Question

If the mass of the moon is $\dfrac{M}{{81}}$ where M is the mass of earth, find the distance of the point from the moon where gravitational force due to earth and moon cancel each other. Given the distance between earth and moon is $60R$ where R is the radius of earth.
(A) 4R
(B) 8R
(C) 2R
(D) 6R

Answer 1

Hint: This question relates to the gravitational field intensity caused due to one body when compared to another body. For this question consider a point with some mass, say ‘m’ and the compare gravitational force experienced by it at that point due to both earth and moon and equate both of them.
Formulas used:
Gravitational field intensity of a body on another body is proportional to the product of mass of both bodies divided by the square of their separation. To get the gravitational force, multiply it by the gravitational constant.
$g = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$ where G is the gravitational constant, ${m_1}$ and ${m_2}$ are the masses of bodies in consideration and ‘r’ is the separation between those bodies.

Complete step by step solution:
By the law of gravitation, we know that the gravitational force $g = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$ where G is the gravitational constant, ${m_1}$ and ${m_2}$ are the masses of bodies in consideration and ‘r’ is the separation between those bodies.
For this question, we need to find the exact point where the gravitational force of earth and moon will cancel each other out. In other words, imagine a body such that it is between the earth and moon. We need to find that point where the gravitational force experienced by the body due to earth and the gravitational force experienced by the body due to the moon will be equal, hence cancelling each other out. Also keep in mind that this distance is being calculated from the moon.
Now, we assume that the mass of that supposed body is ‘m’ while the distance it has from the moon is ‘r’. As such the gravitational force experienced by it due to the moon will be
$\
  {g_{moon}} = \dfrac{{G{M_{moon}}m}}{{{r^2}}} \\
   = \dfrac{{G\dfrac{M}{{81}}m}}{{{r^2}}} \\
   = \dfrac{{GMm}}{{81{r^2}}} \\
\ $
Now if the distance between earth and moon, as given in the question is 60R and the point is located between the earth and moon at a distance r from the moon then its distance from earth will be $60R - r$
As such, the gravitational force experienced by it due to the earth will be
$\
  {g_{earth}} = \dfrac{{G{M_{earth}}m}}{{{{\left( {60R - r} \right)}^2}}} \\
   = \dfrac{{GMm}}{{{{\left( {60R - r} \right)}^2}}} \\
\ $
According to the question, gravitational forces of earth and moon cancel each other out. This means that ${g_{moon}} = {g_{earth}}$
$\
   \Rightarrow \dfrac{{GMm}}{{81{r^2}}} = \dfrac{{GMm}}{{{{\left( {60R - r} \right)}^2}}} \\
   \Rightarrow 81{r^2} = {\left( {60R - r} \right)^2} \\
\ $
Now distance can never be negative so we omit the negative variable in the square root and only take the positive variable. As such,
$\
  9r = 60R - r \\
  10r = 60R \\
  r = 6R \\
\ $

Note:
This question could also have been solved by directly equating the gravitational field intensities of the earth and moon. Gravitational field intensity of a body is given by the formula $\overrightarrow g = \dfrac{{Gm}}{{{r^2}}}\widehat r$ where G is the gravitational constant, ‘m’ is the mass of body in consideration and ‘r’ is the distance of point at which the intensity of the field is to be calculated. $\widehat r$ here is the unit directional vector.