Question

On the way to the moon, the Apollo astronauts reach a point where the Moon’s gravitational 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.

b) What would the acceleration of the astronaut 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/s 2 .

Answer 1

The mass of the Earth M = 5.98 × 10²⁴ kg

Mass of Moon m = 7.36 × 10²² kg

The distance from the Earth to the Moon r = 3.84 × 10⁸ m

At the required distance r ' ,

Force on astronaut due to earth = Force on astronaut due to Moon

GMm ' / (r ') ² = Gmm ' / (r -r ') ²

M/ (r ') ² = m / (r -r ') ²

M/m = [r '/(r-r')] ²

[r '/(r-r')]=[M/m]

= [(5.98x10 ²⁴)/(7.36x10 ²²)]

=9.013

r ' = 9.013 (r-r ')

10.013 r ' = 9.013 r

= 9.013 x3.84 x10 ⁸

r ' = 345.683 x10 ⁶ m

(b).  the acceleration of the astronaut be due to the Earth’s gravity at this point if the moon was not there is

g ' = GM / r ' 2

Where G = The universal gravitational constant = 6.672 × 10⁻¹¹ N · m² /kg² .

Substitute values you get ,

g ' = (6.67x10 ^-11)(5.98x10 ²⁴ ) /(345.683 x10 ⁶) ²

= 3.337x10 ^-3 m/s ²