Question

A mass M is broken in two parts m and (M-m).Relation between m and M so that the force of gravitation between the two parts is maximum is
A). $mM = 2$
B). $m = \dfrac{M}{2}$
C). $m = {M^2}$
D). None of these

Answer 1

Hint- This question is simply based on the gravitational force of attraction and the formula given by Newton will be used which gives the magnitude of this attraction force between any two objects.

Complete step-by-step solution-
Mathematically this force between any two bodies is given by
$F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$
Where G is the gravitational constant
$m_1$ is the mass of one body,
$m_2$ is the mass of another body.
Also from the theory of mathematics we know that for finding maximum value or minimum value of any value we need to go to differentiate the equation.
Thus in this question we have: M - mass of the previous body .
m - a part of the body after breaking ,
M-m - is the mass of the second body.
 Since the gravitational force between two bodies is given by
$F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$
$ {m_1} = m \\
  {m_2} = M - m \\ $
In order to find the maxima, we need to differentiate above equation then equate it equal to zero
$\dfrac{{dF}}{{dm}} = 0$
$ \dfrac{{d\dfrac{{Gm\left( {M - m} \right)}}{{{r^2}}}}}{{dm}} = 0 \\
  \Rightarrow \dfrac{{d\left( {mM - {m^2}} \right)}}{{dm}} = 0 \\
  \Rightarrow M - 2m = 0 \\
  \Rightarrow m = \dfrac{M}{2} \\ $
Hence force between them is maximum when
$m:M = 1:2$
Therefore the correct option is “B”.

Note- The gravitational law of Newton is similar to the law of electric forces of Coulomb, which is used to measure the magnitude of the strength between two charged bodies. These are both reverse-square rules, in which power correlates inversely to the quadrate of the distance between the legs. Coulomb's law has the product of two charges in place of the product of the masses, and the Coulomb constant in place of the gravitational constant.