What happens to the gravitational force between two objects, if the mass between them is doubled

Solution

According to the laws of gravitational, F=Gm1m2r2. Where, m1andm2 are the masses of the object, r= distance between both masses. 2. Force is directly proportional to the product of masses. 3. On increasing the mass of one body the gravitational force of attraction will also increase proportionally.Hence, if the mass of one object is doubled then the force between them also gets doubled.

Answer

Verified

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 masses
r 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.

NCERT Question 6 - Chapter 10 Class 9 - Gravitation (Term 2)

Last updated at May 30, 2019 by

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Transcript

NCERT Question 6 What happens to the force between two objects, if (i) the mass of one object is doubled? (ii) the distance between the objects is doubled and tripled? (iii) the masses of both objects are doubled? We know that gravitational force between two objects is given by, F = 𝐺𝑀𝑚/𝑟^2 where G = Gravitational constant M = Mass of object 1 m = Mass of object 2 r = Distance between the two objects When mass of one object is doubled Let Mass of Object 1 be doubled New Mass of Object 1 = 2M Thus, New Force = (𝐺 × 2𝑀 × 𝑚)/𝑟^2 = 2𝐺𝑀𝑚/𝑟^2 = 2 × Old Force ∴ If mass of one object is doubled, the force is also doubled Distance between object is doubled and tripled Distance is doubled So, New Distance = 2r New Force = 𝐺𝑀𝑚/(2𝑟)^2 = 𝐺𝑀𝑚/(4𝑟^2 ) = 1/4 × Old Force Distance is tripled So, New Distance = 3r New Force = 𝐺𝑀𝑚/(3𝑟)^2 = 𝐺𝑀𝑚/(9𝑟^2 ) = 1/9 × Old Force Therefore, When distance is doubled, Force becomes 𝟏/𝟒 times of Old Force When distance is tripled, force becomes 𝟏/𝟗 times of Old Force (iii) When mass of both objects is doubled New Mass of Object 1 = 2M New Mass of Object 2 = 2m Thus, New Force = (𝐺 × 2𝑀 × 2𝑚)/𝑟^2 = 4𝐺𝑀𝑚/𝑟^2 = 4 × Old Force ∴ If mass of both objects is doubled, the force becomes four times

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