First Year Laboratory
2001-2002
Big G: Measurement of Newton's Constant
1. Introduction
The motion of planets and satellites confirms to great precision that gravitational attraction obeys Newton's Law:
where F is the force of attraction between masses M and m which are separated by a distance D, and G is Newton's constant. However the motion gives no information on how strong the attraction is. The reason is that without knowing G we do not know the masses of planets or stars. It was realised long ago that a measurement of the attraction between known masses would be very valuable. That measurement has been done of course, but only with moderate precision because G is very small.
The instrument invented to measure such a small force is the torsion balance. It is still used today, though with more sophisticated instrumentation. Its essential feature is that a long thin wire can be quite strong in tension and very flexible to twisting (torsion). So a reasonably massive test mass can hang from the wire and experience a very weak restoring force when the gravitational attraction is arranged to twist the wire. Its principle is thus the same as the spring balance, which balances the gravitational attraction of the Earth with the restoring force due to the extensionof the spring.
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2.The Experiment
Warning: Apparatus for measuring G has to be sensitive to tiny forces - so do yourselves a favour by being cautious in how you disturb it. It can take many hours to settle down if jogged.
The torsion balance (Fig. 1) consists of an armature hanging on a thin wire. The armature has two masses m which respond to the gravitational pull exerted by larger masses M a distance D away. The gravitational forces F and lever arm R produce a torque on the armature. When the armature is in equilibrium it has turned through an angle such that the torque due to the stiffness of the wire balances the gravitational torque. ( is the stiffness of the wire in Nm/radian.) So, if is measured and can be found, we can find the strength of the force and hence G. A mirror fixed to the armature reflects a light beam and when the armature turns through , the reflected beam will deflect through = 2.
Thus = 4FR/ = 4GMmR/(D2) so that
G = D² / (4MmR)(1)
If the armature is turned to an angle different from the equilibrium, it will experience a torque . By Newton's second law this torque is equal to the rate of change of the angular momentum dL/dt = I d²/dt² ,where I is the moment of inertia of the armature.
When released the armature will thus undergo simple harmonic motion of angular frequency (/I).
For the armature provided, I is almost entirely due to the masses m so that can easily be found by measuring the period of oscillation 2/, so that
= (2mR²)4 ² / ²(2)
In order to make the effect easier to measure, the angle of the light beam is measured with the torque acting in one direction and then again with the torque reversed. This also makes it unnecessary to determine the "zero" angle. If reversing the torque moves the light spot by S at a distance L from the mirror, then using equations (1) and (2),
G = SD² ²R / (LM ²)(3)
According to the manufacturer's specifications:
large mass M = 1.5kg; radius = 32mm
armatureR = 50mm
enclosure width = 29mm
All quantities on the right-hand side of equation (3) are now accessible - m is not needed.
3.Getting started
As the balance must be sensitive to a weak force, the suspension wire must twist easily. We see from equation (2) that the period of oscillation will thus be long. Time is limited however, so a compromise has been made. Even so you will need to work efficiently to be able to fit all the measurements into the time available. You are advised to complete section 4 within the first 3hr session. At this stage it is worth investing enough time to follow 2 or 3 cycles of oscillation. This will give a value for and an idea of how the oscillations are attenuated by damping forces.
Check that the armature is not locked by the screws X (see Fig. 2)
Follow the light spot for a few minutes (taking readings at least every minute) to see what motion is induced by "background noise" (traffic, wind, etc). If you want to use the data-logger for doing this, the Appendix will tell you how.
This motion is not very regular because the armature is frequently kicked by the disturbances. To find the period of oscillation, it is necessary to make the armature swing with an amplitude that is much bigger; so that the effect of the background noise is comparatively small. The simplest way to do this is to reverse the torque on the armature due to the large masses. They are mounted on a revolving support.
4.Plan of action and preliminary measurement
If the torque acting on the armature changes with time as in Fig.3a the spot position S would ideally respond as in Fig.3b. Here S1is the equilibrium position with the torque acting in one direction and S2 is the equilibrium position with the torque reversed.
Their difference is the S which appears in equation(3).
In practice there are unavoidable disturbances which are indicated in Fig.3c:
- The value for S1 is not constant but may "drift" over a long time.
- The balance is not in equilibrium to begin with but oscillates around S1.
- The act of reversing the torque induces some oscillation about S2– if you are skilful you can minimise this.
