Oscillations and Gravitation Cheat Sheet

The core ideas of Oscillations and Gravitation distilled into a single, scannable reference — perfect for review or quick lookup.

PiqCue — piqcue.com/oscillations-gravitation/cheatsheet

Quick Reference

x(t) = A cos(omega t + phi), where omega = sqrt(k/m).

d^2x/dt^2 = -omega^2 x.

E = (1/2)kA^2 = (1/2)mv^2 + (1/2)kx^2.

T = 2 pi sqrt(L/g) for small angles.

T = 2 pi sqrt(I/(mgh)).

F = GMm/r^2, attractive along line of centers.

U = -GMm/r. Zero at infinity.

v = sqrt(GM/r), T^2 = (4 pi^2/GM)r^3.

Simple harmonic motion (SHM):Periodic motion where restoring force is proportional to displacement: F = -kx.

Amplitude (A):Maximum displacement from equilibrium in SHM.

Angular frequency (omega):omega = 2 pi f = 2 pi/T. For spring-mass: omega = sqrt(k/m).

Phase constant (phi):Initial phase angle in x(t) = A cos(omega t + phi). Set by initial conditions.

Period (T):Time for one complete oscillation cycle. T = 2 pi/omega.

Damping coefficient (b):Parameter in damped SHM. Larger b means faster amplitude decay.

Resonance:Maximum amplitude response when driving frequency equals natural frequency.

Gravitational constant (G):G = 6.674 x 10^-11 N m^2/kg^2. Universal constant in Newton law of gravitation.

Gravitational field (g):g = GM/r^2. Force per unit mass at distance r from mass M.

Escape velocity:Minimum speed to escape a gravitational field: v = sqrt(2GM/R).

Kepler laws:Three laws of planetary motion: elliptical orbits, equal areas, T^2 proportional to r^3.

Geosynchronous orbit:Circular orbit with T = 24 hours. r = 42,164 km from Earth center.

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