Derivation of Gravitational Potential Energy

October 2, 2024

A derivation of gravitational potential energy using Newton's law of universal gravitation. The gravitational force is integrated to obtain the gravitational potential energy between two masses.

To derive the gravitational potential energy, we begin with Newton’s law of universal gravitation. This law states that the gravitational force between two masses m1m_1 and m2m_2, separated by a distance rr, is given by:

F=Gm1m2r2F = G \frac{m_1 m_2}{r^2}

where GG is the gravitational constant.

To find the potential energy, we integrate the gravitational force with respect to distance. Gravitational potential energy is defined as the work done by the gravitational force to move a mass from infinity to a distance rr from another mass:

U=−∫∞rF dr=−∫∞rGm1m2r2 dr=Gm1m2∫∞r1r2 dr=Gm1m2[−1r]∞r=Gm1m2(−1r−(−1∞))=−Gm1m2r\begin{align*} U &= -\int_\infty^r F \, dr \\ &= -\int_\infty^r G \frac{m_1 m_2}{r^2} \, dr \\ &= G m_1 m_2 \int_\infty^r \frac{1}{r^2} \, dr \\ &= G m_1 m_2 \left[-\frac{1}{r}\right]_\infty^r \\ &= G m_1 m_2 \left(-\frac{1}{r} - \left(-\frac{1}{\infty}\right)\right) \\ &= -G \frac{m_1 m_2}{r} \end{align*}

Thus, the gravitational potential energy between two masses m1m_1 and m2m_2, separated by a distance rr, is:

U=−Gm1m2rU = -G \frac{m_1 m_2}{r}

This equation represents the work required to move the two masses from their current position to infinity. The negative sign indicates that gravitational force is attractive, meaning work must be done against this force to separate the masses.

It is important to note that the zero point of potential energy is arbitrarily chosen at infinity. This convention simplifies calculations and aligns with the fact that gravitational force approaches zero as the distance between objects becomes infinitely large.