Two solid spherical planets of equal radii R having masses 4M and 9M their centre are separated by a distance 6R. A projectile of mass m is sent from the planet of mass 4 M towards the heavier planet. What is the distance r of the point from the lighter planet where the gravitational force on the projectile is zero ?
The point where the gravitational force on the projectile is zero occurs when the gravitational forces from both planets are equal.
\[
\frac{G \cdot 4M \cdot m}{r^2} = \frac{G \cdot 9M \cdot m}{(6R - r)^2}
\]
Simplifying,
\[
\frac{4}{r^2} = \frac{9}{(6R - r)^2}
\]
Taking the square root:
\[
\frac{2}{r} = \frac{3}{6R - r}
\]
Cross-multiplying:
\[
2(6R - r) = 3r
\]
Solving:
\[
12R - 2r = 3r
\]
\[
5r = 12R
\]
\[
r = \frac{12R}{5}
\]
So, the distance from the lighter planet is \( \frac{12R}{5} \).