What happened to the force between two object if the mass of one of the body is doubled and separation is half?

Text Solution

Solution : As gravitational force between two object, `F prop (m_(1)m_(2))/r^(2)` thereforce, <br> (i) when mass of one object is doubled, the force become twice. (ii) when distance between the objects is doubled, force become `(1//4)` of its previous value. When distance between the objects is tripled, the force become `(1//9)` of its previous value . (iii) when masses of both object are doubled, force becomes 4 times.

Answer

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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.

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