證明於地球內g正比於r

The force of gravity at distance r inside the earth is only contributed by the sphere of mass centred at the earth centre with radius r. Hence, by Newton's Law of Gravitation,

mg = GMm/r^2
where m is the mass of an arbitrary object
M is the mass of sphere with radius r centred at the earth centre
G is the Universal Gravitational Constant
g is the acceleration due to gravity at distance r from earth centre

But M = (4/3)(.pi.r^3).d
where d is the density of earth (assume uniform independent of r), and pi = 3.14159.....

Thus, g = GM/r^2 = (G/r^2).(4/3)(pi.r^3).d = [4G.pi.d/3].r
i.e. g is proportional to r

Q: 雙星系統的軌跡是不是週期一致??
Yes. Binary stars are stars that revolve around a common centre of mass. Just imagine if the periods of the two stars are not the same, the motion of the stars would make their distance of cloest approach or distance of farther approach be different for each period of time. This means their common centre of mass would shift (i.e oscillate) from time to time. This would violate the Law of Conservation of Momentum as the gravitational force between the two stars is an internal force. The centre of mass of the system should not be moved.

You may refer to the following web-page for the calculation of period for a binary star system. Note that the same period of revolution for the two stars is taken as a pre-requisite.

http://www.egglescliffe.org.uk/physics/gravitation/binary/binary.html