Constant Acceleration
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1 Objecive Consan Acceleraion To deermine he acceleraion of objecs moving along a sraigh line wih consan acceleraion. Inroducion The posiion y of a paricle moving along a sraigh line wih a consan acceleraion a is given by he following equaion y y 0 = ½ a 2 + v 0 where is ime, y 0 and v 0 are he posiion and speed a = 0, respecively. In his lab, you will find he acceleraion by measuring he displacemen of an objec as a funcion of ime. A freely falling objec is an objec moving under he influence of graviaional force alone. I has a consan acceleraion called free-fall acceleraion. This acceleraion is direced downward and is magniude is denoed by g. The acceped value of g is 9.80 m/s 2. If here is anoher consan force, such as ension force, acing on he objec in addiion o he graviaional force hen he acceleraion of he objec will be differen from he free-fall acceleraion. You will deermine he acceleraion of a freely falling objec and an objec moving under he influence of ension and graviaional forces. Exercise 1 Freely falling objec Cauion: Do no ouch he high-volage wire while he spark imer is operaional! If you need help ask he lab echnician or your insrucor o assis you in collecing daa. Figure 1 shows he apparaus ha you will use for measuring acceleraion. An elecromagne is used o hold and release he objec. The elecromagne acs like a magne only when here is elecriciy passes hrough i. If he elecriciy is swiched off, he objec is released. While he objec is falling i passes beween wo verical hin wires. A waxed ape is mouned jus in fron of one wire. Pulses of high volage are applied o he oher wire. The meallic par of he objec makes he gap beween he wires small. As a resul, he pulses creae sparks ha jump from one wire o he oher hrough he meallic par of he objec and he waxed ape. When a spark passes hrough he waxed ape, i makes a spo on he ape. The ime beween any wo successive pulses is he same. In his exercise, you will choose he ime inerval beween any wo successive spos on he ape o be 1/60 s by selecing he frequency of he spark imer a 60 Hz. 1
2 Elecromagne Falling objec Wire Waxed ape Figure 1: Free-fall Apparaus Collecing daa 1. Moun a new ape in place. Laer, you need o idenify he op of he ape. So wrie on he op of he ape he word Top. 2. Suspend he objec from he elecromagne. 3. Make sure he frequency of he spark imer is seleced o be 60 Hz. See Figure 2. This means ha he ime inerval beween successive spark poins is 1/60 s. 4. Swich on he spark imer and sar sparking by pressing he rigger swich. Simulaneously release he objec by swiching off he elecriciy o he elecromagne. Keep pressing he rigger swich unil he objec reaches he ground. Trigger swich Frequency selecor Figure 2: Spark Generaor 2
3 5. Afer you have performed he experimen, remove he ape from he apparaus and place i on a able wih a ruler as shown in Figure 3. Fasen he ape o he able using sicky ape. Noe ha he ruler is placed such ha he numbers are decreasing owards he boom of he ape. This is because he posiive y-axis is aken along he upward direcion. I is possible ha he objec remained wih he magne for a very brief momen before falling, afer he firs spo is made on he ape. In ha case he disance beween he firs and he second spos would be shorer han wha i should be. Therefore, i is advisable o avoid he firs few spos especially if you see any irregulariies in hem. Noe ha he posiion y 0 for he spo where you choose = 0 does no have o be a cm. Y-axis Avoid he firs few spos Sicky ape y = cm a = 3/60 s y = cm a = 2/60 s y = cm a = 1/60 s y = cm a = 0 s Selec his spo as y 0 and assume =0 s a his spo. Figure 3: Daa collecion explained 6. When you ake he readings, look a he reading sraigh on o avoid any parallax error (see Figure 4). This is also he reason why he meer scale is kep sanding on is edge han lying fla; o minimize parallax error. 3
4 Figure 4: Parallax error explained 7. Open Microsof EXCEL and ener 0 in cell A2 and he corresponding firs y value in meers in cell B2. 8. In cell A3, wrie his formula: =A2+1/60 and press he Ener key. Remember ha you have se he ime inerval beween any wo successive spark spos o be 1/60 s. In cell B3, measure he posiion of he spo a =1/60 and record i. 9. To wrie he oher values of, click on cell A3 and use he lef mouse buon o press on he small square a he lower righ corner of he cell and drag i down o as many cells as you have daa for. This will add 1/60 o he value of he previous cell. See Figure 5. 4
