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

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.