Determination of the gravitational constant LEP Related topics Law of gravitation, torsional vibrations, free and damped oscillations, forced oscillations, angular restoring moment, moment of inertia of spheres and rods, Steiner`s theorem, shear modulus. Principle Two small lead balls of equal mass are positioned one at each end of a beam which is held suspended by a thin tungsten thread, so that it can swing freely across its equilibrium position. When two further, but larger, lead balls held on a swivel arm are now brought near to the small lead balls, forces of attraction resulting from gravitation effect acceleration of the small balls in the direction of the larger balls. At the same time, the twisted metal thread generates a restoring moment of rotation, so that the beam is subjected to damped oscillation across a new equilibrium position.. The gravitational constant can be determined both from the difference in the angle of rotation of the different equilibrium positions and from the dynamic behaviour of the swinging system during attraction. An integrated capacitive sensor produces a direct voltage that is proportional to the angle of deflection. This can be recorded over time by an interface system, and the value of the angle of rotation that is required be so determined. Equipment Cavendish-balance, computerized 02540.00 Data cable, 2 x SUB-D, 9 pin 4602.00 Circular level 0222.00 Tasks. Calibrate the voltage of the capacitive angle sensor. 2. Determine the time of oscillation and the damping of the freely swinging torsion pendulum. 3. Determine the gravitational constant, using either the acceleration method, the final deflection method or the resonance method. Set-up and Procedure. Preparation of the system Place the Cavendish-balance on a stable surface and align it by means of the levelling feet and the circular level. Carefully move the balance beam of the pendulum from which the small lead balls are suspended to arrange it both as symmetrical as possible and parallel between the two fixed plates of the sensor. It is absolutely necessary to ensure that the beam can swing freely. Turn the swivel arm so that it is exactly at right angles to the beam, then place the large lead balls on the swivel arm. Important note: Always place the two large lead balls simultaneously on the swivel arm, or simultaneously remove them from it, otherwise the system will tip over and the tungsten thread may snap. Large fluctuations in the ambient temperature must be avoided during the experiment. Connect the Cavendish-balance to the interface with the signal cable, and the interface to a PC with the data cable. Fig. : Experimental set-up PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen 2309
LEP Determination of the gravitational constant The following is valid for a: a = arctan 2(w s -d b )/ l b. Using the measured values for U, we have: F = a / U F 2 = a 2 / U 2 = 0.024 rad /0.5379 V = 3.949 0-2 rad/v (small slit) = 0.0378 rad /0.7974 V = 3.985 0-2 rad/v (large slit) Use the average value for F obtained from the two measurements in the further evaluation. F = 3.967 0-2 rad/v () Fig. 2: Voltage dependence for the determination of the calibration factor F (left part of the curve: oscillation of the small slit across the calibration pin; the curved right part (oscillation across the large pin) shows that the slit did not touch the calibration pin here. 2. Calibration of the system With the balance beam at its rest position, check that one if its slits lies opposite the calibration pin. Should this not be the case, adjust to this position by very gently turning the thread holder. Balance the signal voltage as near to 0 volts as possible with the help of the off-set adjustment. The following settings are recommended: Range: ± 0.5 V (small slit); ± V (large slit); Sampling rate: 0 Hz; Number of points: 4096 (6 min. and 49.6 s); LPF: no Now allow the beam to swing past the calibration pin several times. Pay attention here that the calibration pin does not protrude too far into the slit. Swinging can be started by alternate swivelling of the large lead balls to bring the pendulum to resonance. Should this not be successful, very gently turn the thread holder to get started. Fig. 2 shows the typical sawtooth course of a calibration curve. A sinusoid curve shows that the calibration pin did not touch the inner