PHYSICS ADVANCED LABORATORY I UNIVERSAL GRAVITATIONAL CONSTANT Spring 2001 (additions for Spring 2005 on last page)

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1 PHYSICS ADVANCED LABOATOY I UNIVESAL GAVITATIONAL CONSTANT Spring 001 (aitions for Spring 005 on last page) Purposes: Deterine the value of the universal gravitation constant G. Backgroun: Classical echanics topics-oents of inertia, central forces, torques, gravitation, an ape haronic otion. MIT an Tel-Atoic laboratory write-ups Skills: Protocol: Statistical analysis. Moving the apparatus is to be avoie. The tungsten wire that suspens the boo is very fragile an ifficult to replace. Shoul you nee to ove the Cavenish apparatus, request assistance an o so very carefully after lowering the boo support so that tension is off the wire. Lowering the boo will necessitate a tie-consuing re-centering process when the boo is raise again. When reoving the test asses, reove both siultaneously to avoi tipping the apparatus. The forces being easure are extreely sall. Changes in the environent aroun the apparatus can prouce effects that swap the effects sought. Be sure to electrically groun the test asses an the boo support before each easureent. The evice to be use is the Tel-Atoic TEL-000 Cavenish Balance syste, illustrate scheatically in Fig. 1. This syste uses a syetric ifferential capacitance transucer that, to first orer, eliinates easureent errors associate with the 'penulu' vibrations of the suspene bea. For this laboratory, you will first calibrate this capacitance transucer to eterine the corresponence between voltage output fro the transucer an angular isplaceent. Using this calibration, you will then easure the angular isplaceent as a function of tie for ifferent configurations of the test asses. Fitting the angular isplaceent functions will enable you to eterine equilibriu positions. These equilibriu positions are relate to the cobine effects of torsional forces fro the suspening tungsten wire an gravitational forces between the test asses. By knowing the positions an agnitues of the asses, G can be eterine. eeber: the forces involve are very sall. Injuicious oveents or changes in the local environent while the ata runs are in progress can coproise your easureents. Part A: Angular isplaceent calibration 1. In preparation for ata taking below, start the Cavenish ata acquisition software application on the PC connecte to the apparatus, but o not start a ata-taking run at this point. Ajust the ifferential capacitance control unit (DCCU) to its highest gain setting. Move the test asses to the neutral position.

2 . You now nee to ensure the suspene boo is near the null position by verifying that the output voltage fro the DCCU is zero. You can also verify this by placing a laser bea along a perpenicular line fro the Cavenish apparatus centerline so that it strikes the irror on the central axis support an reflects a spot back to the laser exit. If the suspension bea is not near the null position, contact either the instructor or the teaching assistant. 3. Use the 'optical lever' etho to eterine the angular isplaceent, siilar to that note in the Tel-Atoic write-up. Arrange the laser so that the bea strikes the irror ounte on the tungsten wire lower support, with the bea as close to horizontal as possible. 4. Position a easuring screen so the reflecte laser bea spot strikes the screen. 5. Move the test asses to the near position. The asses on the suspene boo will be attracte towars the lea balls, an the reflecte laser spot will ove across the easuring screen. The perio of the boo otion is about 00 secons. 6. Ajust the orientation an location of the easuring screen so the full otion of the reflecte laser spot reains on the screen. Once this orientation is fixe, carefully eterine the istances neee to accurately eterine the angular isplaceents as a function of the place where the spot strikes the easuring screen. 7. Begin a ata-taking run covering about 10 full cycles of otion. ecor in your notebook the positions for the axiu excursions of the laser spot on the easuring screen. The ata acquisition syste will recor the voltage output of the DCCU as a function of tie. Your easureents of the axiu excursions can be correlate with the axia of the voltage swings fro the DCCU. By eterining the angular changes (in raians!) corresponing to these axia, eterine the calibration of the DCCU voltage. Note that a shift of the irror by Θ about the central axis prouce an angle shift of Θ. 8. After finishing a ata run of about 10 full swings, fit the ata obtaine to the for V() t = Asin( ωt+ ϕ)exp( βt) The factor β is the aping constant for the syste. You shoul copare this value with the aping constants easure in subsequent analyses below. Part B: Measureent of G 1. Overview In the near position, the test asses prouce torques on the suspene boo ue to the gravitational attraction of the suspene asses an the boo. The torque prouce by the test asses on the nearest corresponing suspene ass is n GM =

3 where the factor of arises ue to the presence of two sets of asses, M is the ass of one of the test asses, is the istance of each of the suspene asses fro the central axis of the boo, an is the separation of the test an suspene asses. This is the priary source of torque in the syste. Clearly, all of the ters in this expression ust be eterine with the greatest accuracy possible in orer to arrive at an accurate value of G. However, two aitional torques exist within this apparatus that ust be taken into account in orer to fin G. One is an aitional torque prouce by the interaction of each test ass with the suspene ass at the istant en of the boo. This prouces a torque which acts in the opposite irection of the torque n. By inspection, this torque is seen to be GM GM = sinθ = r + ( ) + ( ) 3 GM = ( + ( ) ) 3/ 3 = n f where f = ( + ( ) ) 3/ An aitional torque arises fro the attraction of the suspension bea to the test ass. By following the analysis etaile in the Tel-Atoic write-up, this aitional torque is given by GM b = fb' ( f b b h) = n fb' where fb' = with the various ters efine in the iscussion above equation 6 of the Tel-Atoic write-up. The cobine effects of these last two ites aounts to about 3-4% of n, an the total torque ue to these gravitational effects is grav total = ( 1 + f ' + f ) b Opposing the su of these torques is the restoring torque Κ( θ) supplie by the tungsten wire. This wire twists an reaches an equilibriu angle given by which eans wire = grav total

