Question

The distance between the centres of the Moon and the earth is \[D\] . The mass of the earth is $81$ times the mass of the Moon. At what distance from the centre of the earth, the gravitational force will be zero?
A. $\dfrac{D}{2}$
B. $\dfrac{{2D}}{3}$
C. $\dfrac{{4D}}{3}$
D. $\dfrac{{9D}}{{10}}$

Answer 1

Hint If net gravitational field intensity due to some bodies at a point is zero, then the gravitational force will also be zero at that point.
The gravitational field intensity due to a body of $M$ at a distance $r$ from its centre is given by $\dfrac{{GM}}{{{r^2}}}$ where $G$ is called Gravitational constant.

 Complete step-by-step solution:As in the question we are asked to find at what distance from the centre of the earth, the gravitational force will be zero
We know that, if net gravitational field intensity due to some bodies at a point is zero, then the gravitational force will also be zero at that point. So, we first calculate the gravitational field intensity due to the earth and the moon at a particular point.
Let the net field at any point P from the earth is zero which is at distance ‘x’ from the centre of the earth. As the distance between the centres of the Moon and the earth is given \[D\] then point P will be at a distance $D - x$ from the centre of the moon.
We know that the gravitational field intensity due to a body of $M$ at a distance $r$ from its centre is given by $\dfrac{{GM}}{{{r^2}}}$ where $G$ is called Gravitational constant.
As given in the question that the mass of the earth is $81$ times the mass of the Moon i.e. ${M_e} = 81{M_m}$
Then the gravitational field intensity at point P due to earth
${E_e} = \dfrac{{G{M_e}}}{{{x^2}}} = \dfrac{{81G{M_m}}}{{{x^2}}}$ towards centre of earth
Now, the gravitational field intensity at point P due to moon
$\Rightarrow {E_m} = \dfrac{{G{M_m}}}{{{{\left( {D - x} \right)}^2}}}$ towards centre of moon
Now, for zero intensity at point P, ${E_e} = {E_m}$ i.e.
$\Rightarrow \dfrac{{81G{M_m}}}{{{x^2}}} = \dfrac{{G{M_m}}}{{{{\left( {D - x} \right)}^2}}}$
On solving the equation we get
$\Rightarrow 81{\left( {D - x} \right)^2} = {x^2}$
On further solving this quadratic equation we have
$\Rightarrow 9\left( {D - x} \right) = x$
On simplifying we get the final answer
$\Rightarrow x = \dfrac{{9D}}{{10}}$

Hence, option D is correct.

Note: The direction of Gravitation field intensity due to a body at a point is towards the centre of gravity of that body.
The gravitational force is a force that attracts any two objects with mass. We call the gravitational force attractive because it always tries to pull masses together, it never pushes them apart. In fact, every object, including you, is pulling on every other object in the entire universe! This is called Newton's Universal Law of Gravitation.