Gravitational Field Equations

Why This Matters

Gravitational field equations are the mathematical backbone of how we describe attraction between masses—from dropping a ball on Earth to satellites orbiting planets. In AP Physics 1, you're being tested on your ability to connect force, energy, and motion through these equations. The exam loves asking you to explain why an orbit is stable, how potential energy changes as objects move apart, and what happens to field strength as distance changes. These aren't isolated formulas; they're all connected through the inverse-square relationship and conservation laws.

Don't just memorize $F = G\frac{m_1 m_2}{r^2}$—understand that this single relationship generates everything else: field strength, orbital velocity, escape velocity, and potential energy. When you see a problem about satellites, free fall, or gravitational binding, ask yourself: Which equation captures the physics here, and how does it connect to the inverse-square law or energy conservation? That conceptual link is what earns you points on FRQs.

---

The Foundational Force: Newton's Law of Universal Gravitation

Every gravitational calculation starts here. The gravitational force between two masses is directly proportional to each mass and inversely proportional to the square of their separation.

Newton's Law of Universal Gravitation

• $\left|\vec{F}_g\right| = G\frac{m_1 m_2}{r^2}$—describes the attractive force between any two masses, always directed along the line connecting their centers
• Universal gravitational constant $G = 6.674 \times 10^{-11} \text{ N·m}^2/\text{kg}^2$—this tiny value explains why you need planetary-scale masses to feel significant gravitational effects
• Inverse-square dependence means doubling the distance reduces the force to one-quarter; this relationship underlies all gravitational field equations

---

Field Strength: Quantifying Gravitational Pull

Gravitational field strength tells you how much force a test mass would experience at any point in space. It's the force per unit mass, giving you a location-dependent property of the field itself.

Gravitational Field Strength

• $g = G\frac{M}{r^2}$—the field strength at distance $r$ from mass $M$, measured in N/kg (equivalent to m/s²)
• Test mass concept: place any mass $m$ at that location, and it experiences force $F = mg$; the field exists whether or not a test mass is present
• Near Earth's surface, $g \approx 9.8 \text{ m/s}^2$ (often approximated as 10 N/kg on the AP exam) because $r$ is essentially constant at Earth's radius

Surface Gravity Approximation

• $g \approx 9.8 \text{ m/s}^2$ at Earth's surface—this is the uniform field approximation valid when altitude changes are negligible compared to Earth's radius
• Weight equals $W = mg$, but apparent weight equals the normal force $N$; in free fall, $N = 0$, so apparent weight is zero (microgravity)
• Elevator problems test this: when accelerating upward, $N = m(g + a)$; when accelerating downward, $N = m(g - a)$

---

Energy in Gravitational Systems

Gravitational potential energy describes how bound two masses are. The negative sign isn't just convention—it reflects that you must add energy to separate masses that attract each other.

Gravitational Potential Energy

• $U_g = -G\frac{m_1 m_2}{r}$—the potential energy of a two-mass system, with zero reference at infinity
• Negative value means the system is bound; as $r$ increases, $U_g$ becomes less negative (increases toward zero), meaning energy was added to separate the masses
• Conservation of mechanical energy: $K + U_g = \text{constant}$ for objects moving only under gravity—this is how you solve orbit and escape problems

Near-Surface Potential Energy Change

• $\Delta U_g = mg\Delta y$—the simplified form when $g$ is constant and height changes are small compared to planetary radius
• Reference point is arbitrary for $\Delta U$; only changes in potential energy matter for energy conservation
• Connects to work: $W_{\text{gravity}} = -\Delta U_g$, so gravity does positive work when objects fall ($\Delta y < 0$)

---

Orbital Motion: Balancing Gravity and Inertia

Circular orbits occur when gravitational force provides exactly the centripetal acceleration needed. The satellite is constantly falling toward the planet but moving forward fast enough to keep missing it.

