Question

A body of mass 5 kg is cut into two parts of masses (a) $\dfrac{m}{4};\dfrac{{3m}}{4}$ (b) $\dfrac{m}{7};\dfrac{{5m}}{7}$ (c) $\dfrac{m}{2};\dfrac{m}{2}$ (d)$\dfrac{m}{5};\dfrac{{4m}}{5}$ . When these two pieces are kept apart by a certain distance; in which case the gravitational force acting is maximum?

Answer 1

In order to answer this question, to know that in which case the gravitational force is maximum we will use the universal gravitation equation. The force between any two objects in the cosmos is described by this equation. You can calculate the force between two things if you know how big they are and how far away they are.

Formula used:
$F = G\dfrac{{M \times m}}{{{r^2}}}$
The gravitational force is denoted by the letter $F$ . (measured in Newtons, $N$ )
$G$ is the universe's gravitational constant, and it is always the same quantity.
The mass of a single object is $M$ . (measured in kilograms, $kg$ )
The mass of the other object is $m$ . (measured in kilograms, $kg$ )
$r$ is the distance between the two items (measured in meters, $m$ )

Complete step by step answer:
The gravitational force is a force that attracts all mass-bearing objects. The gravitational force is referred to as attractive because it always strives to pull masses together rather than pushing them apart. In reality, everything in the cosmos, including you, is tugging on every other item. Newton's Universal Law of Gravitation is the name for this.

Now, coming to the question; if $r$ is the distance between mass $m$ and $\left( {M - m} \right)$ objects, the gravitational force between them will be:
$F = G\dfrac{{m\left( {M - m} \right)}}{{{r^2}}} \\\ \Rightarrow F = \dfrac{G}{{{r^2}}}\left( {mM - {m^2}} \right) \\\$
Where, $M$ = Mass of one object
For $F$ to be maximum
$\dfrac{{df}}{{dm}} = 0\,\,\left[ {As\,M\,and\,r\,are\,const} \right] \\\ \Rightarrow \dfrac{d}{{dM}}\left[ {\dfrac{G}{{{r^2}}}\left( {mM - {m^2}} \right)} \right] = 0 \\\ \Rightarrow M - 2m = 0 \\\ \therefore \dfrac{m}{M} = \dfrac{1}{2}$
So, the force will be maximum when the parts are equal

Hence, option C has equal parts.

Note: The resultant (vector sum) of two forces on Earth is the force of gravity: (a) Newton's universal law of gravitation, and (b) the centrifugal force, which derives from the choice of an earthbound, rotating frame of reference. Because of the centrifugal force created by the Earth's rotation and because sites on the equator are farthest from the Earth's core, the force of gravity is weakest at the equator.