What happens to the gravitational force when the mass each of the two masses doubles?

1. The force would remain the same
2. The force would be doubled
3. The force would be halved
4. The force would increase by a factor of 4

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Question 5 Exercise 3E

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Answer:

The gravitational force reduces to one-fourth if the distance between two masses is doubled.

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Hint: Relation between gravitational force, mass and distance is,$F=G\dfrac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}$ Where G is Newton’s gravitational constant${{m}_{1}}$ and ${{m}_{2}}$ are the massesr is the distance.

Complete step by step solution:

Newton stated that in the universe each particle of matter attracts every other particle. This universal attractive force is called “Gravitational”.Newton’s law:- Force of attraction between any two material particles is directly proportional to the product of masses of the particles and inversely proportional to the square of the distance between them. It acts along the line joining the particles.$F\propto \dfrac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}$ $F=G\dfrac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}$ Where G is the proportionality constant and it is universal constant.(i) If the mass of an object is doubled:$m{{'}_{1}}$ = ${{m}_{1}}$ $m'_{2}$ = $2{{m}_{2}}$ $F'=G\dfrac{{{m}_{1}}'{{m}_{2}}'}{{{\left( r{{'}^{{}}} \right)}^{2}}}$ $F'=G\dfrac{{{m}_{1}}\left( 2{{m}_{2}} \right)}{{{r}^{2}}}$ $F'=2\times G\dfrac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}$ $F'=2\times F$ When the mass of an object is doubled then the force between them is doubled.(ii) The distance between object is doubled and tripled:When $r'=2r$ Then $F'=G\dfrac{{{m}_{1}}{{m}_{2}}}{r{{'}^{2}}}$ $F'=G\dfrac{{{m}_{1}}{{m}_{2}}}{{{\left( 2r \right)}^{2}}}$ $F'=G\dfrac{{{m}_{1}}{{m}_{2}}}{4{{r}^{2}}}$ $F'=\dfrac{G}{4}\dfrac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}$ $F'=\dfrac{F}{4}$ When the distance between the objects is doubled then force between them is one fourth.When $r'=3r$ Then $F'=G\dfrac{{{m}_{1}}{{m}_{2}}}{{{\left( r' \right)}^{2}}}$  $F'=G\dfrac{{{m}_{1}}{{m}_{2}}}{{{\left( 3r \right)}^{2}}}$ $F'=G\dfrac{{{m}_{1}}{{m}_{2}}}{9{{r}^{2}}}$  $F'=\dfrac{F}{9}$ When the distance between the objects is tripled then force between them is one ninth.(iii) The masses of both objects are doubled:When $\begin{align}& m{{'}_{1}}=2{{m}_{1}} \\ & m{{'}_{2}}=2{{m}_{2}} \\ \end{align}$ Then $F'=G\dfrac{m{{'}_{1}}m{{'}_{2}}}{{{r}^{2}}}$ $F'=G\dfrac{2{{m}_{1}}\times 2{{m}_{2}}}{{{r}^{2}}}$ $F'=4G\dfrac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}$ $F'=4F$ When the masses of both objects are doubled then the force between them is four times.

Note: This law is true for each particle of matter, each particle of matter attracts every other particle. Students should use the gravitational force formula carefully and write its term properly.