4 grav GM K( θ) = total = ( 1+ f + fb' ) K( θ) G = M ( 1+ f + f ') Thus, the easureent of K an θ, along with the asses an iensions of the apparatus, will suffice to perit a eterination of G.. Measureent protocol In practice, eterining the equilibriu angular shift θ is ifficult an tie-consuing because the eflection is sall an the tie neee to reach equilibriu can be long. For that reason, we will eterine the equilibriu angle θ by fitting the angular eflection as a function of tie when the asses are in the near position to the expression θ() t near = θ + θ 1 sin( ωt+ θ)exp( βt) Note that the aping coefficient β eterine in Part A enters here as well. You shoul nonetheless vary β as a paraeter in your fits here an copare the results to those in Part A. This expression approaches the equilibriu angle θ as t becoes large. When the suspene bea finally coes to rest, the DCCU shoul inicate this angle; you shoul note whether this final resting angle inee correspons to the results of your fit. You shoul repeat this process for the test asses ove to the alternate near position. Then repeat the easureent at both near positions a secon tie, for a total of four easureents. 3. The torsion constant K The torsion constant K can be eterine two ways an you shoul try both ethos: a) The torsion constant can be estiate fro the properties of the tungsten wire irectly using K = πµφ 4 3l where φ is the iaeter of the wire (5 µ), Young's oulus µ for tungsten is 1.57 x N/ an the length of the wire is l. b) Fro your fit to the oscillations of the suspene bea, you will have eterine the oscillation frequency ω. This frequency of oscillation correspons to K = ( ω + β )I where I is the oent of inertia of the suspene syste of bea an asses. Since this is a coposite syste, you will have to eterine the total oent of inertia by suing the iniviual oents of inertia an using the parallel axis theore. 4. Deterining For all gravitational torques, enters in ters as secon or thir power. Thus, eterining accurately is iportant. To easure, use the following etho, estiating uncertainties at each step. b

5 a) Measure the with between the glass plates W. b) Deterine the iaeters of each of the lea test asses an fin their average D. c) Deterine the size of the gaps between the test asses an the glass plates when the test asses are in the near position. Fin the average of this gap g. ) Deterine the approxiate reuction in ue to the rotation of the boo ( θ ). e) Using these values is given by = W + D + g ( θ) 5. ecoring ata Since you will be oing the sae calculation of G for all four trials, you shoul transfer your ata to a spreasheet to autoate your calculations. 6. Uncertainties You shoul accept the values inicate in Figure 1 with their state uncertainties. Do not isasseble the apparatus to ake any easureent beyon reoving the test asses for weighing. Asie fro those uncertainties inicate in the figure, you will nee to easure quantities an ascertain their associate uncertainties. Since nearly all quantities associate with the eterination enter as siple factors or sus, you shoul use propagation of uncertainties to eterine the final uncertainty in your value of G. Again, you shoul use a spreasheet to autoate these calculations.

6 In Spring 005, we will o this lab as a class project. Your groups will o only a fraction of Part B: Measureent of G. Ji will o all of Part A an will ake the ynaic easureent of the torsion constant (section 3b in Part B). You ay copare your calculation of the torsion constant in section 3a with his value. The value of W to be use in your calculation of (section 4 of Part B) will be given on the whiteboar. The value of g in section 4 will be zero (the large balls will be touching the glass). Your groups will ake two easureents, one with the balls in the near position (as escribe in the vieo) an one with the balls in the far position. DO NOT EXCITE THE VIBATION WITH THE STICK AS DESCIBED IN THE VIDEO. Instea, rive the vibration by soothly reversing the position of the outer balls at the instant the inner balls are closest to the outer balls. Five or six reversals shoul result in vibrations of sufficient aplitue for a useful ecay spectru. After five or six reversals, leave the balls in either the near or far position an allow the vibration to ecay. Once ecay has begun, recor the positions of the first 6-10 turning points on the whiteboar; this will allow a eterination of a calibration constant which can be copare with Ji s value. The ecay spectru is what you will fit to the equation in section of Part B. After you have the ecay spectru leave the roo an allow the vibration to ecay copletely, peritting a static eterination of θ (this ay take quite a long tie). epeat this process for the other position, either near or far. Analysis: The six groups fro Tue an Thur will share their ynaic an static eterinations of θ. Discuss pros an cons of the static-vs-ynaic eterinations of θ. Deteine G with proper uncertainty using the share values. Table I. Soe aitional iensions of the Tel-Atoic apparatus not given in Figure 1. Ite aius of sall ass Thickness of support bea With of support bea Diension 6.75(1) 1.65(1) 1.8(1) c

7 L = 14.9(1) c l = 6.656(1) c = 14.7(1) g M M L Figure 1. Diension labels for eleents of the Cavenish apparatus for easuring G. The enclosing glass case is not shown in this sketch. For aitional etails, see Figure 1 in the Tel-Atoic write-up.