Answer:
[tex]r = 3.84 \times 10^7 m[/tex]
Explanation:
Let say the distance at which gravitational force due to both Earth and moon is zero is given by force force balance
so it is given as
[tex]F_{moon} = F_{Earth}[/tex]
[tex]\frac{GM_{moon}}{r^2} = \frac{GM_{earth}}{(d - r)^2}[/tex]
so we have
[tex]\frac{M_{moon}}{r^2} = \frac{M_{earth}}{(d - r)^2}[/tex]
[tex]\frac{7.35 \times 10^{22}}{r^2} = \frac{5.97 \times 10^{24}}{(3.84 \times 10^8 - r)^2}[/tex]
now we have
[tex]\frac{2.71}{r} = \frac{24.4}{3.84\times 10^8 - r}[/tex]
[tex]10.4 \times 10^8 - 2.71 r = 24.4 r[/tex]
now we have
[tex]r = 3.84 \times 10^7 m[/tex]
The distance from the centre of the Moon to the point where gravitational pulls of Earth and Moon are equal = 3.83× 10^7 m
The law of universal gravitation states that every object in the universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and irreversibly proportional to the square of the separation between the two objects.
The formula to solve the gravitational force between the earth and the moon;
GM(moon) / r² = GM(earth) / (d – r)²
G = 6.67 × 10-11 Nm2/kg2.
Mass of moon = 7.35 × 10^22 kg,
Mass of earth = 5.97 × 10^24 kg,
Center-to-center distance(d) = 3.84 × 108 m
Therefore r²= ?
Substitute the above variables into the formula after canceling out G at both sides of the equation.
7.35 × 10^22/r² = 5.97 × 10^24/ ( 3.84 × 10^8– r)²
now, we take the square root of both sides, we have
2.71× 10^11 / r² = 24.4 × 10^11 / (3.84 × 10^8 – r)
2.71 / r² = 24.4 / (3.84 × 10^8 – r)
If we cross multiply, we have
24.4r = 1.04064× 10^9 – 2.71r
24.4r + 2.71r = 1.04064 × 10^9
27.11r = 1.04064× 10^9
r = 1.04064× 10^9 / 27.11
r = 3.83× 10^7 m
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