10. Gravitation — formulas and resources

viz

F=G\frac{m_1m_2}{r^2}

Vector form

\vec F_{12} = -G\frac{m_1m_2}{r^2}\hat r

The gravitational force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance.

viz

g=\frac{GM}{r^2}

Gravitational field of a point mass, of magnitude GM/r².

viz

U(r)=-G\frac{Mm}{r}

Universal gravitational potential energy, inversely proportional to the distance.

\Phi=-\frac{GM}{r}

and

U=m\Phi

The gravitational potential is minus GM/r; the potential energy is mΦ.

viz

v_{\text{orb}}=\sqrt{\frac{GM}{r}}

Speed of a circular orbit around a central mass.

viz

T=2\pi\sqrt{\frac{r^3}{GM}}

Equivalently

T^2=\frac{4\pi^2}{GM}r^3

Period of a circular orbit; the square of the period is proportional to the cube of the radius.

viz

v_{\text{esc}}=\sqrt{\frac{2GM}{R}}

Escape speed from a distance R of a mass M.

viz

E=-\frac{GMm}{2r}

Total mechanical energy of a circular orbit, equal to minus GMm/(2r).