Mechanics. LD Physics Leaflets. Determining the gravitational constant with the gravitation torsion balance after Cavendish P m 1.

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1 Mechanic Meauring ethod Deterining the gravitational contant LD Phyic Leaflet P..3. Oject of the experient Deterining the gravitational contant with the gravitation torion alance after Cavendih Recording the excurion and evaluating the eaureent with the IR poition detector and PC g To record the daped ocillation around the final equiliriu poition of the torion pendulu a a function of tie. g To calculate the gravitational contant G uing the end-deflection ethod. g To calculate the gravitational contant G uing the acceleration ethod. Principle The heart of the gravitation torion alance according to Cavendih i a light tranvere ea, horizontally upended y a thin torion tring, which upport a all lead all with the a at each end at a ditance d fro the upenion point. Thee two all are attracted y two large lead phere with the a. Although the force of attraction i le than 0 9 N, it i till poile to deontrate thi attraction uing the extreely enitive gravitation torion alance. The otion of the all lead all i oerved and eaured uing the IR poition detector (ee Fig. ). The IR poition detector ha four infrared diode which illuinate a concave irror which i ounted in a fixed poition on the tranvere ea of the torion pendulu. The reflected light i projected onto a row of phototranitor to regiter the ocillation of the a. On the ai of the otion over tie, the a and the geoetry of the etup, it i poile to deterine the gravitational contant uing the enddeflection ethod or in an areviated eauring procedure y ean of the acceleration ethod. Fig. : Gravitation torion alance according to Cavendih (left), and cheatic diagra of the experiental etup (right). d L α ΙΙ α Ι Bi 005 S II S S I LD Didactic GH. Leyoldtrae. D Huerth / Gerany. Phone: (033) Fax: (033) e-ail: info@ld-didactic.de

2 P LD Phyic leaflet Apparatu Gravitation torion alance IR poition detector (IRPD) Optical ench, tandard cro ection Optic rider 60/ Optic rider 90/ Stand rod, 5 c additionally required: PC with Window 98 or higher The end-deflection ethod i aed on the oervation that the gravitational force etween two lead phere with the ae and at a ditance i given y (Fig. ): F = G (I) Thu, the oent of oentu M I acting on the torion pendulu i = F d = G d (II) MI when the two large lead phere of a are in poition I (ee Fig. ). The oent of oentu i copenated y the righting oent of the torion cord. The torion pendulu thu aue the equiliriu poition I. By wiveling the large lead phere in poition II the force are yetrically inverted. The oent of oentu acting on the odie i now M II = M I. The pendulu execute daped ocillation around the equiliriu poition II. For the difference of the two oent of oentu we have with the correponding angle α I and α II : D ( α α ) = M M = M (III) I II I II I The angular directional quantity D can e deterined fro the ocillation period T and the oent of inertia J of the torion pendulu: 4 π = J (IV) T D Safety note Mind the intruction heet for the gravitation torion alance and the infrared poition detector: g Protect the enitive and of the gravitation torion alance fro uncontrolled echanical loading. g Alway look the ocillation yte of the gravitation torion alance when the device i not ued. In particularly, ake ure the ocillating yte i locked during tranport and aely. g The enitive electronic of the IR poition detector can e ipaired or daaged y electrotatic dicharge. g Choe an experient area where no electrotatic charge can e uild up on either the operator or the device. The oent of inertia J i equivalent to the oent of inertia of the two all all: J = d (V) Thu, equation (IV) i tranfored to D 8 π = d (VI) T Fro equation (I), (III) and (IV) we otain π d G = ( αi αii) (VII) T Fro geoetry follow the relationhip (here for poition) SI tan α = L fro which follow for all angle: S I α = (VIII) L By uing thi equation (VIII) equation (VII) can e written a follow (for further detail ee intruction heet 33 or leaflet P..3.): π d (SI SII) G = (IX) T L ) Acceleration ethod Directly after the large lead phere are lewed fro poition I to poition II, the all all are ujected to an acceleration a 0 which follow fro the equation of otion: a0 = G (X) Thu the gravitational contant i given y: a0 G = (XI) The acceleration a 0 applied to the ae can e deterined fro the acceleration a 0 of the light reflection fro the geoetry relationhip a follow: ' d a0 = a0 (XII) L A the path i given y (t) = a 0 t (XIII) the acceleration a 0 can e otained y fitting a paraola of the general for (t) = A t + B t + C (XIV) to the firt phae of the otion. The coparion of equation (XIII) with equation (XIV) give: a 0 = A With equation (XII) the gravitational contant i given y: ' d G = a0 (XV) 4 L LD Didactic GH. Leyoldtrae. D Huerth / Gerany. Phone: (033) Fax: (033) e-ail: info@ld-didactic.de

