Question

证明于地球内g正比于r

Answer 1

更新1:

如果识呢条可以答埋 双星系统的轨迹是不是周期一致??

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: 双星系统的轨迹是不是周期一致?? Yes. Binary stars are stars that revolve around a mon centre of mass. Just imagine if the periods of the o 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 me their mon 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 beeen the o 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 o stars is taken as a pre-requisite. egglescliffe.uk/physics/gravitation/binary/binary