physics quest.?

Question

At what distance does the force of gravity between the moon and earth negate each other and equal zero. give answer in percentage or fraction ie 60% of earth & 40% of moon or 50%of moon 50 %of earth

Answer 1

In quite close terms, the answer is at a point 90% of the distance to the Moon. So, at that point you're 90% of the full Earth-Moon distance away from the Earth, and 10% of it away from the Moon.

The reason for this is that gravitational attraction towards any one body is proportional to M / D^2. (We can ignore G, the Gravitational Constant, which is common to both attractions in this problem.) So, to find the "balance point" where the two attractions of the Earth and the Moon precisely cancel one another out, you need to solve:

M_e / (D_e)^2 = M_m / (D^m)^2 or (D_e / D_m)^2 = M_e / M_m,

where suffixes 'e' and 'm' stand for Earth and Moon respectively.

Now, extremely conveniently, the Earth is very closely 81 times as massive as the Moon, and 81 is a perfect square! So, clearly,

D_e / D_m is very closely 9.

How convenient in a decimal system!: the distance from the Earth to the balance point, D_e, is therefore ~ 90% of the full Earth-Moon distance, and the distance to the Moon, D_m is only ~ 10% of that full distance.

(More accurately, Earth is ~ 81.3 times more massive than the Moon. Sqrt(81.3) is ~ 9.02. So the fractions are ~ 9.02/10.02 and ~1.00/10.02; to 3 sig. figs., the answers found above are almost unaltered: 90.0% and 9.98%. [They don't quite add up to 100% because of the next, but "insignificant" figure in the first one making it more like 90.02%.] )

Live long and prosper.

Answer 2

This is why I am failing honors physics. Mr S. only gives us a binder full of **** to study, feel lucky!