Gravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)


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1 Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng mass are attracted by the masses of other objects. Ths means that near the surface of the earth all objects are attracted by the mass of the earth tself, and wll fall towards the center of the earth under the nfluence of gravty. Because the attracton s proportonal to mass, all objects wll fall wth the same constant acceleraton, whch s denoted by the letter g. Equpment spark tmers (/class) thermal tape to record dscharge pts. ruler & meter stck 500g mass & hanger rods & clamps to hold tmers pad to protect floor scotch tape level maskng tape (to hold thermal tape n place whle measurng) vertcal rod & bench edge clamp graphng software graph paper (back up) Note: just setups at front of room for entre class Before the Lab Read the sectons n your text descrbng onedmensonal moton and constant acceleraton. Be famlar wth the concepts of poston, velocty, acceleraton and the slope of a graph. Theory An object fallng near the surface of the earth experences a unform (constant) acceleraton. Because acceleraton s constant, n a downward drecton, and equal to g, ts moton can be descrbed by the equatons: 1 y( t) = y0 + v0t + gt v( t) = v0 + gt Here we have defned down as beng the postve y drecton. These equatons predct that a plot of velocty versus tme wll start at the value v o at tme t=0 and show a straght lne wth postve slope. The value of that slope wll be equal to acceleraton (the rate of change of velocty, a=dv/dt). Thus we can measure the value of g by determnng the slope of a graph that plots velocty versus tme for a fallng object. Procedure In ths lab you wll measure the freefall acceleraton of a mass. The freefall apparatus consst of a spark tmer and thermal tape. A freely fallng mass attached to one end of the tape pulls the tape through the spark tmer. Voltage pulses between the two electrodes of the spark tmer leave marks on the conductve (.e. shny) sde of the thermal tape. Ths records the changng poston of the object as a functon of tme. The voltage pulses are appled 60 tmes per second, so the tme between pulses s 1 60 of a second. Your lab nstructor wll make a tape for you or assst you n makng one. (WARNING: STAND CLEAR OF IMPACT AREA TO PREVENT INJURY TO FEET!) Carefully examne the tapes and draw small crcles around the spark marks so ther locatons are easly seen. (Do not cover the marks themselves, just outlne them). Occasonally a spot n the sequence may be mssng. If you thnk there s a mssng spot, draw a queston mark about where you thnk t should be. 1
2 Choose one spot near the top to be pont 0 where the poston s y o =0 at tme t=0. Ths should not be the frst spot. Do not choose the large spot where the object rested pror to beng released! Choose a spot, that s easy to measure and where all the followng ponts appear to travel n a nearly straght lne, wth no mssng spots. (Note: Because ths s not the frst spot we cannot assume that the ntal velocty, v o, s equal to zero. Usng the data table (at the end of ths handout) begn numberng ponts 0,1,,3 and record tmes for each pont so that pont 0 s t0=0, pont 1 s t 1 =1/60 sec= sec, pont s t =/60 sec = sec etc. If there are mssng spots you should nclude a poston for them n your table and wrte mssng besde these ponts. Draw a straght lne through each spot perpendcular to the edge of the tape. You wll use these lnes to measure the dstance to each spot. Carefully measure the dstance from spot number zero to each followng spot on your tape. Record these postons n your table as poston y. You wll now compute the velocty at for each spot. At pont (where means 1,,3 etc.), the poston s gven by: 1 y = y0 + v0t + gt (Equatons 1) v = v0 + gt It can also be shown for constant acceleraton only that the average velocty between two tmes s equal to the nstantaneous velocty at a tme halfway n between. Thus, for equally spaced tme ntervals, the nstantaneous velocty at tme t s equal to the average