Clarification of the term investigate:
When a problem asks you to investigate something experimentally, it means that you must control variables and test hypotheses. This is the process of inductive reasoning.
When a problem asks you to investigate something theoretically, it means that you must start with a clear and simple set of postulates, and derive from those what you expect to measure. This process is called deductive reasoning.
Good science combines inductive and deductive reasoning. Good physics is both theoretically derivable and experimentally testable.
A note about sources of data:
When a problem asks you to measure a fundamental quantity, you may ask what assumptions you may make and what data it is appropriate to use. This is somewhat subtle, and, of course, the judges especially consider your understanding of the process of science. That said, there are a few rules:
- All sources of data must be transparent. In other words, either you took them yourselves, or you have clearly cited where the data came from.
- You may use public scientific data, so long as you analyze them yourselves and clearly cite where they came from. Use the databases that are provided for scientists to facilitate their research. As a student doing research, you are now part of the scientific community, so do not be afraid to file a help ticket to ask technical questions when you get stuck.
It is important, however, to understand that the more you do yourselves, the more impressed the jury will be. That said, many branches of physics, especially fields such as astrophysics and geophysics, fund centralized data collection, and then make the data public.
Measuring the AU: (1) Check an astronomical calendar, or ephemeris, early for any astronomical events that you could use as a natural experiment, (2) Consider using a public astronomical database, such as the GONG network of solar telescopes, in addition to any measurements you make yourselves, (3) Consider requesting observing time on a publically funded professional telescope, such as through the Skynet Junior Scholars program.
The Archimedes Death Ray: You will want to start by watching the Mythbusters episodes and looking at the web side made by the MIT students for their class. You might even find other work investigating this question, before you start your own work in physical history.
Spherical magnets: Remember how spherical planets act like point masses for Newton’s law of gravity. You can show that spherical magnets act like point magnets, each characterized by its magnetic moment vector.
The Apparent Weight of an Hourglass: Read the American Journal of Physics article referenced in the problem before starting your investigations.