What happens to the force between two objects if one of the masses is doubled and the distance between them is halved?

To do:

To find 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?

Solution:

We know the formula for gravitational force between two objects:

$\boxed{F=G\frac{mM}{d^2}}$

Where,

$F\rightarrow$gravitational force

$G\rightarrow$gravitational constant

$M\rightarrow$mass of object 1

$m\rightarrow$mass of the object 2

$d\rightarrow$distance between object 1 and object 2

$(i)$. When the mass of one object is doubled:

Then, the mass of object 1 becomes $2M$

Then, the gravitational force between object 1 and object 2

 $F'=G\frac{m(2M)}{d^2}$

Or $F'=2(G\frac{mM}{d^2})$

Or $F'=2F$

Therefore, if the mass of one object is doubled, then the force is also doubled.

$(ii)$ When the distance between the objects is doubled and tripled:

If the distance between the objects is doubled

Then distance becomes $2d$

Then gravitational force $F'=\frac{(GmM)}{(2d)^2}$

Or $F'=\frac{1}{4}(\frac{GmM}{d^2})$

Or $F'=\frac{F}{4}$

Therefore, gravitational force becomes one-fourth of its initial force when the distance between two objects is doubled.

Now, if it’s tripled

$F'=\frac{(GmM}{(3d)^2}$

$F'=\frac{1}{9}(G\frac{mM}{d^2})$

Or $F'=\frac{F}{9}$

Therefore, gravitational force becomes one-ninth of its initial force when the distance between two objects is tripled.

$(iii)$. When the masses of both objects are doubled:

If the masses of both the objects are doubled, then

$F'=G\frac{(2m)(2M)}{d^2}$

$F'=4F$ 

Therefore, gravitational force will become four times greater than its actual value.

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If the masses of two objects are doubled, and the distance between them is halved, then the gravitational force acting between them becomes

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