## 15.5 GRAVITATIONAL POTENTIAL We all learn that lifting an object gives it potential energy, which we calculate with the simple formula $mgh$. But what happens when you go so high up that gravity itself gets weaker? For rockets, planets, and stars, we need a better way to measure this stored energy. This section introduces **Gravitational Potential Energy**, a more powerful concept that works anywhere in the universe. We'll explore why, in space, this energy is always a negative number and how the related idea of **Gravitational Potential** helps us describe the strength of a gravitational field itself. ### Limitations of $\Delta \mathrm{P} \cdot \mathrm{E}=\mathrm{mgh}$ * The formula $\Delta \mathrm{P} \cdot \mathrm{E}=\mathrm{mgh}$ is only accurate for potential energy near the Earth's surface. * It becomes invalid as we move away from the surface because gravitational force weakens, and thus $g$ is not constant. ### Gravitational Potential Energy (U) * **Zero-Reference Point:** To calculate gravitational potential energy, a zero-reference point is established. Conventionally, this is set at **infinity** ($r = \infty$). * **Consequence of Zero-Reference:** Choosing infinity as the zero-reference point means potential energy is **negative** for any finite distance $r$ from the gravitating body. This is because potential energy decreases as objects move closer to the zero-reference point (if starting from infinity) and increases as they move apart. * **Derivation Principle:** * It is calculated based on , accounting for variable gravitational force. * The change in potential energy ($\Delta P \cdot E$) is equal to the net work done ($W_{net}$) in moving a mass. * Work done against a variable force over a distance is calculated by summing work over infinitesimal segments ($\Delta r$). * For a mass $m$ being moved from an initial distance $R_E$ to a final distance $r_n$ from the center of Earth (mass $M_E$), the total work done is: $$ W_{\text {net }}=-\mathrm{GM}_{\mathrm{E}} \mathrm{~m}\left(\frac{1}{\mathrm{R}_{\mathrm{E}}}-\frac{1}{\mathrm{r}_{\mathrm{n}}}\right) $$ * **Absolute Gravitational Potential Energy:** * Defined as the work done in bringing a mass $m$ from **infinity** ($r_n = \infty$) to a point at a distance $r$ from the center of the Earth. * The general expression for gravitational potential energy $U$ at a distance $r$ is: $$ U=-\frac{G M_{E} m}{r} \quad (15.9) $$ * **Definition:** The amount of work done in moving a body from Earth's surface (or any point $r$) to a point at infinite distance where the value of $g$ is negligible. * **Variation of U:** $U$ becomes less negative (increases) as $r$ increases, approaching zero at infinity. ### Gravitational Potential (V or $\phi$) * **Definition:** The work done per unit mass in bringing a small test mass from infinity to a specific point. * It is the gravitational potential energy $U$ per unit mass $m$: $$ V=\frac{U}{m} \quad (15.10) $$ * **Expression for V:** By substituting the formula for U: $$ V=-\frac{G M_{E}}{r} \quad (15.11) $$ * **Key Characteristics:** * Gravitational potential $V$ does **not** depend on the mass of the object being moved ($m$), only on the mass of the gravitating body ($M_E$) and the distance $r$. * Its value is always negative and approaches zero at infinity. * **Calculation at Earth's Surface:** Using constants ($G = 6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} \mathrm{kg}^{-2}$, $M_E = 6 \times 10^{24} \mathrm{~kg}$, $R_E = 6.4 \times 10^{6} \mathrm{~m}$): $$ V = -\frac{6.67 \times 10^{-11} \times 6 \times 10^{24}}{6.4 \times 10^{6}} = -62.3 \times 10^{6} \mathrm{~J} \mathrm{~kg}^{-1} = -62.3 \mathrm{~MJ} \mathrm{~kg}^{-1} $$ * **Universality:** The derived equations for U and V are valid for any gravitating object, not just Earth. ### Relationship between g and V The gravitational field strength $g$ is the negative gradient of the gravitational potential $V$. $$ g = -\frac{\Delta V}{\Delta r} $$ This indicates that the field strength points in the direction of decreasing potential. | Concept | Formula | Description | | :------------------------------- | :-------------------------------------------------------------------- | :------------------------------------------------------------------- | | **Gravitational Potential Energy** | $U = -\frac{G M m}{r}$ | Work done to bring mass $m$ from infinity to $r$. | | **Gravitational Potential** | $V = -\frac{G M}{r}$ or $V = \frac{U}{m}$ | Work done per unit mass to bring a test mass from infinity to $r$. | | **Constants** | $G = 6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} \mathrm{kg}^{-2}$ | Universal Gravitational Constant | | | $M_E = 6 \times 10^{24} \mathrm{~kg}$ | Mass of Earth | | | $R_E = 6.4 \times 10^{6} \mathrm{~m}$ | Radius of Earth |