edge of the slit. Record calibration curves for each of the two slits to obtain a reliable value for the calibration factor. Determine the average calibration factor F for the beam for both slits from the difference in voltage U and the angle of swivel a (Fig. 3). Theory and Evaluation. Determination of the time of oscillation and the damping of the freely swinging beam and the damped beam. Start the pendulum swinging gently. The connecting line between the two large lead balls is hereby at right angles to the balance beam, so that they do not exert any attractive force on the small lead balls. Fig. 4 shows the change in voltage over time with damped swinging of the torsion pendulum. The following settings are recommended: Range: ± V; Sampling rate: 2 Hz; Number of points: 892 ( h and 8 min.); LPF: no The oscillating time determined from several periods in this measurement example is T = 239.3 s. U(t) = U 0 + U*e -lt cos (v t) (2) Fig. 4: Change in amplitude over time for free and damped oscillations of the balance beam Fig. 3: Geometry for determining the angle calibration factor F 2 2309 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen
Determination of the gravitational constant LEP Equation (2) describes the change in voltage over time U(t). U 0 = the voltage of the equilibrium position, generally U 0 0; U* = initial amplitude, l = damping constant) Using x = e -/2lT (3) we obtain from equation (2): t = 0 S U = U o + U* t 2 T 2 S U 2 = U 0 - (U - U 0 ) x t 3 = T S U 3 = U 0 + (U - U 0 ) x 2 t n n 2 T 2 S U n = U 0 + (U - U 0 ) x (n-) (-) n- Using equation (3) for the three successive reversal points U n, U n+ and U n+2 we can find from them the zero line U 0 of the damped oscillation. U 0 U n x U n x U n 2 x U n x Using equation (4) for three successive reversal points we can therefore determine both the dimension x = exp(-/2 l T) and also, with x known, the zero line U 0 of the damped oscillation. x e 2 lt U n 2 U n U n U n A satisfactorily accurate average value for x can be obtained from several triplets of reversal points, whereby a start can be made both from the upper reversal points and from the lower reversal points. An average value of x = (0.845 ± 0.035) is obtained for the free and damped oscillations in Fig. 4. Using equation (3) and with T = 239.3 s, a damping constant of l =.4 0-3 s - is obtained. Fig. 5: Amlitude of the oscillation as a function of time during the acceleration phase (4) (5) (6) 2. Determination of the gravitational constant 2. The dimensions and values listed below have been used in the following calculations: M = Average mass of a large lead ball =.049 kg m = Average mass of a small lead ball = 4.50 0-3 kg r sb = Average radius of a small lead ball = 0.67 0-2 m r lb = Average radius of a large ball = 2.82 0-2 m d = Distance of the axis of rotation to the centre point of a small lead ball = 6.66 0-2 m R = Distance between the centre points of small and large lead ball in an extreme position = 4.62 0-2 m m b = Mass of the balance beam = 7.05 0-3 kg l b = Length of the balance beam = 4.9 0-2 m w b = Width of the balance beam =.29 0-2 m d b = Thickness of the balance beam = 0.4 0 2 m W = Distance between the outer surfaces of the glas plates = 3.5 0 2 m r h = Radius of the beam hole for the ball holding = 0.45 0 2 m r AL = Density of the aluminium beam = 2.7 0 3 kg m -3 2.2 Determination using the acceleration method Start with the pendulum carefully positioned in the rest position, and with the large lead balls in their neutral middle position. The following settings are recommended: Range: ± 0. V; Sampling rate: 0 Hz; Number of points: 2048 (3 min. and 24.8 s); LPF: 5 s Rapidly change the positions of the large lead balls without allowing then to hit the glass walls. The force of gravitation will now cause the small lead balls to be accelerated in the direction of the large ones, the small ones fall on the fixed large ones, so to speak. Fig. 5 shows the displacement of the beam during this acceleration. Using the assumptions, that are valid here, that in each case the masses of the pairs of lead balls are approximately equal, and that the acceleration is negligible because of the distance of the large balls to the small balls, then we have for the accelerating force F: F 2G mm R 2 ar2 m a S G 2M (7) Where G = gravitational constant and a = acceleration. Due to not ideal surface of the large lead ball one of the balls can be in direct contact with the glas plate and the other only 0. 0 2 m far from the glas plate. Is W the distance between the outer surfaces of the glas plates and is the balance beam with small balls in the middle between the glas plates, then we receive for R: R 2 W r lb 2 0.002 4.62 0 2 m PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen 2309 3
