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## Main Question or Discussion Point

I've got a Physics Olympiad coming up in a couple of months. Here is one paper from it: http://www.physics.ox.ac.uk/olympiad/Downloads/PastPapers/BPhO_Round_1_2013_prt_2.pdf If you would look through a couple of the questions I'm sure you'll get the flavour. I'll reproduce one of them here:

Q3.

(a) The Earth can be approximated by a perfect homogeneous stationary solid sphere, radius RE. A straight smooth tunnel is drilled along a diameter. Show that a particle, mass m, released from the entrance, will perform simple harmonic motion. Determine its period, T1, in terms of R

R

(b) A straight smooth tunnel is drilled through the Earth, in any direction, from any point on the Earth’s surface, not passing through the centre of the Earth. Determine the motion of a particle released from the entrance and obtain its period of oscillation, T2.

(c) Compare the period of a satellite orbit, TS, in very close Earth orbit, with T2.

(d) If the particle is given an increased velocity, when in the middle of the tunnel, what can be deduced about its trajectory when it emerges from the tunnel into space?

The theory I know already, regarding this question, is F=GmM/r

Where can I get a book, or something online, with many more problems like this and their worked solutions? A book would be ideal because then it would presumably be grouped by topic (e.g. this is gravitation).

Thanks for any recommendations.

Q3.

(a) The Earth can be approximated by a perfect homogeneous stationary solid sphere, radius RE. A straight smooth tunnel is drilled along a diameter. Show that a particle, mass m, released from the entrance, will perform simple harmonic motion. Determine its period, T1, in terms of R

_{E}and g. Hence evaluate T1.R

_{E}= 6.38 x 10^{3}km(b) A straight smooth tunnel is drilled through the Earth, in any direction, from any point on the Earth’s surface, not passing through the centre of the Earth. Determine the motion of a particle released from the entrance and obtain its period of oscillation, T2.

(c) Compare the period of a satellite orbit, TS, in very close Earth orbit, with T2.

(d) If the particle is given an increased velocity, when in the middle of the tunnel, what can be deduced about its trajectory when it emerges from the tunnel into space?

The theory I know already, regarding this question, is F=GmM/r

^{2}, basic mechanics (simple harmonic motion, centripetal motion, etc., if that was ever needed), Newton's laws etc. So not that much. Everything else has to be worked out.Where can I get a book, or something online, with many more problems like this and their worked solutions? A book would be ideal because then it would presumably be grouped by topic (e.g. this is gravitation).

Thanks for any recommendations.