# How accurately can I expect to measure the gravitational constant with a club of college students?

I am a math instructor with almost no experimental physics background, but I run a math and engineering club that is interested in doing some experiments.

I have read up a bit and see some obvious plans for calculating g with pendulums or falling objects, and a more complicated plan (Cavendish experiment). Maybe there are others out there.

If I implement these methods with a group of college students, how accurate is my result likely to be?

I'm particularly interested in getting a low-budget experiment that would be able to tell the difference between daytime gravity and nighttime gravity ([Our gravitational attraction toward the sun is approximately $0.006 \frac{m}{s^2}$; please excuse my issues with frames of reference]) but maybe that is completely out of reach. Thanks for your help or references.

-
Probably depends on how accurately you can measure the mutual attraction between the students. Can you suspend one of them from a rope and measure g from the pendulum period? – Martin Beckett Jul 27 '14 at 13:56
"the difference between daytime gravity and nighttime gravity" ummm... what ? – ticster Jul 27 '14 at 13:59
What do you want to measure: the local gravitational acceleration (e.g. Pendulum experiments) or the universal gravitational constant (e.g. Cavendish experiment)? – Johannes Jul 27 '14 at 14:03
If you assume that earth is a perfect sphere, and that no other sources of gravity need be taken into account, you can deduce $G$ from $g$ using only the mass of the earth. If you want to, as you seem to, consider the small impact of various effects that slightly change the apparent surface gravity of the earth, then obviously it's not that simple. – ticster Jul 28 '14 at 8:03
@ChrisCunningham - Measuring g and G are anything but the same thing. The value of g varies from place to place, depending on latitude, altitude, and the nature of the underlying rock. g is lower in Mexico City than it is in Helsinki. It also varies with time. There are instruments that can measure the local value of g to twelve places of accuracy. On the other hand, G is a constant and is the least well known of the fundamental constants. Physicists only know G to four places of accuracy. – David Hammen Jul 28 '14 at 20:36

## Background

You would need a very sensitive instrument to measure the daytime vs nighttime difference in g. It is not 0.006 m/s2. It is much, much smaller than that, about $6\times10^{-11}$ m/s^2.

Your 0.006 m/s2 is the gravitational acceleration toward the Sun at distance of 1 AU. The Earth as a whole is accelerating sunward at 0.006 m/s2. You cannot measure that acceleration with any local experiment, and a pendulum most certainly is a local experiment.

What you can measure is tidal gravity, but you will need a very sensitive instrument. At noon, on object on the surface of the Earth is a bit closer to the Sun than is the center of the Earth and thus the object experiences a slightly greater sunward acceleration than does the Earth as a whole. The Sun pulls the object away from the Earth at noon, decreasing the sensed value of g.

The difference between these two accelerations is the source of the tides caused by the Sun. Tidal forces are approximately a 1/r3 force. This is why the tides raised by the Moon are about twice those of the tides raised by the Sun, even though the Sun is much, much more massive than the Moon.

What about midnight? At midnight, the object is a bit further from the Sun than is the center of the Earth and thus the object experiences a slightly reduced sunward acceleration than does the Earth as a whole. The Sun pulls the Earth away from the object away at midnight, once again decreasing the sensed value of g. The difference between the daytime and nighttime is extremely small, about $6\times10^{-11}$ m/s2. Detecting that small a change would require a superconducting gravimeter.

What about sunrise or sunset? Now the tidal gravitational force points toward the center of the Earth, but with about half the magnitude of the outward tidal force at noon/midnight. The difference between the solar tidal force at sunset and at noon is measurable without needing a superconducting gravimeter; you only need to be able to measure to eight significant digits. There is a problem, however: the Moon. The lunar tides are about twice as strong as the solar tides. It might be easier to conduct your experiment when the Moon is new or full, making the lunar and solar affects additive and easier to measure.

## Measurement

Measuring this won't be easy. A seconds pendulum nominally has a period of two seconds. The length of such a pendulum is a bit shy of one meter. (Aside: This was the definition of a meter recommended by French scientists. There was one problem: It was a "placist" definition because gravitational acceleration varies somewhat over the face of the Earth.) Assuming a seconds pendulum with a nominal, non-tidal period of two seconds, a seconds pendulum will have a period of 2.000000167 seconds at noon at the sub-moon point during a solar eclipse and a period of 1.999999917 seconds at sunset of the same day. An hour as measured by the clock will be 0.45 milliseconds shorter at sunset than at noon.

That's not much of a difference, and you'll have to wait until Oct 23, 2014 for a solar eclipse, and you'll have to go to central America to be close to the sub-moon point. Fortunately, the effect isn't significantly reduced at a non-eclipse new moon, and the latitudinal effects aren't terrible so long as you don't live in a far northern city such as Anchorage or Helsinki.

That 0.45 millisecond difference is measurable, and with relatively inexpensive equipment. Having students start and stop a millisecond accuracy stopwatch won't work. You'll need something more sophisticated than that. Set your physics and engineering students to the challenge.

## Calculations

The two links above are the WolframAlpha calculations that result in the 2.000000167 and 1.999999917 second periods of a nominally seconds pendulum. There are some magic numbers in those calculations.

