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by: Garett Kovacek


Garett Kovacek
GPA 3.95

Horst Wahl

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Horst Wahl
Class Notes
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This 4 page Class Notes was uploaded by Garett Kovacek on Thursday September 17, 2015. The Class Notes belongs to PHY 3802L at Florida State University taught by Horst Wahl in Fall. Since its upload, it has received 35 views. For similar materials see /class/205518/phy-3802l-florida-state-university in Physics 2 at Florida State University.




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Date Created: 09/17/15
Mechanics lEYBOLDMERAEUS Measurement determining the gravitation constant 113 2 V 33210C Gravitational Torsion Balance Determining the gravitation constant in accordance with the full deflection method The gravitation constant G will be determined in accordance with the full deflection method using a torsion balance I The gravitational force with which two lead balls with masses of m1 15 kg and m2 0015 kg at tract each other at a distance of 45 cm is less than 10399 N This force can only be verified with an extremely sensitiVe torsion balance The core of the torsion balance is a light trans verse beam suspended horizontally from a thin torsion wire and with a lead ball of mass m 0015 kg attached at each end These balls are attracted by two large lead balls with a mass of m 15 kg The resultant defledtion of the transverse beam from its initial position is de termined by means of a light pointer cf Fig 1 Fig 2 Both the initial and final equilibrium positions of the balance measuring system and the damped intermediate oscillation motion must be observed to determine the gravitation constant This in volves an observation time of approximately 45 minutes Carefully check the initial position of the balance and then start the experiment by moving the large balls to the Opposing position that is to say from position I to position 11 After a few oscillations the measuring system moves from the initial position designated as one limit position to the new limit position The angle between these two limit positions is given by Fig 1 Gravitation torsion balance after Cavendish K2 L l l l l l I I m1 1 a i b quot9 quot 52 rn1f3 3 M 13 it d D a This can be calculated from the dimensions of the setup and the light pointer deflection in accordance with the details given in Fig 1 The torque M acting on the measuring system in one limit position due to mass attraction is M 2 Fd where F is the force of attraction between each pair of balls and d is the axis distance of the small balls in the balance from the torsion band The equilibrium of this torque is main tained by the twisting of the torsion band by the angle If the angular writing force of the torsion band is designated as D this torque M D Symbol designations refer to Fig 1 b the distance between the center points of the large and small balls the path of the small balls in the balance distance of the small balls from the axis the path of the light pointer on the wall theldistance of the balance mirror from the wa r mam The following applies because of angle doubling due to reflection o a S 2 d 2 2 EFT ta 39 2 L The angular righting force D can only be deter mined from the oscillation period T of the tor sional oscillation of the measuring system w T2 412 D 4I2J or D 39 12 The moment of inertia J here can be equated with the moment of inertia of the two small balls J Zmzdz because the suspension with the mirror contributes nothing to the moment of inertia in practice Thus 8112m2d2 T2 mlmzd If the value M 2Fd 26 7 and the values b obtained above for D and a are inserted in the equation M D E it follows where 2M Da that V 2 mlm2 8n dm2 S w 6V r0 30 ummuumu E Fig 2 Gravitation torsion balance with additional equipment set up ready for use Note The following formula applies for the force of attraction between two masses m1 and m2 m m 1 2 F G 2 r r distance between the masses From this we can calculate anZdS mlT2L I This formula for the gravitation constant only contains measurable variables The mass of the small balls is not included in the calculation so it is not necessary to know this Apparatus 1 gravitation torsion balance 332 10 1 steel tape measure 311 77 1 lamp housing 450 60 1 lamp 6 V 30 w 450 51 1 aspherical condenser 460 20 1 transformer 6 V 30 w eg transformer 6 V 30 w 562 73 1 stopclock g 313 05 1 pair of magnets 510 48 2 saddle bases 300 11 1 Leybold multiclamp 4 301 01 1 stand rod bent at right angles 300 51 1 stand rod bent by 15 300 52 1 large stand base 300 01 1 pair of levelling screws 300 06 Setting up Stick the selfadhesive scale included in the scope of