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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?
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.
Updated on: 10-Oct-2022
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