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1) On the way to the moon, the Apollo astro-

nauts reach a point where the Moon’s gravi-

tational pull is stronger than that of Earth’s.

Find the distance of this point from the

center of the Earth. The masses of the

Earth and the Moon are 5.98 × 1024 kg and

7.36 × 1022 kg, respectively, and the distance

from the Earth to the Moon is 3.84 × 108 m.

Answer in units of m


2) What would the acceleration of the astro-

naut be due to the Earth’s gravity at this

point if the moon was not there? The

value of the universal gravitational constant

is 6.672 × 10−11 N · m2/kg2.

Answer in units of m/s2

Verified Answer :

Hagrid

(1) You must find the point of equilibrium between the two forces,


G  MTms / (R−x)^2 = G  MLms / x^2
MT / (R-x)^2 = ML / x^2

So,

x = R sqrt(ML MT) - ML / (MT - ML)
R = is the distance between Earth and Moon.

The result should be,

x = 3.83 10^7m
from the center of the Moon, and 

R - x = 3.4610^8 m
from the center of the Earth.


(2) As the distance from the center of the Earth is the number we found before,
d = R - x = 3.4610^8m
The acceleration at this point is
g = G  MT / d^2
g = 3.33*10^-3 m/s^2

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