LEP Determination of the gravitational constant When s is the displacement of a small lead ball during the acceleration process over the time t, then we have: s t2 s 0 n 0 t t 0 2 2 a t t 02 2 On changing the position of the large lead balls at the time t 0 and with a displacement of s 0, then n o = 0. Plotting s(t)-s 0 as a function of (t-t 0 ) 2, as shown in Fig. 6, then the acceleration a can be determined from the slope of the straight line. After a time of (t-t 0 ) = 30 s, the acceleration curve deviates from linearity, which can be attributed to the growing influence of the angular restoring moment of the torsion thread. The value for the displacement s(t)-s 0 is obtained from Fig. 7: s (t)-s 0 = d tan a (a = voltage value for the displacement x calibration factor F) From (4) and (5) we have: G s t2 s 0 t t 0 2 R2 2 M 35.4 0 6 m.3 0 3 s 4.622 0 4 m 2 5.54 2 0.049 kg (literature value: G = 6.672 0 m 3 kg - s -2 ) 2.3 Determination using the equilibrium method Here, the gravitational constant is determined from the difference in the rest positions of the pendulum at the two extreme positions of the large lead balls. With the large lead balls in their neutral position and the pendulum at rest, the equilbrium position is given by the angle 0. When the large lead balls are brought to their other extreme position then, because of the attractive force, the pendulum swings across a new rest position 0. When, following this, the large lead balls are brought to the other extreme position, the pendulum swings across the rest position 02. The angle of deflection D required for the calculation of the gravitational constant G is so given by D = /2 ( 02-0 ). m 3 kg s 2 (8) (9) Should it not be possible to reach stable rest positions because of external influences (temperature fluctuations, vibrations), these can be determined from the observation of at least three reversal points of the oscillating torsion pendulum. Condition for the rest position of the pendulum: if the moment t G of the gravitation force F G is equal to the torsional moment t T of the filament. t G F G d t T D D 4 p2 T 2 I D (0) (D = directive moment; I = moment of inertia of the oscillating system) Inserting the moments of inertia of the balance beam I b and the small lead balls 2 I s in (0), then it follows that (the beam has two holes for ball holding): I 2 m b l 2 b w 2 b2 2m a 2 5 r2 sb d 2 b 2m h a 2 r2 h d 2 b.404 0 4 kg m 2 () The mass of the balance beam is calculated from the dimensions of the beam and the density of the beam material. The gravitaion force F G has three components: F = attraction force between the large balls and the close-by small balls; F 2 = attraction force between the large balls and the father small balls; F 3 = attraction force between the large balls and the beam with two holes. For the moment of gravitational force t G we receive: t G 2 F d F 2 d2 F 3 d2 2 G M R 2 3m m h 2 f 2 m b F 2 4 d From (2), we finally have, for the gravitational constant G: G a 2p 2 T b R 2 I 2M d 3m m h 2 f 2 m b f 2 4 d 9.726 0 8 D m 3 kg s 2 (2) (3) Fig.6: The distance (s-s 0 ) as a function of (t-t 0 ) 2 acceleration phase during the Fig.7: The geometry for determination of the distance s during the acceleration phase 4 2309 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen
