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  4. Two solid spherical planets of equal radii R have masses 4 M and 9 M. Their centres are separated by a distance 6 R. A projectile of mass m is sent from the planet of mass 4 M towards the heavier planet. If the distance r of the point from the lighter planet where the gravitational force on the projectile is zero is k R, then calculate k.

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Q. Two solid spherical planets of equal radii $R$ have masses $4\, M$ and $9\, M$. Their centres are separated by a distance $6\,R$. A projectile of mass $m$ is sent from the planet of mass $4 \,M$ towards the heavier planet. If the distance $r$ of the point from the lighter planet where the gravitational force on the projectile is zero is $k R$, then calculate $k$.

Gravitation

Solution:

The gravitational force on projectile of mass $m$ at a distance $r$ from a planet of mass $4 M$ is zero.
$\therefore \frac{G(4 M) m}{r^{2}}=\frac{G(9 M) m}{(6 R-r)^{2}}$
or $4(6 R-r)^{2}=9 r^{2}$
or $2(6 R-r)=3 r$
or $r=2.4 \,R$