Gravitation: every key term you need (+ practice quiz)

More AP Physics C guides

• Kinematics
• Newton's Laws of Motion
• Work, Energy and Power
• Systems of Particles and Linear Momentum
• Rotational Motion

Newton's Law of Universal Gravitation

F = Gm₁m₂/r²; every pair of masses attracts along the line joining them; an inverse-square central force.

Gravitational Constant

G = 6.67 × 10⁻¹¹ N·m²/kg²; the universal constant setting the strength of gravity; small, so gravity is weak on lab scales.

Gravitational Field

g = F/m = GM/r²; the gravitational force per unit mass at a point; a vector pointing toward the source mass.

Surface Gravity

g = GM/R²; the field at a planet's surface; gives 9.8 m/s² for Earth using its mass and radius.

Inverse-Square Dependence

Gravitational force and field fall off as 1/r²; doubling the distance cuts the force to one quarter.

Gravitational Potential Energy

U = −GMm/r; the energy of a two-mass system taking U = 0 at infinite separation; always negative for a bound pair.

Why U is Negative

Work must be done against attraction to separate masses to infinity, so bound configurations have U < 0.

Relation Between F and U

F = −dU/dr; differentiating U = −GMm/r gives F = −GMm/r², the attractive gravitational force.

Circular Orbit Condition

Gravity supplies the centripetal force: GMm/r² = mv²/r, giving orbital speed v = √(GM/r).

Orbital Speed

v = √(GM/r); the speed for a circular orbit of radius r; smaller orbits require higher speeds.

Orbital Period

T = 2π√(r³/GM); the time for one circular orbit; follows from v = 2πr/T and v = √(GM/r).

Kepler's First Law

Planets move in ellipses with the Sun at one focus; circular orbits are the special case of zero eccentricity.

Kepler's Second Law

A line from the Sun to a planet sweeps equal areas in equal times; a consequence of angular momentum conservation.

Kepler's Third Law

T² ∝ r³ (or a³); the square of the period is proportional to the cube of the orbital radius (semi-major axis).

Total Energy of an Orbit

E = KE + U = −GMm/(2r) for a circular orbit; the total mechanical energy is negative for a bound orbit.

Kinetic Energy in Orbit

KE = ½mv² = GMm/(2r); exactly half the magnitude of the potential energy (virial relation).

Escape Velocity

v_esc = √(2GM/R); the minimum launch speed for total energy zero, allowing escape to infinity from the surface.

Escape vs Orbital Speed

v_esc = √2 × v_orbit; escape speed is √2 times the circular orbital speed at the same radius.

Gravity Inside a Shell

A uniform spherical shell exerts no net gravitational force on a mass inside it; the field there is zero.

Gravity Inside a Uniform Sphere

Only mass within radius r contributes, so g ∝ r inside a uniform planet, rising linearly from zero at the center.

Weightlessness in Orbit

Orbiting astronauts are in continuous free fall; both they and the station accelerate together, giving zero apparent weight.

Angular Momentum of a Particle

L = r × p; magnitude mvr sinφ; a particle moving in a straight line has constant L about any fixed point.

Angular Momentum of a Rigid Body

L = Iω about a fixed axis; for general motion L = L_cm(orbital) + I_cmω(spin).

Torque and Angular Momentum

τ_net = dL/dt; the rotational form of Newton's second law valid even when I changes.

Conservation of Angular Momentum

If net external torque is zero, L is constant; I₁ω₁ = I₂ω₂ when a skater pulls in her arms.

Rotational Kinetic Energy

K_rot = ½Iω² = L²/(2I); reducing I at constant L increases KE (work is done pulling mass inward).

Total KE of Rolling Body

K = ½Mv_cm² + ½I_cmω²; for rolling, K = ½Mv²(1 + I/MR²).

Rolling Energy Split

Fraction of KE that is rotational = (I/MR²)/(1 + I/MR²); 1/3 for a solid cylinder, 2/7 for a solid sphere.

Inelastic Rotational Collision

A putty blob sticking to a rotating rod: conserve L about the pivot, not p (pivot exerts an external force).

Angular Momentum in Orbits

Central gravitational force gives zero torque, so L = mvr sinφ is constant — Kepler's second law.

Orbital Energy

For a circular orbit K = −E = GMm/(2r); total E = −GMm/(2r); bound orbits have E < 0.

Elliptical Orbit Speeds

v_perihelion r_p = v_aphelion r_a from angular momentum conservation; fastest at closest approach.

Gyroscopic Precession

Torque from gravity changes the direction of L, not its magnitude; precession rate Ω = τ/L = Mgd/(Iω).

Rolling Up an Incline

A rolling object climbs higher than a sliding one at the same v_cm because rotational KE also converts to PE.

Sliding-to-Rolling Transition

A ball launched sliding on a rough surface: friction reduces v and increases ω until v = Rω; L about the contact point is conserved.