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)

Answer 1

Answer: (C)

If both the mass and radius of the earth decrease by $1 \%$,
the value of acceleration due to gravity
$g'=\frac{g\left(0.99 M_{e}\right)}{\left(0.99 R_{e}\right)^{2}} \approx 1.01 \frac{G M_{e}}{R_{e}^{2}}=1.01 g$
So, $g$ would increase by
$(1.01-1) \times 100 \%=1 \%$
New escape velocity would be
$v_{e}'=\sqrt{\frac{2 \times 0.99 M- G_e}{0.99 R_{e}}}$
$=\sqrt{\frac{2 G M_{e}}{R_{e}}}=v_{e}$
So, it will remain unchanged.
Also gravitational potential energy
$=\frac{G m\left(0.99 m_{e}\right)}{0.99 R_{e}}$
$=-\frac{G M_{e} m}{R_{e}}$
Hence, it also will remain same.