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A planet of volume ‘ $V$ ’ and mass ‘ $m$ ’ has gravitational acceleration ‘ $g$ ’ on its surface. If it expands to $8$ times its original volume, what will be the acceleration due to gravity?

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4 g

2 g

g / 4

g / 8

answer is C.

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Detailed Solution

A planet of volume ‘

V

’ and mass ‘

m

’ has gravitational acceleration ‘

g

’ on its surface. If it expands to

8

times its original volume, the acceleration due to gravity will become

g / 4

.
Volume

(V) = \frac{4}{3} π r^{3},

where

r

is the radius of the planet
If a planet expands to

8

times its original volume, then-

\frac{4}{3} π r^{3} \times 8 = \frac{4}{3} π R^{3}

(

R

is the new radius of the planet) After solving above equation, we get-

R = 2 r

. We know that,

g = \frac{GM}{r^{2}}

and

g^{'} = \frac{GM}{R^{2}}

(

g

is the acceleration due to gravity when the radius is ‘

r

’ and

g^{'}

is the acceleration due to gravity when the radius is ‘

R

’. So,

\frac{g^{'}}{g} = \frac{r^{2}}{R^{2}}

\Rightarrow \frac{g^{'}}{g} = \frac{r^{2}}{{(2 r)}^{2}}

\Rightarrow \frac{g^{'}}{g} = \frac{1}{4}

Therefore,

g^{'} = \frac{g}{4}

.
So the gravity becomes one-fourth of the original.

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