Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces. Besides grvity, these re; the electromgnetic force, the wek force, nd the strong (nucler) force. In ddition to these four known forces, there is strong evidence for the existence of fifth force which is long-rnged, repulsive, nd depends on mss in some mnner sort of nti-grvity. The exct source nd nture of this force, nd even definitive proof of its existence, remin elusive. Grvity, by contrst, ws the first force to be given correct mthemticl description. Sir Isc Newton discovered universl mthemticl model for the grvittionl force tht is vlid for ll clssicl (non-quntum mechnicl), non-reltivistic interctions where the field is not too lrge. This model ws extended by Einstein in the erly 20 th century in his Generl Theory of Reltivity to cover the cses of reltivistic velocities (velocities comprble to the speed of light) nd very lrge msses. Interestingly, lthough Grvity ws the first of the fundmentl forces to be given mthemticl description, it is the only force for which reltivistic, quntum-mechnicl model hs not yet been developed. The problem of quntum grvity is one of the most theoreticlly chllenging nd importnt unsolved problems in physics tody. As importnt s reltivistic, quntum-mechnicl description of grvity is for fundmentl understnding of the workings of the universe, s prcticl mtter Newton s universl grvittionl force lw is sufficient for nerly every circumstnce likely to be encountered in the ordinry course of events. This model sttes tht the force between ny two msses m 1 nd m 2, seprted by displcement vector r is given by F G = Gm 1m 2 r 2 ˆr (1) Like ll fundmentl forces, the grvittionl interction is binry, mening tht it lwys cts between pirs of objects. One cn rbitrrily identify one object s the source of the force field, nd the other s the object of the force field. For exmple, if one tkes m 1 s the source of the grvittionl field, then the force exerted by m 1 on m 2 is described by eqution 1, with the unit vector ˆr pointing from m 1 to m 2. The negtive sign, consequence of the ttrctive chrcter of grvity, mens tht the force on m 2 due to m 1 cts in the direction opposite to the unit vector ˆr; i.e., the force on m 2 due to m 1 points from m 2 to m 1. Conversely, one could tke m 2 s the source of the field, nd m 1 its object. In this cse, the unit vector ˆr points from m 2 to m 1 nd the force cts in the opposite direction, so tht the force of m 2 on m 1 cts long the line seprting them, in the direction from m 1 to m 2. In generl terms, the unit vector ˆr points from the source mss to the object mss, nd becuse grvity is ttrctive, the force cts in the opposite direction, from the object mss to the source mss. Figure 1 depicts the directions of the forces of two mssive objects on one nother. We cn therefore decompose the grvittionl force tht source mss m s exerts on n object mss m o into the product of the object mss nd the grvittionl field due to the source mss, so tht where the grvittionl field creted by the source mss is given by [ F msonm o = Gm ] s r 2 ˆr m o = Γ s m o (2) Γs = Gm s ˆr (3) r2 nd the unit vector ˆr points from m s to m o. It is in this form tht the grvittionl force exerted by the erth on mss m ner its surfce is probbly most fmilir to you:
FIG. 1: Directions of grvittionl forces of two mssive objects on one nother F GE = m Γ E = mgŷ (4) where the unit vector ŷ points in the locl direction of up. This direction, of course, is just the direction of the unit vector pointing from the center of the erth to the locl position of the mss m. We cn then identify the grvittionl field of the erth ner its surfce s ΓE = gŷ (5) The numericl vlue of the erth s grvittionl field constnt g cn be found by evluting the mgnitude of Γ from eqution 3, using the vlues of the universl grvittionl constnt G = 6.67 10 11 N m2, the kg 2 mss of the erth, M E = 5.94 10 24 kg, nd the erth s rdius[1], R E = 6.37 10 6 m: g = GM E R 2 E = (6.67 10 11 )(5.94 10 24 ) (6.37 10 6 ) 2 = 9.76 m s 2 In tody s lb we re going to mesure the locl vlue of g by dropping slotted plte through photogte timer. The timer will give us series of mesurements of the velocity v of the plte s function of the verticl position y. Recll tht we hve two kinemtic equtions to describe the motion of n object under constnt ccelertion: nd y = y 0 + v 0 t + 1 2 t2 (6) v = v 0 + t (7) Since we do not hve ccess to experimentl informtion bout the time t, we re going to hve to use these two equtions to eliminte it s vrible. Solving eqution 7 for t, we get