-
Regarding the original definition of the metre, I had been led to believe that it was $1/40,000,000 th$ of the earth's circumference thought the poles. But the pendulum definition works pretty well. – Dr Chuck Jul 27 '14 at 17:31
@DrChuck -- The seconds pendulum was the original definition of the meter as proposed by a vast majority of French scientists who escaped the guillotine. That definition was rejected by non-French scientists as "placist". The 1/40000th of the Earth's polar circumference was a backup, non-placist definition. C.P. Snow's Two Cultures strikes again. – David Hammen Jul 27 '14 at 17:35
The quarter-circumference of the earth is not "placist"? They had to include the particular polar great circle (through Paris, of course) – DJohnM Jul 28 '14 at 0:15
@User58220 Originally they hoped the Earth was uniform enough that any North-South line would work. – Martin Beckett Jul 28 '14 at 5:47
@DavidHammen might be interested in this: physics.stackexchange.com/questions/128691/… – Ben Crowell Jul 29 '14 at 21:26

A couple of decades ago, I put quite a bit of time into building a high-precision differential gravimetry setup using a pendulum. It was a lot more difficult than I'd expected, and I was never very successful.

If you want to do differential gravimetry, it would probably be much more doable to measure differences between different heights above the earth's surface. The fractional change per unit height is $-2/r$, where $r$ is the radius of the earth. This works out to be about 0.3 ppm per meter, which is not bad at all if you have access to a tall building. A further advantage is that this could be done during a class period.

A high-precision pendulum is not at all easy to build. The first issue is that you need an accurate sensor to pick off the timing. I did this using a photogate interfaced to a computer.

The period depends on amplitude in proportion to $K(\sin(\theta/2))$, where $K$ is the complete elliptic integral of the first kind. This function is pretty flat for small $\theta$, but not flat enough to allow the variation to be ignored completely. I doubt that a mechanical escapement is going to work.

One approach that I tried was to release the pendulum using a magnet from an accurately controlled initial height, and then to allow its amplitude to decay naturally. If the decay was completely reproducible, then this would allow a good differential measurement. In reality, I found that the decay was not highly reproducible. In general, any experiment involving static and kinetic friction tends to be hard to reproduce accurately.

Another possible approach is to use the photogate to determine the velocity of the pendulum individually for each swing. From the velocity you can determine the amplitude, and thereby correct the period on a swing-by-swing basis.

I believe that professionals no longer use a pendulum for this type of thing. For absolute measurements, they use an apparatus in which a weight is dropped in a small, portable vacuum column, with interferometry to measure the motion. For relative measurements, I think they use a simple mass hanging from a spring.

By the way, it's possible to get a surprisingly good measurement of $g$ with just an index card, a pin, and a stopwatch. Put the pin through a point on the diagonal of the card, at a distance $L$ from the center. Let $d$ be the length of the diagonal. Measure the period $T$ and estimate the quality factor $Q$ of the oscillations. Use a small enough amplitude so that $K(\sin(\theta/2))\approx 1$.

$$g=(4\pi^2L/T^2)[1+(1/12)(d/L)^2][1-1/(4Q^2)]^{-1}$$

I found it pretty easy to get within 5% by this technique. The main problem was the tendency of the card to flutter rather than rotating purely in its own plane.

-
Thanks very much for this. In particular the point that static and kinetic friction both appear is a very good point, and I appreciate the explicit inclusion of how the quality factor appears. Unfortunately I think this kind of thing is probably what drove me toward math and away from physics -- every time I decide I want to "do" physics, friction and air resistance prevent me from doing it! – Chris Cunningham Jul 27 '14 at 20:45

I understand you want to measure or demonstrate the tidal effect of the Moon (stronger than from the Sun), changing the local gravity acceleration. The gravitational constant is a universal constant, it does not change in time.

It is achievable/worth trying, if you have a good workshop. The mid 20th century pendulum gravimeters, which you are trying had the following main catches.

1/ Air resistance: you will need the demonstration type mechanical vacuum pump and glass "recipient". (You do not need a diffusion vacuum pump.)

2/ Temperature: you need to keep the whole thing at a stable temperature should be easy with 2014 technology.

3/ Good bearings. Sharp wedges from high quality steel or, better, clockwork like hard gems.

4/ Structure rigidity: It is difficult to compensate for the whole structure softness. The pendulum tends to pull the bearing horizontally. The trick used was having an even number of pendulums (usually four) swinging in opposite phase.

5/ Electrostatic force. This may distort your measurement. Everything needs to be grounded. They used to use a small piece of a mildly radioactive isotope to ionize the residual gas in the vacuum chamber. Probably not a good idea today for a college experiment.

6/ Good precision of time measurement. Should not be a problem with atomic clocks available on the internet and quartz clocks in every shop.

My rough guess is that you can make it, but in is not a trivial task.

-
I know G doesn't change; my initial impression was that g would change from day to night because of the effect of the sun (a newbie frame of reference error). Anyway, yikes. As a math professor with no laboratory experience, almost every single one of your six points is wayyy out of my reach. Thanks very much!! – Chris Cunningham Jul 27 '14 at 23:29
If you want to see what Professor Pippard of the Cavendish Laboratory in Cambridge had to say about a very careful (Foucault) pendulum experiment, read allais.info/priorartdocs/pippard.pdf . It makes you realize what a really good experimental physicist thinks about... Or see here for a description of a lovely experiment (falling prism & interferometer) to do this "the right way"... – Floris Jul 29 '14 at 22:38