## A Problem of Dimensions

Posted in Cute Problems, Maynooth, The Universe and Stuff with tags , , , on August 21, 2019 by telescoper

We’ve more-or-less sorted out who will be teaching what next term in the Department of Theoretical Physics at Maynooth University next term (starting a month from now) and I’ll be taking over the Mathematical Physics module MP110, which is basically about Mechanics with a bit of of special relativity thrown in for fun. Being in the first semester of the first year, these is the first module in Theoretical Physics students get to take here at Maynooth so it’s quite a responsibility but I’m very much looking forward to it.

I am planning to start the lectures with some things about units and dimensional analysis. Thinking about this reminded me that I posted a dimensional analysis problem (too hard for first-year students) on here a while ago which seemed to pose a challenge so I thought I would post another here for your amusement.

The period P for an elliptical orbit of semi-major axis a of  a moon of mass m around a planet of mass M, depends only on the quantities  a, m, M and G (Newton’s Gravitational Constant).

(a). Using dimensional analysis only, determine as completely as possible the relationship between P and these four quantities.

(b). How would the period P compare with the period P′ of a system consisting of a moon of mass 2m orbiting a planet of mass 2M in an ellipse with the same semi-major axis a?