Inserting this result into eqution 6, we get t = (v v 0) Solving for v 2, we obtin ( (v v0 ) y = y 0 + v 0 = y 0 + 1 = y 0 + 1 2 (v2 v 2 0) ) + 2 ( (v v0 ) (v 0 v v 20 + v2 2 + v2 0 2 vv 0 ) 2 ) v 2 = 2(y y 0 ) + v 2 0 nd with = g, we hve the desired reltion between the velocity nd g. v 2 = 2g(y y 0 ) + v 2 0 (8) Inspection of eqution 8 revels two fcts pertinent for our nlysis. The first is tht only the chnge in verticl position, y = (y y 0 ) is relevnt for determining g, so tht we cn rbitrrily choose the origin of the y-xis. The second is tht the initil velocity ppers only s constnt offset. If we rewrite the eqution s v 2 = 2g( y) + v 2 0 (9) it is cler tht the slope m of the v 2 vs y curve is proportionl to g, nd tht the squre of the initil velocity, v 2 0, which is of no physicl interest, ppers only s the offset constnt, so tht we don t require priori knowledge of this prmeter nd cn, in fct, neglect it. Procedure A schemtic digrm of the experimentl setup is shown in figure 2A. The slotted br is dropped verticlly through the perture of photogte timing device. The timing module will be set to the s2 functionl mode. In this mode of opertion, timing cycle begins when the photogte bem is interrupted nd ends upon being interrupted gin. The next cycle begins when the bem is once gin interrupted, nd so on. The slotted br, depicted in figure 2B, consists of series of 1 cm wide slots, ech 1 cm prt. The distnce over which timing cycle occurs is therefore 2 cm. Becuse new cycle doesn t begin until the bem is gin interrupted, the timing cyles themselves re sptilly seprted by 4 cm, s shown in the figure. Prior to conducting the experiment, you should verify tht the function mode is set to s2. The function mode cn be ltered by repetedly pushing the function button. The device cycles through ech of the functions; n LED indictes which function is ctivted. There is lso n LED indictor tht identifies the units in which the result will be output on the disply pnel. You should press the cler button, to cler out ny previous results. After dropping the br through the timing gte, press the stop button. The disply unit will then cycliclly red out the time intervls. Preceding ech time intervl redout, the disply will indicte which time intervl is bout to be shown; i.e., 1 st, 2 nd, 3rd,.... You should record these times, strting from # 1 through #7. This is your rw dt. Figure 3 depicts n pproprite coordinte system for nlyzing the dt. The verge velocity in the j th time intervl t j is equl to[2]
Figure 2A Figure 2B FIG. 2: Physicl setup nd dimensions for the mesurement of g v j = w t j (10) where w = 2 cm, s shown in figure 2B. Becuse the sptil distnce between the middle of ech timing cycle is equl to h = 4 cm, the vlue of the verticl displcement for the j th time intervl is equl to y j = (jh + y 0 ) (11) where h = 4 cm. The vlue of y 0 is rbitrry. The simplest choice is to mke it equl to zero, so tht In summry, the procedure is s follows: y j = jh (12) 1. Set up the timing device so tht the functionl mode is s2. Cler ny previous dt. 2. Crefully drop the slotted br verticlly through the photogte timer perture. 3. Press the stop button on the timing module. 4. Record the time intervls t j. Py ttention to units. You should hve 7 time intervls. If you hve fewer thn this, repet the mesurement. Convert the times to seconds. 5. Clculte the velocities v j using eqution 10. Leve the dimensions s cm/s. 6. Clculte the displcements y j using eqution 12. Leve the units s cm. 7. Squre ech of the velocities to obtin v 2 j (in cm2 /s 2 ).
FIG. 3: Coordinte system for nlyzing dt from the mesurement of g 8. You should now hve tble tht looks like the following. j t j y j v j ] [ vj 2 [s] [cm] m ] s 2 1 t 1 4 v 1 v 2 1 2 t 2 8 v 2 v 2 2 3 t 3 12 v 3 v 2 3 4 t 4 16 v 4 v 2 4 5 t 5 20 v 5 v 2 5 6 t 6 24 v 6 v 2 6 7 t 7 28 v 7 v 2 7 [ m s 9. Repet this mesurement 5 more times (for totl of 6 sets of dt) 10. Record your dt into Word file nd sve it on your flsh drive. [1] This is the men rdius of the erth. The erth is somewht flttened t the poles, nd bulges t the equtor so tht, for instnce, the equtoril rdius is lrger thn the men rdius. [2] Note tht this is not equl to the instntneous velocity t the midpoint of the intervl. This is becuse the slotted br is ccelerting s it flls. However, the pproximtion is sufficient for us to obtin the requisite ccurcy.