Gravitational Interaction Equations

PCE PHYSICS

Newton’s Law of Gravitation

F = G(m1m2)/r^2 (force goes up w/ mass, down w/ distance squared)

Gravitational Field (g) def

g = F/m (force per unit mass; "strength" of gravity)

Field of a Mass (M)

g = GM/r^2 (field at distance r from mass M)

Gravitational Potential Energy (Ep)

Ep = -GMm/r (always negative b/c gravity attracts)

Gravitational Potential (V)

V = -GM/r (potential per unit mass; doesnt depend on test mass)

Work & Potential Energy

(W = -Delta Ep) Work is energy "spent" by gravity; as an object falls, Ep decreases (-) so gravity does positive (+) work.

Work & Potential Difference

W = m(Vb - Va) (work to move mass m between 2 points)

g at Earth surface (g0)

g0 = GMe/Re^2 (approx 9.8 m/s^2)

g vs Height (h)

g = g0 / (1 + h/Re)^2 (gravity weakens as u go higher)

Linear approx of g (small h)

g approx g0(1 - 2h/Re) (used when h << Re)

Kepler’s 2nd Law

dA/dt = constant (equal areas in equal time)

Angular Momentum (L)

L = r x mv (constant in orbit; explains Keplers 2nd)

Orbital Velocity (v)

v = sqrt(GM/r) (found by gravity = centripetal force)

Centripetal Accel (ac)

ac = v^2/r (inward accel for circular motion)

Orbital Period (T)

T = 2pi * sqrt(r^3/GM) (time for 1 full orbit)

Kepler’s 3rd Law

T^2 prop. r^3 (period squared proportional to radius cubed) K = T²/R³ where k is a constant for each orbiting body.

Kinetic Energy in orbit (Ek)

Ek = 1/2 mv^2 (always + energy of motion)

Total Energy in orbit (E)

E = -GMm/2r (Ek + Ep; negative = bound orbit)

What negative energy means

E < 0 means object is stuck in orbit & cant escape

Escape Velocity (ve)

ve = sqrt(2GM/R) (min speed to reach E = 0 & leave)

Earth escape velocity #

v approx 11.2 km/s

Condition for escape

E total >= 0 (enough energy to reach infinity)

Orbit speed vs Height

v = sqrt(GMe / (Re + h)) (higher orbit = slower speed)

Perigee vs Apogee

Perigee = closest/fastest. Apogee = farthest/slowest.

Orbit Eccentricity (e)

e = sqrt(1 - b^2/a^2) (0 = circle, closer to 1 = oval)

Deterministic Chaos

laws are set/predictable but outcome is sensitive to start conditions