Physics 110A: Chap 8 Extra Problems
Required Additional Problems
A. Develop an expression for the orbital period of a satellite skimming the
(airless) surface of a spherical body as a function of that body's average
density, ρ. Assume a gigantic mass disparity (so μ is effectively
the mass of the little thing, and M is the mass of the parent
body). You will ultimately need to use
G ≈ 6.67×10−11
m³/s²/kg.
- First, balance centripetal acceleration against gravitational acceleration
written in terms of the density rather than the mass (can assume uniform
density) of the parent body.
- Now form a relation for the period as the circumference over
velocity. This should start out looking basically like Kepler's third law, but reducing
to a function only of density (and constants).
- Form a table for orbital time (in minutes) as a function of the
average density for several different compositions: gas giant or normal
star with average density around that of water (1000 kg/m³), asteroid
or moon with the density of rock (3000 kg/m³), the Earth (5500
kg/m³), and a white dwarf star with density one million times that of
water.
For orbits not skimming the surface, you can stil play the same
game for a circular orbit just by considering the average density
within the sphere fitting within the orbit.
B. What would happen to the orbit of the Earth if the Sun unexpectedly collapsed into a black hole, without losing any of its mass in the process?
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