Gravitation?

Question

A mass M is split into two parts, m and M - m, which are then separated by a certain distance. What raio m/M maximizes the magnitude of the gravitational force between the parts?
Ans: 1/2

Answer 1

Given 2 parts of mass {a,b} the gravitational force is given by

Gab/r^2

now a = m and b = M-m
so F = Gm(M-m)/r^2
for F to be Max/Min df/dm should be equal to 0 so we have

df/fm = 0
or d/dm[Gm(M-m)/r^2]
= G/r^2d/dm(mM-m^2)
= G/r^2(M-2m)=0
or
M-2m=0
or 2m=M
or m=M/2
or m/M=1/2
now to ensure that for m=M/2 the Force would be maximum we have to differentiate on second order so we get


d2f/dm = d/dm(G/r^2(M-2m))
=G/r^2d/dm(M-2m)
=-2G/r^2

and if the value is < 0 as we get above so F is maximum

Alternatively considering the hypothesis that a*b to be maximum a should be equal to be so for
Gm(M-m)/r^2
to be maximum
m = M-m
or 2m=M
or m = M/2
or m/M = 1/2

Answer 2

If you take the mass of a globe it would contain a structural energy which acts in the opposite direction of the Gravity energy holding it together . (Newtons 3rd law=To every energy there is an equal an opposite reacting energy.)
Now ,if we split the gobe in half the Gravity energy would also be divided in half.The Reason is the energy due to gravity is proportional to mass(E=MC^2).
So the ratio is 1/2. =is correct.

Answer 3

K?