D.1.8 (HL)—Gravitational potential

Define potential at a point

Gravitational potential $V_g$ at a point is the work done per unit mass in bringing a small test mass from infinity to that point. Set $V_g=0$ at infinity. Its SI unit is $\mathrm{J\,kg^{-1}}$, and it is a scalar quantity.

Worked example — orbital potential

At $r=7.9\times10^6\,\mathrm{m}$ from Earth’s centre, with $M=6.0\times10^{24}\,\mathrm{kg}$, $V_g=-(6.67\times10^{-11})(6.0\times10^{24})/(7.9\times10^6)=-5.1\times10^7\,\mathrm{J\,kg^{-1}}$. The negative value is potential energy per kilogram relative to zero at infinity.

Interpret the sign

At finite distance the potential is negative because the field does work as an attracting mass is brought inward from infinity. Moving outward increases $V_g$ toward zero; moving inward makes it more negative. The potential difference between two points is what determines work: $W=m\Delta V_g$.

Common trap

Do not confuse potential $V_g$ in J kg⁻¹ with potential energy $E_p$ in joules, and do not use $r^2$: potential follows $1/r$, whereas field strength follows $1/r^2$.