If both the mass and radius of earth decrease by 1 %, the value of (I) acceleration due to gravity would increase by 1 % (II) acceleration due to gravity would decrease by 1 % (III) escape velocity from earth's surface would decrease by 1 % (IV) the gravitational potential energy of a body on earth's surface will remain unchanged

Question

If both the mass and radius of earth decrease by 1 %, the value of (I) acceleration due to gravity would increase by 1 % (II) acceleration due to gravity would decrease by 1 % (III) escape velocity from earth's surface would decrease by 1 % (IV) the gravitational potential energy of a body on earth's surface will remain unchanged (A) (I) and (III) (B) (I) and (IV) (C) (II) and (IV) (D) (III) and (IV)

Tardigrade Admin · Accepted Answer

If both the mass and radius of the earth decrease by

1%

,
the value of acceleration due to gravity

g^{'} = \frac{g ( 0.99 M _{e} )}{( 0.99 R _{e} ) ^{2}} \approx 1.01 \frac{G M _{e}}{R _{e}^{2}} = 1.01 g

So,

g

would increase by

(1.01 - 1) \times 100% = 1%

New escape velocity would be

v_{e}^{'} = \frac{2 \times 0.99 M - G _{e}}{0.99 R _{e}}

= \frac{2 G M _{e}}{R _{e}} = v_{e}

So, it will remain unchanged.
Also gravitational potential energy

= \frac{G m ( 0.99 m _{e} )}{0.99 R _{e}}

= - \frac{G M _{e} m}{R _{e}}

Hence, it also will remain same.