Work and Energy
Beginner
gravitational-potential-energy
Source: High school physics (Chinese)
Problem Sets:
work - dynamics 1106
Problem
A satellite of mass $m$ moves in a circular orbit of radius $r$ around a central body of mass $M$. The gravitational potential energy of the system is defined to be zero when the satellite is infinitely far away.
Find the mechanical energy of the system when the satellite is in this orbit.
$E = -\frac{GMm}{2r}$
The gravitational force provides the centripetal force for the satellite's circular motion. Let $G$ be the gravitational constant.
$$ \frac{GMm}{r^2} = \frac{mv^2}{r} $$From this, we find the kinetic energy $E_k$:
$$ E_k = \frac{1}{2}mv^2 = \frac{GMm}{2r} $$The gravitational potential energy $E_p$ at radius $r$ is:
$$ E_p = -\frac{GMm}{r} $$The total mechanical energy $E$ is the sum of the kinetic and potential energies.
$$ E = E_k + E_p = \frac{GMm}{2r} - \frac{GMm}{r} $$