Unit 7: Gravitation

📚Study Guide: Gravitation

Unit 7: Gravitation

Gravitation in AP Physics C: Mechanics is analyzed with calculus and vector methods, going far beyond the algebraic treatment of Physics 1. Newton's Law of Universal Gravitation states that every mass attracts every other mass with a force F = −G(m₁m₂/r²) r̂, where the negative sign indicates attraction along the line connecting the masses. The gravitational field g = F/m_test = −GM/r² r̂ describes the force per unit mass at a point in space. For spherically symmetric mass distributions, the field outside the mass is identical to that of a point mass at the center—a result derived using Gauss's law for gravity or direct integration. Inside a uniform solid sphere, the gravitational field increases linearly with distance from the center: g(r) = GMr/R³. The gravitational potential energy for two masses is U = −GMm/r, chosen to be zero at infinite separation. This negative potential energy means gravitational forces are binding; work must be done to separate the masses to infinity. The total mechanical energy of an orbiting body is E = K + U = −GMm/(2a) for an elliptical orbit, where a is the semi-major axis. This negative total energy indicates a bound orbit. If E = 0, the orbit is parabolic (escape trajectory); if E > 0, it is hyperbolic. Circular orbits are a special case where the orbital velocity v = √(GM/r) and the period T = 2πr/v. Kepler's laws are derived from Newton's mechanics: the Second Law (equal areas in equal times) is a consequence of angular momentum conservation for a central force, and the Third Law (T² ∝ r³) follows from equating gravitational force to centripetal force. Escape velocity from a planet's surface is v_esc = √(2GM/R). On the AP Exam, gravitation questions require vector integration for non-uniform mass distributions, energy analysis of orbits, and derivations of Kepler's laws.

Key Concepts

• Newton's Law of Gravitation (Vector Form): F = −G(m₁m₂/r²) r̂. The force is central, attractive, and follows an inverse-square law.
• Gravitational Field: g = −GM/r² r̂. For a spherical shell, the external field is identical to a point mass; the internal field is zero.
• Gravitational Potential Energy: U = −GMm/r. Negative because gravity is binding. U → 0 as r → ∞.
• Total Orbital Energy: E = −GMm/(2a) for an ellipse. E < 0 means bound; E = 0 means parabolic escape; E > 0 means hyperbolic.
• Escape Velocity: v_esc = √(2GM/R). The minimum speed needed to escape a planet's gravity from its surface with no residual speed.
• Kepler's Laws Derived: Second law from conservation of angular momentum; third law from F_grav = F_centripetal for circular orbits.
• Inside a Uniform Sphere: g(r) = GMr/R³. Field strength increases linearly from zero at the center to GM/R² at the surface.

Vocabulary

• Central Force: A force that is always directed toward or away from a fixed point.
• Conservative Field: A field for which the work done in moving between two points is independent of path, allowing definition of potential energy.
• Gravitational Potential: Gravitational potential energy per unit mass: V_g = −GM/r.
• Escape Velocity: The minimum initial speed required for an object to escape from the gravitational influence of a massive body.
• Bound Orbit: An orbit with negative total energy, in which the object remains gravitationally attached.
• Eccentricity: A parameter describing how much an orbit deviates from being circular.
• Aphelion: The point in an orbit farthest from the central body.
• Perihelion: The point in an orbit closest to the central body.

Essential Formulas

• F = -G*m1*m2 / r² * r̂
• g = G*M / r²
• U = -G*M*m / r
• E = -G*M*m / (2*a) (ellipse)
• v_esc = sqrt(2*G*M / R)
• T² = (4π² / G*M) * a³ (Kepler's Third Law)
• v_orb = sqrt(G*M / r) (circular)

Common Mistakes

• Using U = mgh Instead of U = −GMm/r for Large Distances: mgh is an approximation valid near a planet's surface. For orbital mechanics, always use the exact form.
• Forgetting Orbital Energy Is Negative for Bound Orbits: Bound systems have E < 0. Zero energy means barely unbound; positive energy means definitely unbound.
• Confusing Circular Orbit Velocity with Escape Velocity: Escape velocity is √2 times larger than circular orbit velocity at the same radius. This is a factor of √2, not 2.
• Assuming All Orbits Are Circular: Most orbits are elliptical. Circular orbits are a special case where eccentricity e = 0.

AP Exam Strategies

• Use Energy Conservation for Orbit Problems: K + U is constant. At any point, ½mv² − GMm/r = E. This lets you find speed at any distance if you know E.
• Remember E < 0 Bound, E = 0 Parabolic, E > 0 Hyperbolic: This classification determines whether an object remains in orbit or escapes.
• Use Angular Momentum Conservation for Central Forces: For any central force, r × p is constant. This gives Kepler's Second Law and constrains orbital motion.
• Derive Kepler's Third for Circular Orbits First: Set GMm/r² = mv²/r, substitute v = 2πr/T, and solve. Extend conceptually to ellipses by replacing r with a.

Real-World Applications

• Satellite Launches: Launch vehicles must achieve precise orbital velocities. Too slow, and the satellite falls back; too fast, and it escapes into a higher orbit or interplanetary space.
• Gravitational Slingshot Maneuvers: Spacecraft steal momentum from planets by flying close by, gaining speed without using fuel, thanks to conservation of energy in the planet's reference frame.
• Black Hole Detection: The extreme gravitational fields near black holes cause orbital velocities approaching c, detected through X-ray observations of accretion disks.