Gravitational Potential Energy

The gravitational potential energy of a pair of point masses separated by distance $r$ is $U=-\frac{G{m}_1{m}_2}{r}$, taking the reference of zero energy at infinite separation. The negative sign indicates the system is bound, meaning energy must be supplied to pull the masses apart. Substituting $m_1=m_2=10$ kg and $r=1$ m gives $U=-\frac{6.67\times 10^{-11}\times 10\times 10}{1}=-6.67\times 10^{-9}$ J. The option $-3.34\times 10^{-9}$ J wrongly inserts a factor of one-half that belongs to kinetic, not potential, energy. The option $-6.67\times 10^{-11}$ J forgets to multiply by the mass product. The positive option $+6.67\times 10^{-9}$ J carries the wrong sign, contradicting the attractive nature of gravity. This is the NCERT pairwise potential-energy formula, where the energy represents the work done by gravity in assembling the configuration from infinite separation. Unlike the near-surface approximation $U=mgh$, which assumes a constant field, this exact expression is valid for any separation and correctly approaches zero as the masses move infinitely far apart. For systems of more than two masses, the total potential energy is obtained by summing such terms over every distinct pair. A plausibility check confirms the energy is negative and minuscule, as expected for small everyday masses bound very weakly by gravity.