delivery of the torsion balance on a steady base or on a wall refer to the instruc tions for use 332 10 for the distance of the scale from the torsional balance and for the height h at which it must be set up Set up the torsion balance in accordance with the instructions for use 332 10 at first without the large lead balls An alternative way of setting up the balance is to fit it to a stand base and then place the latter 9n a flat rough concrete or stone sur face Two saddle bases can be used here as coun terweights for the illuminating equipment The setup must remain completely still during mea surement 39 Setting up the light pointer Equip the condenser with the diaphragm holder and insert the 1 mm slit Place the lamp in the cor rect position for the experiment refer to Fig 2 or Fig 3 and focus the lamp filament on the torsion balance mirror by moving the lamp insert The light filament must be set up vertically Finally focus the slit on the scale by moving the whole stand setup along the angled stand rod Always make readings on the scale from the same shadow edge of the slit To do this focus the shadow edge so that it is possible to make readings to within 05 mm also refer to the in structions for use 332 10 for how to set up the light pointer Fig 3 Gravitation torsion balance setup 39 39ialternative Adjust the zero point of the light pointer in accordance with the torsion balance instructions for use After adjusting the zero point place the lead balls in position and move to one limit position position I or II Fig 1 You must not touch the housing with your fingers or with the lead balls under any circumstances Allow theappara tus to stand touching it Beforestartingmeasurement5 observe the stabili39 ty of the zero point for at least IO minutes and recordthis Carrying out the experiment Measurement must always be carried out by at least two people one of these should exclusive ly follow the light pointer on the scale and the other should displace the lead balls call out the measurement times quotREADY NOWquot and record the measurements made 39 At the time t 0 move the lead balls quickly from one limit position to the other but en suring that the housing is not touched by either your fingers or the lead balls At the same time start the stopclock rRead off the pointer position every 5 seconds and note this After one complete oscillation period increase the measurement interval to 10 seconds and after two further periods to 30 seconds The total measurement time must be at least 3 oscil lation periods Then measure the rdistance k between the light pointer mirror and the scale using a tape measure Measurement example Technical data of the torsion balance need not be checked and are as follows m 15 kg d 005 m see Fig 1 b 00465 m fbr at Feast39 two hourS without L Note b 00465 m as the distance between the center points of the large and small balls is an approx imate value The distance is smaller in the in itial and final equilibrium position if the small balls are exactly in the center of the metal housing of the torsion balance for the selected zero point 39 initia l position to a the painter39 before the startof measurement 604 cm x 039 437 cm Fig 4 Measurement example oscillation of the gravitation balance about the final equilibrium position I 39 Evaluation and results From the graph we can see that one oscillation period T 245 seconds 39 The equilibriumposition x of the light pointer after oscillation has died out is determined from three successive amplitudes eg x1 x3 in accordance with the algorithm approximate value x x x x 2 3581 cm 4 2 4 It is possible to check this by means of the amplitudes x2 x4 and x3 x5 In accordance with this the pointer deflection S is equal to S x0 x 23 cm By inserting this in equation 1 we obtain the gravitation constant 2 00452 005 0023 15 2452 437 5237 1039l1m3kg391s392 Fig 4 Deflection S of the light pointer as a function of time 61 60 cm 59 5839 5739 56 360 560 The determined value is subject to the f011owing systematic error the smai ba11 is aiso attrac ted by the more distant second large ba11 aibe it with a much iower force refer to Fig 5 Fig Sz Caicuiating the correction factor In accordance with the gravitation law this force is given by 39 m m F 26 LZ Q b2 4d2 and has a component f opposed to the force F0 to be measured m m h f 25 1 2 F b2 4d2 7h5 4d539 8 if B is the fraction by which the observed force F0 is reduced b3 5 e ee b2 4d2 lbz 4d2 B 0075 where d 005 and b 00465 m The vaiue for G obtained without this correction must therefore be increased by 75 For our measurement exampie we thus obtain G 6698 1011m3kg1s2 instead of the theoreticai value G 5705 10 1m3kg1s2


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