newtonian gravity - 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.

Answer

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.