Question

If the distance between centers of earth and moon is D and the mass of earth is 81 times the mass of moon, then at what distance from centre of earth the gravitational force will be zero

• A. $\frac{D}{2}$
• B. $\frac{2D}{3}$
• C. $\frac{4D}{3}$
• D. $\frac{9D}{10}$

Answer 1

The correct option is D

$\frac{9D}{10}$

If P is the point where net gravitational force is zero then

$F_{PA}=F_{PB}$

$\frac{G{m}_{1}m}{x^2}=\frac{G{m}_{2}m}{(d-x{)}^2}Bysolvingx=\frac{\sqrt{m_1}d}{\sqrt{m_1}+\sqrt{m_2}}$

For the given problem d = D, $m_1$ = earth, $m_2$ = moon and $m_1$ = 81 $m_2$ $\therefore m_2=\frac{m_1}{81}$

$Sox=\frac{\sqrt{m_1}D}{\sqrt{m_1}+\sqrt{m_2}}=x=\frac{\sqrt{m_1}D}{\sqrt{m_1}+\sqrt{\frac{m_1}{81}}}=\frac{D}{1+\frac{1}{9}}=\frac{9D}{10}$