Question

At what distance from the center of the Moon is the point at which the strength of the resultant of the Earth's and Moon's gravitational fields is equal to zero? The Earth's mass is assumed to be η = 81 times that of the Moon, and the distance between the centers of these planets n = 60 times greater than the radius of the Earth R.

Answer 1

Hint: To find the distance of the point where gravitational fields is zero, we consider the distance from moon to this point be some variable ‘x’, and we equate the gravitational strengths of earth and moon at this point in order to make a relation, we solve this equation to find ‘x’.

Formula used: Gravitational Field of a body of mass M to a point which is at a distance ‘r’ from the center of the body is \[{\text{E = }}\dfrac{{{\text{GM}}}}{{{{\text{r}}^2}}}\], where G is the gravitational force constant.

Complete Step-by-Step solution:
Now let us assume the distance between the centers of earth and moon to be ‘d’. Along the line of their radii earth exerts a gravitational pull towards itself and moon towards itself respectively. At a point ‘P’ where the resultant of their gravitational forces is equal to zero, the gravitational force exerted by the earth is equal to that of the moon. Let this point be at a distance of ‘y’ from the moon.

At point P the gravitational force of the earth is equal to that of the moon.
\[
   \Rightarrow {{\text{E}}_{\text{e}}}{\text{ = }}{{\text{E}}_{\text{m}}} \\
   \Rightarrow \dfrac{{{\text{G}}{{\text{M}}_{\text{e}}}}}{{{{\text{r}}_{\text{e}}}^2}}{\text{ = }}\dfrac{{{\text{G}}{{\text{M}}_{\text{m}}}}}{{{{\text{r}}_{\text{m}}}^2}} \\
\]
From the given,
Mass of earth ${{\text{M}}_{\text{e}}}$= 81${{\text{M}}_{\text{m}}}$
Distance between earth and the point P is “d – y” and distance between moon and point P is “y”.
\[ \Rightarrow \dfrac{{{\text{G81}}{{\text{M}}_{\text{m}}}}}{{{{\left( {{\text{d - y}}} \right)}^2}}}{\text{ = }}\dfrac{{{\text{G}}{{\text{M}}_{\text{m}}}}}{{{{\text{y}}^2}}}\]
\[
   \Rightarrow 81{{\text{y}}^2} = {\left( {{\text{d - y}}} \right)^2} \\
   \Rightarrow 9{\text{y = d - y}} \\
   \Rightarrow {\text{d = 10y}} \\
\]
Distance between earth and moon d = 10y
Given the distance between the centers of earth and moon is 60 times the radius of earth ‘R’.
Hence 10y = 60R
⟹y = 6R
Distance to the point where the net gravitational force is zero, from the moon is six times the radius of earth, i.e. 6R
We know the radius of earth is 6371km.
Hence the distance from the center of the Moon is the point at which the strength of the resultant of the Earth's and Moon's gravitational fields is equal to zero is
y = 6 × 6371km = 38226 km =${\text{3}}{\text{.8}} \times {\text{1}}{{\text{0}}^{\text{4}}}{\text{ km}}$.

Note – In order to answer this type of questions the key is to know the concept of gravitational force exerted by a body on another body at a certain distance from its center.
G is the universal gravitational constant, it is an empirical physical constant. It is used to show the force between two objects caused by gravity and has a value of $6.67259 \times {10^{ - 11}}{\text{ N}}\dfrac{{{{\text{m}}^2}}}{{{\text{K}}{{\text{g}}^2}}}$.