3 LD Phyic leaflet P..3. Setup Iportant: atifactory eauring reult are only poile when the torion alance i adjuted properly. In addition, the torion ocillation caued y the attraction etween the ae ut not e ditured y undeired pendulu otion. The torion pendulu i extreely enitive to hock tranitted to the houing of the torion alance. Change in teperature caue convection current in the houing of the torion alance, which in turn reult in unwanted oveent of the torion pendulu. Chooe a tale experient etup on a olid wall. Select an experient ite which i not expoed to direct unlight or draft. When lewing the phere upport avoid hock to the houing. e.g. y knocking at the etup with the lead phere. Fig. how the experiental etup. Aeling the gravitation torion alance efore uing it the firt tie: - Aele a enchtop for the experiental etup on the wall like hown in Fig. (ee alo intruction heet 33 0 and 3 ). - Setup the optical ench with the gravitation torion alance. - Poition the gravitation torion alance (without the large lead phere) o that there i ufficient pace to turn the upporting ar of the lead phere. - Looen the arreting echani of the torion pendulu and correct the alignent o that the pin at the end of the pendulu hang in the iddle of the rod hole and ake ure that the torion pendulu can wing freely. - Allow the torion pendulu to hang for one to two day and readjut the zero point if neceary (ee intruction heet 33 0). Setting the ditance etween the gravitation torion alance and the IR poition detector for the firt tie: - Clap the IR poition detector on the rear panel to tand rod which i inerted in the rider on the optical ench. - Setup IR poition detector on the optical ench in uch a anner that the ditance etween the front window of the gravitation alance and the IR poition detector i 70 c. - Connect power upply V AC to the IR poition detector and adjut the window with infrared LED roughly to the ae level a the irror of the gravitation torion alance. - The two red adjutent LED now light up o rightly that their irror iage can e een in the plane of the device, either on the device itelf or on a heet of white paper which i held eide it: If the iage i to the left or right of the window, ocillate the gravitation torion alance lowly o that the irror iage of the right LED pae acro the front panel. If the iage i aove or elow the window, project it onto the center y raiing or lowering the IRPD. - Enure that the row of photo tranitor i ituated in the plane which i covered y the irror o that all of the photo tranitor are ued for eauring. - Adjut the height y uing the green and red LED. Auing that the photo tranitor row i in the plane illuinated y the irror thee LED are witched on and off depending on the luinou intenity of the phototranitor: Red LED flicker: illuination/adjutent ufficient Green LED flicker: illuination/adjutent good Fig. : Experiental etup: Benchtop aely of the gravitation torion alance with electronic recording of the ocillation curve uing the IR poition detector. LD Didactic GH. Leyoldtrae. D Huerth / Gerany. Phone: (033) Fax: (033) e-ail: info@ld-didactic.de

4 P LD Phyic leaflet Recording the ocillation - Connect the IR detector to the coputer via the RS3 port. - If not yet intalled intall the oftware and open the oftware (preconfigured CASSY La uer interface) - Open the window Setting uing the tool ox utton or function key F5 fro the top utton ar: Thee adjutent of the experiental etup have to e perfored only once ecaue the ret poition i retained after the gravitation torion alance ha een locked. - Select the ta General in the window Setting and et the appropriate COM port y electing the IR poition detector: Carrying out the experient Firt Preparation - Allow the etup to tand for at leat two hour unditured y hock, o that the pendulu can coe to ret in one of the equiliriu poition. Note: Typical equiliriu poition for a good adjutent are approxiately 0 and 50, repectively. If thi i not fulfilled the gravitation torion alance ha to e rotated y a all angle. It ight e neceary to repeat thi tep everal tie. If the arreting crew ha een looened after a long period of diue, the torion pendulu ay require ore tie to ettle into an tale equiliriu poition. - Check the taility of the zero point y recording the aeline. Started data acquiition y the utton. - Meaure the zero-point fluctuation for at leat 0 inute. After electing the IR poition detector for the appropriate Co port the oftware open a tale, diplay and variou window: indicator for the eauring quantity, i.e. the window Poition, the window Setting and the window Meauring Paraeter : - Wait until the yte ettle into a tale equiliriu poition (ee aove firt preparation). - You ay clear the aeline eaureent with the utton or the function key F4. - Start data acquiition y preing the utton or F9. - Move the upporting ar with the lead phere rapidly (ut carefully!) fro poition I to poition II. - Swivel the lead phere fro poition II ack to poition I and repeat the eaureent for the ocillation around the equiliriu poition I. - Stop the data acquiition y preing the utton or F9. Note: You ay ave your eaureent y preing the utton or uing the function key F. - Accept the preet value y cloing all window inide the ain window. - Pre the utton or function key F9 to tart recording the ocillation of the gravitation torion alance. Note: The utton work a a toggle witch. The data acquiition can e topped y preing or F9. ) Acceleration ethod - Wait until the yte ettle into a tale equiliriu poition (ee aove firt preparation). - You ay clear the aeline eaureent with the utton or the function key F4. - Start the data acquiition y preing the utton or F9. - Move the upporting ar with the lead phere rapidly (ut carefully!) fro poition I to poition II and record the firt phae of the otion. - Stop the data acquiition y preing the utton or F9. Note: A thi experient ue the IR poition detector for recording the ocillation of the gravitation torion alance the acceleration ethod i autoatically included when perforing the end-deflection ethod. The eaureent interval i et to an appropriate value, e.g. 5. LD Didactic GH. Leyoldtrae. D Huerth / Gerany. Phone: (033) Fax: (033) e-ail: info@ld-didactic.de

