Question

A spacecraft is on a journey to the moon. At what point, as measured from the center of the earth, does the gravitational force exerted on the spacecraft by the earth balance that exerted by the moon? This point lies on a line between the centers of the earth and the moon. The distance between the earth and the moon is 3.85 × 108 m, and the mass of the earth is 81.4 times as great as that of the moon.

Answer 1

Now remember that gravitational force is given by:

F = G*m1*m2/R^2

Suppose mass of spacecraft is M, and net gravitational force on spacecraft is balanced at distance d from the center of earth, then at this point

Force on spacecraft due to earth = Force on spacecraft due to moon
G*M*Me/d^2 = G*M*Mm/(D - d)^2

Me = Mass of earth = 81.4*Mm

Mm = Mass of moon

D = distance between moon and earth = 3.85*10^8 m

So,

G*M*Me/d^2 = G*M*Mm/(D - d)^2

81.4*Mm/d^2 = Mm/(D - d)^2

[(D - d)/d]^2 = 1/81.4

(D - d)/d = sqrt (1/81.4) = 0.1108

D - d = d*0.1108

d = D/1.1108 = 3.85*10^8/1.1108

d = 3.47*10^8 m = distance of spacecraft from earth

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