Question

Mass \[m\] is divided into two parts \[Xm\] and \[\left( {1 - X} \right)m\]. For a given separation the value of \[X\] for which the gravitational force of attraction between the two pieces becomes maximum is:
A. \[\dfrac{1}{2}\]
B. \[\dfrac{3}{5}\]
C. \[1\]
D. \[2\]

Answer 1

Hint: In this question, the mass \[m\] is divided into two parts \[Xm\] and \[\left( {1 - X} \right)m\] , and we need to find the value of \[X\] for which the gravitational force of attraction between the two pieces becomes maximum. To solve this, find the gravitational force of attraction between the two masses. For maximum value \[\dfrac{{dF}}{{dX}} = 0\]. Differentiate force with respect to \[X\] and equate it to zero. Solve the equation, to find out the value of \[X\]

Formula used:
The magnitude of Gravitational force \[F\] between two particles \[{m_1}\] and \[{m_2}\] placed at a distance \[r\] is given by,
\[F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}\]
Where \[G\]is the universal constant called the Gravitational constant.
\[G = 6.67 \times {10^{ - 11}}{\text{ N - m/k}}{{\text{g}}^2}\]

Complete step by step answer:
Mass \[m\] is divided into two parts \[Xm\] and \[\left( {1 - X} \right)m\].
Let the distance between them be \[R\] meters
Therefore, Gravitational force between them is given by \[F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}\]
Substituting the values in the formula we get,
\[F = \dfrac{{GXm\left( {1 - X} \right)m}}{{{R^2}}}\]
\[ \Rightarrow F = \dfrac{{GX\left( {1 - X} \right){m^2}}}{{{R^2}}}\]
The gravitational force for a given distance will be maximum when \[X\left( {1 - X} \right)\] will be maximum
Thus, for maxima, \[\dfrac{{dF}}{{dX}} = 0\]
Differentiating the gravitational force between the two masses \[F\] with respect to \[X\] we get,
\[\dfrac{{dF}}{{dX}} = \dfrac{{G{m^2}}}{{{R^2}}}\dfrac{{d\left( {X\left( {1 - X} \right)} \right)}}{{dX}}\]
\[ \Rightarrow \dfrac{{dF}}{{dX}} = \dfrac{{G{m^2}}}{{{R^2}}}\dfrac{{d\left( {X - {X^2}} \right)}}{{dX}}\]
\[ \Rightarrow \dfrac{{dF}}{{dX}} = \dfrac{{G{m^2}}}{{{R^2}}}\left( {1 - 2X} \right)\]
For maxima, \[\dfrac{{dF}}{{dX}} = 0\]
\[ \Rightarrow \dfrac{{dF}}{{dX}} = \dfrac{{G{m^2}}}{{{R^2}}}\left( {1 - 2X} \right) = 0\]
\[ \Rightarrow \left( {1 - 2X} \right) = 0\]
On solving we get,
\[X = \dfrac{1}{2}\]
The gravitational force between the masses has a maximum value at \[X = \dfrac{1}{2}\]
The mass \[m\] should be divided into \[\dfrac{m}{2}\] and \[\dfrac{m}{2}\] for maximum gravitational force.
Hence the correct option is option (A).

Note:
Unlike the electrostatic force, Gravitational force is independent of the medium between the particles. It is conservative in nature. It expresses the force between two-point masses (of negligible volume). However, for external points of spherical bodies, the whole mass can be assumed to be concentrated at its center of mass.