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Given such non-ideal behaviour, it is necessary
(i)to measure S in the presence of oscillations
(Fig.4 gives an idea of the problem; the equilibrium positionS0has to be found from the damped sinusoid — a simple estimate of S0 is given by
Ŝ0 = Si / 4 + Si+1 / 2 + Si+2 / 4
where S0 is the true value and Ŝ0 the estimate.)
(ii) to restore the torque at t2 in order to check the drift.
Now that you know you will see that a measurement of S will take about an hour.
!!!!!It is very desirable to reach this point at least an hour before the end of the first session so you have time for a preliminary measurement of S.
Then the balance can settle overnight (leave the laser ON, why ? ), ready for a definitive measurement during the second session. Even if there is not a whole hour left it is still valuable to get some idea of S.
5. Review of the preliminary measurement
Make a table showing the values of all the quantities needed for finding Gand their errors. Since G is a product of these quantities the fractional errors are appropriate. For which quantities is it worth investing effort in more careful measurement?
Is the value of G that your measurements give reasonable ? [ In the spirit of the introduction, the value of G to compare your measurement with is NOT the "official" value – the pioneers did not have one! But you (and they) could make a rough estimate of G from the gravitational field strength at the Earth's surface,
Use the radius of the Earth and guess the Earth's density to be between that of rock (2500) and that of iron (7900 kg/m3. ]
6. Definitive measurement
Try to devote at least half of the second session to this.
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7.Extracting a value for G
In quoting your final value for G and its error, you should consider both corrections and errors. As a guide, you may wish to consider the following points.
The manufacturers of the balance give an example of a correction. They consider the attraction between each M and the "wrong" m (see Fig.5).
- An obvious source of error is the attraction exerted by people around. This can be random or systematic depending on where they are during the measurements. Calculate the torque exerted by a (spherical) 70kg mass at one metre on a line at 45° to the direction of R. Use this to estimate the effect on G.
- The value of was obtained using the damped period. Strictly, the undamped period should be used. Your measurement of the attenuation allows an estimate of this correction.
- The simple estimator Ŝ0= Si/4 + Si+1/2 + Si+2/4 is not perfect. If each excursion away from the equilibrium position is actually a factor smaller than the previous one then
(S0 - Si+1) = (Si - S0) and (Si+2 - S0) = 2 (Si - S0)
You can use this to check how close Ŝ0 is to the true equilibrium S0.
Can you think of other effects, even if there is no time to measure them ? Are any of them automatically taken into account by the way the measurements are made?
Your summary of the experiment should include the "error budget", updated from
section 5.
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Appendix: The data-logger
Experience has shown that the need to record the position of the light spot every minute leaves too little time for thinking about the experiment. A data-logger has been developed by
first-year students[1] to deal with this problem.
It uses a PC to move a carriage carrying light-sensitive resistors in such a way that the carriage follows the light spot. At the same time, it records the position of the carriage every so often (normally 30sec).
How to use it:
1)Laser ON (Make sure that the moving spot is at the right height to be ‘seen’ by the data-logger and that other spots are at the wrong height.)
2)LV supply ON
3)PC ON
4) At the PC:
Click on the NEWTON icon (on the desktop).
The program will ask you to give a name to the file where your data will be stored; use a unique name – such as yours.
The program will then become ready, but will not go into action until you make it – and may need help in finding the spot.
Control keys:
ReturnToggles data-saving OFF/ON
SpaceCycles through NOTHING/SPOT-FINDING/SPOT-TRACKING
dChanges direction when spot-finding
-Shifts the carriage 10mm left
=Shifts the carriage 10mm right
_Shifts the carriage 1mm left
+Shifts the carriage 1mm right
tabReminder of the control key functions
qexit from the NEWTON program.
NB: If the program is restarted, the present position is taken as 250mm on the x scale; ie. the previous zero is lost.
You will need to check that what the PC takes as metres and seconds correspond to real metres and seconds.
You will also need to check how well the carriage follows the spot – there will be some error here.
If you want to display the data, you can use CURVE EXPERT. It can be called from the desktop; your data file will be there too.
If you want to keep your data, copy them to a floppy disk; don’t rely on their staying in the PC.
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[1] The data-logger was developed in two first-year projects by J. Austin, J. Dixon, P. Kasprowicz and N. Moloney.