veloctes between tmes t 1 and t 3 : y3 y1 v = v = (Equaton ) t3 t1 Use Equaton to fnd the nstantaneous velocty for ponts 1,, 3 etc. Use the postons and tmes for ponts 3 and 1 to calculate v, for ponts and 4 to calculate v 3, for ponts 3 and 5 to calculate v 4, etc. Record each n your data table. You can use a smlar equaton to calculate and record average acceleratons: v3 v1 a = a = (Equaton 3) t3 t1 Make a graph of velocty versus tme for your data. Your nstructor wll show you how to do ths on the computer. If your graph makes t apparent that you faled to take nto account mssng ponts, adjust the tmes accordngly. (See Appendx E for notes f you are usng Excel.) Usng the computer program agan, ft a straght lne to your data. The slope of ths lne wll be your expermental value for g, whch you can call g expermental. Determne the slope of your lne and record t wth your data. Calculate and record the percent dfference between the expermental value and the commonly accepted value, g=9.8 m/sec, usng the equaton: gexp ermental g Percent dfference = g = 100% (Equaton 4) g If v o s n fact not zero at t=0, then your graph wll ntercept the vertcal axs at some velocty other than zero. Determne v o from your graph. Make a plot of poston versus tme usng the computer. If poston ncreases as tme squared then ths wll be a quadratc curve. Use the quadratc curve ft to ft ths data and fnd a second value for g.
3 For your report Prnt copes of the graphs that you made for each student n your group. On each graph you should label the axes and record the slope and ntercept. Wrte a cover page wth a summary descrbng the experment. Explan how you obtaned the values of g and v 0 from your data. Compare the value for g to the accepted value and dscuss possble sources of dscrepancy. You must explan why you expect your velocty graph to be a straght lne and why ts slope should be g. Comment on how the value of "g" determned from your graphs compares to the varaton n values calculated pontbypont n your table. Suppose the tme dfference between sparks were not precsely 1/60 second. How would that affect your results? (Would g expermental be too large? too small? the same?) 3
4 Data Table: Free Fall wth Spark Tmer Pont Tme from t=0, t Dstance from y=0, y sec 0 cm Calculated velocty, v Calculated acceleraton, a Remember measurements should nclude unts! Value of g from velocty graph, g expermental Percent dfference from accepted value, % g Intal speed from velocty graph, v o Value of g from poston graph, g expermental Percent dfference from accepted value, % g Intal speed from poston graph, v o 4
5 ADDENDUM TO GRAVITATIONAL ACCELERATION LAB ALTERNATIVE CALCULATIONAL INSTRUCTIONS (provded by Dr. Flores) 1. Make sure that no ponts are mssng on the tape.. Fasten tape to table usng maskng tape. 3. Skp the frst two ponts; the frst pont you use should be marked as x=0 cm (and t=0). Place a meter stck along the tape matchng the frst pont used wth 0 on the meter stck. 4. Read the postons of all other ponts usng the meter stck and record n a table. Use about 0 consecutve ponts. Note that an error as small as a mllmeter can result n a bg fluctuaton n the acceleraton. 5. Make a table lke the one below but wth at least 0 ponts: Tme x (cm) x v= x/ t v a= v/ t *60= *60=3600 1/60 53*60= *60 / *60= *60 3/ *60=40 4/ IMPORTANT NOTE: at tme t=0, poston x=0, velocty v=10, and acceleraton a= Fnd the average and the devaton of the acceleraton. Wrte your result as a = a ± d a 7. Is your result n agreement wth a=980 cm/s? If your acceleraton n free fall s not n agreement wth accepted value repeat experment. 8. Plot usng DataStudo x vs. t v vs. t a vs. t a. Note: f the value of R n your plots s 1 you have a perfect ft 9. Fnd the equaton of the straght lne n your v vs. t graph. Compare the slope wth a n step Sample calculaton of average and devaton: Gven: 3,4,6,5,7,4 average: a = ( a )/ N = ( )/ 6 =4.8 The devaton: d a = ( a a )/ N = ( )/ 6=1.16 (the vertcal bars mean absolute value) Result: a = 4.8 ±
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