Gravitational potential energy is the energy an object possesses due to its position within a gravitational field. It is defined as the work done by an external force to move an object from a reference point (usually infinity) to a specific point in the gravitational field without accelerating it. Derivation of the expression for gravitational potential energy: Step 1: Define the work done by an external force. The work done $dW$ by an external force to move a mass $m$ by an infinitesimal distance $dr$ against the gravitational force is given by: $$dW = F_{ext} \cdot dr$$ The gravitational force $F_g$ between Earth (mass $M$) and an object (mass $m$) at a distance $r$ from Earth's center is given by Newton's Law of Universal Gravitation: $$F_g = -\frac{GMm}{r^2}$$ The negative sign indicates that the force is attractive, directed towards the center of Earth. To move the object without acceleration, the external force $F_{ext}$ must be equal in magnitude and opposite in direction to the gravitational force: $$F_{ext} = -F_g = \frac{GMm}{r^2}$$ Step 2: Calculate the total work done. The gravitational potential energy $U(r)$ at a distance $r$ from the center of Earth is the total work done to bring the mass $m$ from infinity (where potential energy is conventionally zero) to the distance $r$. $$U(r) = \int_{\infty}^{r} dW = \int_{\infty}^{r} F_{ext} \cdot dr$$ Substitute the expression for $F_{ext}$: $$U(r) = \int_{\infty}^{r} \frac{GMm}{r^2} dr$$ Step 3: Perform the integration. $$U(r) = GMm \int_{\infty}^{r} r^{-2} dr$$ $$U(r) = GMm \left[ \frac{r^{-1}}{-1} \right]_{\infty}^{r}$$ $$U(r) = GMm \left[ -\frac{1}{r} \right]_{\infty}^{r}$$ $$U(r) = GMm \left( -\frac{1}{r} - \left(-\frac{1}{\infty}\right) \right)$$ Since $\frac{1}{\infty} = 0$: $$U(r) = GMm \left( -\frac{1}{r} - 0 \right)$$ $$U(r) = -\frac{GMm}{r}$$ The expression for gravitational potential energy at a distance $r$ from the center of Earth is: $$\boxed{U(r) = -\frac{GMm}{r}}$$