Find expressions for the potential energy (V) and the kinetic energy (K) of the moon in the gravitational field of the earth. Hence find the total energy of the moon and state the significance of its negative sign.

Explanation

Let mass of the earth =M , mass of the moon =m, distance between centres of earth and moon =r, orbital speed of moon `v=sqrt((GM)/r)`.
So, kinetic energy of moon `=K=1/2mv^2=(GMm)/(2r)`
Again, gravitational potential energy of moon
`=V=-(GMm)/r`
So, total energy `E=K+V=(GMm)/(2r)-(GMm)/r=-(GMm)/(2r)`
Significance of negative sign is that , due to gravitational attraction between moon and the the earth, a closed system has formed. if `(GMm)/(2r)` amount of energy can be given to moon from outside , total energy of moon will be zero, i.e., it will be free from earth.s attraction force.