Circular Orbital Velocity

• $v_c = \sqrt{\frac{GM}{r}}$—the speed required for a stable circular orbit at radius $r$ from the center of mass $M$
• Derived from setting gravitational force equal to centripetal force: $G\frac{Mm}{r^2} = \frac{mv^2}{r}$
• Larger orbits require slower speeds—counterintuitive but essential; satellites in higher orbits move more slowly

Escape Velocity

• $v_{\text{escape}} = \sqrt{\frac{2GM}{r}}$—the minimum speed to escape a gravitational field entirely, reaching infinity with zero final speed
• Exactly $\sqrt{2}$ times orbital velocity at the same radius—this ratio appears frequently on exams
• Derived from energy conservation: set $K + U = 0$ (just enough energy to reach infinity with $v = 0$)

Orbital Period and Kepler's Third Law

• $T = 2\pi\sqrt{\frac{r^3}{GM}}$—the period of a circular orbit, showing that $T^2 \propto r^3$ (Kepler's Third Law)
• Farther orbits take longer—both because the path is longer and because orbital speed is slower
• Useful for comparing orbits: if you know one satellite's period and radius, you can find another's using the ratio $\frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3}$

---

Conservation Laws in Orbits

Orbital motion is governed by what stays constant. In circular orbits, everything is constant; in elliptical orbits, only total mechanical energy and angular momentum are conserved.

Mechanical Energy in Circular Orbits

• Total mechanical energy $E = K + U_g = -\frac{GMm}{2r}$—always negative for bound orbits, with magnitude equal to kinetic energy
• Kinetic energy $K = \frac{1}{2}mv^2 = \frac{GMm}{2r}$—exactly half the magnitude of potential energy in circular orbits
• Potential energy $U_g = -\frac{GMm}{r}$—twice the magnitude of kinetic energy; this 2:1 ratio is testable

Angular Momentum Conservation

• $L = mvr$ for circular orbits—constant because no external torques act on the satellite
• In elliptical orbits, $L = mvr\sin\theta$ remains constant, so satellites move faster at periapsis (closest approach) and slower at apoapsis (farthest point)
• Kepler's Second Law (equal areas in equal times) is a direct consequence of angular momentum conservation

---

Superposition and Special Cases

Real gravitational problems often involve multiple masses or symmetric distributions. The total field is always the vector sum of individual contributions.

Superposition of Gravitational Fields

• Vector addition: the net gravitational field at any point equals $\vec{g}_{\text{net}} = \vec{g}_1 + \vec{g}_2 + \cdots$
• Direction matters—fields from different masses may partially cancel or reinforce depending on geometry
• Lagrange points and other equilibrium positions exist where fields from multiple bodies sum to zero (or balance centrifugal effects)

Spherical Shell Theorem

• Inside a uniform spherical shell: $g = 0$—all gravitational effects cancel due to symmetry
• Outside a uniform spherical shell: $g = \frac{GM}{r^2}$—the shell acts as if all mass were concentrated at the center
• Planetary interiors: only mass inside your radius contributes to the field; mass in shells above you cancels out

---

Quick Reference Table

---

Self-Check Questions

1. Inverse-square check: If a satellite moves from radius $r$ to radius $2r$, by what factor does (a) gravitational force change? (b) orbital velocity change? (c) orbital period change?

2. Energy comparison: Why is gravitational potential energy negative, and what would it mean physically if $U_g$ were positive?

3. Orbit analysis: A satellite in circular orbit fires its engines briefly to speed up. Explain what happens to its orbit shape, total energy, and angular momentum.

4. Compare and contrast: Both escape velocity and orbital velocity depend on $\sqrt{GM/r}$. Explain why escape velocity is larger and derive the exact ratio between them.

5. FRQ-style application: Two planets have the same density but Planet B has twice the radius of Planet A. Compare their surface gravitational field strengths and escape velocities. (Hint: Express mass in terms of density and volume.)