5 LD Phyic leaflet P..3. Meauring exaple Fig. 3 how an illutration of daped ocillation of the gravitation torion alance and the two correponding final poition. While wiveling the lead phere fro poition II ack to poition I the data acquiition wa not witched off. - Calculate the ean average of e.g. poition I y electing the tool Draw the Mean - Select the end part of the recorded ocillation y dragging over data the oue pointer (elected data ecoe lue). - Plot the reult of the calculation into the diplay y electing the tool Text for the Set Marker popup enu (or Alt-T). Alternatively, the reult can e dragged fro the tatu line (left window otto) into the diplay (Fig. 5). - Repeat the evaluation for poition II. Fig. 3: Ocillation at the equiliriu poition I and II of the gravitation torion alance according to Cavendih. Evaluation The oftware allow an eay evaluation of the eaured data. To acce the data evaluation tool click with right oue utton into the diplay (plot) to open the pop up enu. Fig. 5: Deterining the equiliriu poition I and II y calculating the ean value S I and S II at the end of the ocillation. Deterining the ocillation period - Select the Meaure Difference tool y Alt-D (or y clicking the right oue utton into the diplay and chooing the uenu Select Marker ) - Meaure the period e.g. over 5 ocillation y clicking at the plot when the ocillation firt pae the ean value S I and then click at the plot when 5 ocillation have een elaped. - The reult of the evaluation can e diplayed uing e.g. Alt-T. Alternatively, the reult can e dragged fro the tatu line into the diplay (Fig. 6). - Repeat the evaluation for the ocillation around the tale poition II. Fig. 4: Acceing the data evaluation tool y clicking with the right oue utton into the diplay. Note: The zoo tool (Alt-Z) ight e ueful to enlarge the plot of the recorded data. Alternatively, the cale of the diplay can e changed y clicking with the right oue utton into the x- or y-axi area. Deterining the equiliriu poition I and II The two tale final equiliriu poition I and II of the two upended all lead all can e deterined a follow: Fig. 6: Deterining the ocillation period over 5 period for the ocillation at poition I and II. (Note: The text of ean value for S I and S II of Fig. 5 wa changed for ake of clarity.) LD Didactic GH. Leyoldtrae. D Huerth / Gerany. Phone: (033) Fax: (033) e-ail: info@ld-didactic.de

6 P LD Phyic leaflet ) Acceleration ethod - Zoo the data around the firt phae of the otion, e.g. at poition II, and fit a paraola to the recorded data. - The reult of the evaluation can e diplayed in the plot y Alt-T. Sutituting thee value into equation (IX) we otain: π G = kg 0.7 G = kg Literature value: G = kg ) Acceleration ethod Fro atching a paraola to firt phae of the ocillation after wiveling the lead phere fro poition II ack to poition I (ee Fig. 7) we otain: A = Fig. 7: Deterining the acceleration for the firt phae of the ocillation after wiveling the lead phere fro poition II ack to poition I (for aller zoo range ee Fig. 8). a 0 = G = kg 0.7 G = kg Suppleentary inforation For a detailed treatent of the geoetrical relation and the reulting their error refer to the alternative light pointer ethod, i.e. leaflet P..3.. Fig. 7: Deterining the acceleration for the firt phae of the ocillation after wiveling the lead phere fro poition II ack to poition I (for larger zoo range ee Fig. 7). Reult The calculation of the aple eaureent of Fig. 5 and Fig. 6 how: Period of ocillation T = 634 The difference etween the two final poition: S = S II S I = 6.5 LD Didactic GH. Leyoldtrae. D Huerth / Gerany. Phone: (033) Fax: (033) e-ail: info@ld-didactic.de