength λ f the. Als cmpute the uncertainty δλ. Part 4. Interfermetry and waelength In an interfermeter, a signal that has taken a direct path frm a surce t a receier is cmpared with a signal that has taken an alternate path. On channel 1 f the scillscpe is the electrical signal directly t the transmitter. On channel 2 is the signal frm the receier that hears the after passing thrugh abut a meter f air. If this air path is a whle number (N) f waelengths, these tw signals will be in phase and will add. If the air path is N + 1/2 waelengths, the signals will be ut f phase and will tend t cancel. The subtractin will nt gie exactly zer unless the tw signals are exactly the same amplitude. The time fr the electrical signals t trael can be ignred in cmparisn with the time fr the t trael thrugh air. Adjust the Channel 1 and Channel 2 lts/diisin knbs n the scillscpe s that the tw signals are nearly three diisins in height (peak-t-peak). (Yu may hae t fiddle with the little "cal" knbs in the middle f the bigger knbs.) Abe the channel 2 knb there is an ADD-ALT-CHOP switch that yu shuld me t the ADD psitin. Nw the scillscpe displays the sum f the direct signal and the signal that has gne thrugh the air. Me the glider with yur hand and ntice the change in amplitude f the summed signals. Switch briefly back t the ALT r CHOP psitin t see the signals me frm in phase t ut f phase. Return t the ADD psitin and measure with a ruler the distance that yu must me the glider t pass thrugh 10 t 20 maxima r minima. Diide this distance by the number f maxima r minima t determine the waelength frm interfermetry λ int. What is δλ int? Hw des it cmpare with the waelength λ that yu calculated in Part 3? Calculate the difference between the tw measurements λ λ int λ and δ λ [δλ int 2 + δλ 2 ] 1/2.

8 M6.8 PreLab Questins: 1. What is the apprximate frequency f the ultrasnic transmitter used in this experiment? What is the waelength f at that frequency? 2. (Cunts as tw questins) Fur things are measured directly in this lab: x, the distance between the phtgates, f0, the frequency f the surce, t i, the time fr the glider t trael between the phtgates (trial i), and f i, the frequency at the receier (trial i). Using Mathcad statements, shw hw t cmpute i, the speed f the glider (trial i), and _ i, the speed f (trial i), frm these measured quantities. 3. Shw that eq n (6) fllws frm eq n (3). 4. Eq n (4) is an exact expressin fr f, while eq n (5) is nly apprximate. Using 345 m / s, 0.30 m/s, and f 40.0 khz, cmpute f using bth equatins (4) and (5). By hw much d the tw alues differ? What is the fractinal difference? (Since we want t cmpare the tw alues, display seeral digits fr the tw f s, rather than runding t significant figures.) 5. Sketch the graph f f s.. What is the apprximate slpe f this graph? Gie a algebraic expressin and a numerical alue fr the slpe. 6. Hw is it determined whether the air track is leel? 7. Assume that the alues f, δ, f, and δf are knwn. Shw hw t cmpute δλ, the uncertainty in the waelength. 8. On Jan. 17, 1930, it was -33F (-36C) in Bulder, with an uncertainty f ±1 C. What was the speed f utside n that day (include the uncertainty in the speed f )? 9. The air track in this experiment is 2 meters lng. If the glider ges faster than 2 m/s, this experiment wn t wrk, at least nt with the existing equipment. Why nt? [Hint: hw is f measured?] 10. When the surce is receding frm a statinary receier and the speed f the surce appraches the speed f, what happens t the Dppler shift f? What happens t f when the surce is appraching a statinary receier and its speed appraches the speed f.