5 Drag down he small square. All he way down o as many cells as you have daa for Figure Coninue enering he posiion values for =2/60, =3/60 and so on. Make sure o use cm-scale of he ruler, no inch-scale. For he example in he figure, a = 0, y = m; = 1/60 s, y = m; = 2/60 s, y = m; = 3/60 s, y = m and so on. Take a leas 15 poins. Wrie he values of y, in unis of meers, ha you recorded from he ape in column B. Noe he values should be decreasing. 11. Copy he enire able and pase i in your repor as a Microsof Excel Workshee Objec. To do his: click Pase in he Home Tab Pase Special.. Choose Pase As Microsof Excel Workshee Objec and click OK. Analyzing Daa You will deermine he acceleraion by ploing y versus, and fiing he curve o a parabola. Ploing y versus means ha you need o plo y on he verical axis and on he horizonal axis. 1. To plo, selec he daa wih he heading, click on Inser ab, hen in he Char group click on Scaer buon and choose he firs opion as you did in Lab 2-Graphing. Then in he Char Layous group choose Layou 1. Change he char ile o Exercise 1, x- 5
6 axis ile o (s) and y-axis ile o y (m). Add gridlines o he x-axis by righ clicking on he x-axis numbers and choose Add Major Gridlines. 2. To fi your daa o a parabola, righ click on he daa poins in your plo and choose Add Trendline. Selec Polynomial wih order 2, ick Display Equaion on char and close he Forma Trendline window. Your graph will look like Figure 6 (he numbers of your rendline equaion may be differen). y (m) Exercise 1 y = x x (s) Figure 6: An example of polynomial fi 3. You can deermine he acceleraion a by comparing y = ½ a 2 + v 0 + y 0 wih he Trendline equaion y = x x Therefore, ½ a = or your measured value of a = 9.75 m/s 2. The negaive sign indicaes ha he free fall acceleraion is downwards. 4. The acceped value of free fall acceleraion in Dhahran area is m/s 2. How well does your measured value ( 9.75 m/s 2 in he example) agree wih he acceped value? Calculae he percen difference beween your measured value and he acceped value using Measured value Acceped value Percen difference = 100 Acceped value 5. Copy your graph and pase i in your repor. Also record your values of a and he percen difference in your repor. Exercise 2 Anoher way of analyzing he daa Anoher way of geing he acceleraion a is ploing he daa such ha you ge a sraigh line. You can do his by redefining your variables. The redefined variables should make he original equaion looks like ha of a sraigh line Y = m X + b. 6
7 Here, Y is he variable ploed on he verical axis and X is he variable ploed on he horizonal axis. The consan m is he slope and he consan b is he y-inercep (recall Lab 1- Graphing). If you divide he original equaion, y y 0 = ½ a 2 + v 0, by you will ge y y 0 = ½ a + v 0 So you need o plo y y 0 on he verical axis and on he horizonal axis. Then he slope will be ½ a and he y-inercep will be v 0. This equaion forms he basis for he daa analysis for his exercise. Your equaion General sraigh line equaion Variable ploed on he verical axis slope Variable ploed on he horizonal axis y-inercep 1. In column C of your Excel workshee of Exercise 1 analysis, calculae y y 0. Remember y0 is he posiion a ime = 0. Tha is he value in cell B2. To calculae y y 0, ype =(B2-B$2)/A2 in cell C2 and press Ener on he keyboard. Since y0 is a fixed value, i needs o be kep as a consan in calculaing column C values. This is done by yping B$2. 2. To wrie oher values of y y 0, click on cell C2 and use he lef mouse buon o press on he small square a he lower righ corner of he cell and drag i down. Noe for he firs daa poin you are dividing 0 by 0, and you will ge #DIV/0!. Remove his and leave he cell empy for his daa poin. 3. Selec only he columns A and C wih he help of Crl key on he keyboard and plo versus, and add linear rendline. y y 0 4. From he slope calculae a. 5. Copy your graph and pase i in your repor. Also record your value of a in your repor. 7
8 Exercise 3 An objec wih small ne force In his exercise, you will deermine he acceleraion of an objec moving under he influence of ension and graviaional forces. You do no need o use he elecromagne here. The falling objec is conneced o a pulley sysem as shown in shown in Figure 7. The objec is no more freely falling as he ension is acing on i in addiion o he graviy. Figure 7: An experimenal se up for measuring smaller acceleraion 1. Moun a new ape in place and mark is op locaed near he elecromagne. 2. The masses in he mass holder are already seleced for you o be slighly (abou 10 g) less han he objec so when he sring is released i will go down wih a smaller acceleraion (much less han 9.80 m/s 2 ). 3. Bring he objec o he op (near he elecromagne) and hold he objec a he op wih your hand. 4. Se he frequency of he spark imer o 10 Hz. This means ha he ime inerval beween any wo successive spark spos is now 1/10 s. This is because he speed of 8
9 he objec will be small, and hus he disance beween wo successive spos will be very small for 1/60 s ime inerval. 5. Sar he spark imer by pressing he rigger swich and immediaely release he objec from he op. You should NOT release he spark imer rigger swich unil he objec reaches he boom. 6. Analyze your daa, plo y versus and deermine he acceleraion of he objec as you did in Exercise Find he raio of his acceleraion wih he free-fall acceleraion: a / g. 8. Copy your graph and pase i in your repor. Also record your values of a and he raio a / g in your repor. 9
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