Discuss the motion of moon around the earth and derive its equation of motion.

Solution

Earth and the moon form a two-body system.
Let M and m be the masses of the earth and moon respectively.
Let $straight r with rightwards harpoon with barb upwards on top$ be the position vector of moon with respect to earth.
Here, the force on the moon is gravitational force of earth and given by,
$rightwards arrow for F of equals negative fraction numerator G M m over denominator r squared end fraction r with hat on top$
The equation of motion of moon is,
$space F with rightwards harpoon with barb upwards on top space equals space straight mu space fraction numerator straight d squared straight r over denominator dt squared end fraction$
where, μ is the reduced mass of the moon.
Reduced mass of the earth is given by,
$straight mu equals fraction numerator Mm over denominator straight M plus straight m end fraction$
Substituting the values of $straight mu space and space F with rightwards harpoon with barb upwards on top$ in the equation of motion, we get
$negative fraction numerator G M m over denominator straight r squared end fraction space stack straight r space with hat on top space equals space fraction numerator M m over denominator straight M plus straight m end fraction fraction numerator d squared r over denominator d t squared end fraction rightwards double arrow space space space space fraction numerator d squared r over denominator d t squared end fraction space space equals space minus space fraction numerator straight G left parenthesis straight M plus straight m right parenthesis over denominator straight r squared end fraction space straight r with hat on top$
This is the required equation of motion.