angular frequency

Meaning: Angular frequency is how fast the phase angle of a repeating motion grows. One full cycle covers 2π radians, so ω is 2π times the frequency f (cycles per second). It equals the angular speed of an object in uniform circular motion, and simple harmonic motion can be pictured as the projection of that circular motion. For a mass on a spring, ω depends on the spring constant and the mass. For a small-angle pendulum, it depends on g and the length, not the mass.

Equation: ω = 2πf = 2π/T, where ω is angular frequency (rad/s), f is frequency (Hz = s⁻¹) and T is period (s). Spring: ω = √(k/m), with k the spring constant (N/m) and m the mass (kg). Pendulum: ω = √(g/L), with g in m/s² and L in m.

On the MCAT: A passage gives x = A cos(ωt), a wave y = A sin(kx − ωt), or an AC voltage V = V₀ sin(ωt). You read off ω, then find f or T. Another common question is a scaling one. If the mass on a spring is quadrupled, ω halves and T doubles.

Common confusion: Students treat ω as if it were f. A value of ω = 10 rad/s is about 1.6 Hz, not 10 Hz, because you must divide by 2π. Also, ω describes how fast the phase advances. It is not the speed of the oscillating object, and a mass on a spring does not rotate.