Determination of the gravitational constant LEP Fig. 8 shows a measurement according to the equilibrium method. During the first three oscillations the large lead balls were in one of their extreme positions. After approx. 800 s they were swivelled to their other extreme position and, after a further four oscillations, back to the initial extreme position. The voltage difference determined from the zero positions of each of the oscillation segments, which were calculated using equation (5), is U = 33.2 mv. From this, q D = /2 U F = 68.57 0-5 rad. With this value for q D, a gravitation constant G = 6.67 0 m 3 kg - s -2 is obtained (literature value: G = 6.672 0 m 3 kg - s -2 ). 2.4 Determination using the resonance method When the large lead balls are brought to their extreme positions in-phase then the oscillation of the balance beam reaches the resonance condition. In the extreme position the distance between the small and large ball is constant (R = const.), gravitation force and its moment are also constant. Is U e the stable rest position of the beam and U D the amplitude if the large ball ball is brought to its extreme position, then (U e -U D ) is the new rest position. For the time dependent amplitude we receive according to Eq. (2): Ut2 U e U D 2 U U e U D 22 e lt cos vt2 (4) (U is the first reversion point). For the second reversion point U 2 = U(/ 2 T), if the large ball is brought to its other extreme position we receive: For the third reversion point U 3 = U(T), if the large ball is brought to its initial extreme position we receive: U 3 U e U D 2 U 2 U e U D 22 x (6) From Eq. (5) und (6) we receive for any reversion points pair: U D 3U n U e 2 U n U e 2 x4 2 n x 3U n 2 U e 2 U n U e 2 x4 2 n x (7) According to Eq.(7) U e can be calculated from three reversion points and for known x-value U D can be calculated: U D 3x U n x2 U n U n 2 4 2 n 2 x 2 (8) Fig.(9) shows the voltage as a function of time for the resonance method. At the point with maximal amplitude the large balls were brought to their other extreme positions. For calculation of U D many values of the reversion points were used: U D = 6.33 mv (see Fig.9 and Eq.8). For the angle of deflection q D we receive: q D = U D F = 67.46 0-5 rad. With this value of q D and according to Eq. (3) the gravitation constant is: G = 6.56 0 m 3 kg - s -2 U 2 U e U D 2 U U e U D 22 x with x e lt (5) (literature value: G = 6.672 0 m 3 kg - s -2 ). The resonance method is rather quick and exact, only tree reversion points are needet for calculation of the gravitation constant. Fig.8: Measurement of gravitation constant G with the equilibrium method. During the first three oscillations the large balls were in their extreme positions. After 800 s the large balls were moved to their other extreme positions and after four oscillations in the initial positions. Fig.9: Measurement of gravitation constant G with resonance method PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen 2309 5
LEP Determination of the gravitational constant 3. Apendix. Consideration of systematic errors 3. Calculation of f ; Influence exerted by the large lead balls on the most distant small lead balls (see Fig.0) t 2 2F 2y d2 2G (4) As the most distant large masses also exert attraction on the small lead balls, and so reduce the influence of gravitation force, then the following correction must be taken into consideration, acc. to Fig 0: The moment t h of the gravitation force between the large balls and the point like masses m h of the beam holes: t h 2 G M m h R 2 d M m R 2 4d 2 d sin a 2G M m R R 2 R 2 2R 2 4d 2 2 d 3 (5) The moment t 2 of the gravitation force between the large balls and the most distant small balls reduced by t h : t 2 2 G M d R 2 m f m h 2 (6) 3.2 Calculation of f 2 ; Influence exerted by the large lead balls on the balance beam (see Fig.) As the large lead balls also exert attraction on the balance beam, this influence must also be taken into consideration: When dm = Q r dx for a mass element of the beam at a distance x from the turning point (Fig. ). (Q = cross-sectional area of the beam; r = density of the beam material (Al = 2.7 g cm -3 )). A large lead ball acts on the mass element with a torsional moment: dt 3 F 3y dx G t 3 G M m b l B R 3 2 l b 2 l b (7) with X = ax 2 + bx + c; = 4ac - b 2 ; a = ; b = -2d; c = R 2 + d 2 Taking both large lead balls into consideration we obtain for t 3 : t 3 2G M m b d R 2 R3 d l b 2 l b x dx 2 d x c a B R b d xdx X2X 2bx 2c2 R3 2X 2 x d a B R b M Q r 2 3 d x c a B R b d x dx 2 3 d x c a B R b d 3 a dx d2 x b R 2 t 3 2 G M m b d R 2 f 2 ; f 2 0.202 x dx x (8) For the whole moment t G we receive (taking both corrections into consideration): t G 2 G M d R 2 3m m h 2 f 2 m b f 2 4 (9) Fig.0: The geometry for calculation of the influence of the large lead balls on the most distant small lead balls Fig.: The geometry for calculation of the influence of the large lead balls on the balance beam 6 2309 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Göttingen