The Angular Frequency Calculator converts between frequency (f), angular frequency (ω), and period (T) using the fundamental relationships ω = 2πf and T = 1/f. Angular frequency measures how fast an oscillation or wave rotates in radians per second. This calculator is essential for physics problems involving simple harmonic motion, AC circuits, waves, and oscillatory systems. Enter any one value to instantly calculate the other two with detailed step-by-step solutions.

1. Choose input: Enter frequency (f), angular frequency (ω), or period (T).
2. Select units: Choose from Hz/kHz/MHz/GHz/rpm, rad/s/deg/s/rpm, or s/ms/μs/min/h.
3. Calculate: Click the corresponding button to compute other values.
4. Review steps: See unit conversions and formula applications.
5. View results: Get all three values simultaneously with precise calculations.

• ω (omega) = angular frequency in radians per second (rad/s)
• f = frequency in Hertz (Hz) or cycles per second
• T = period in seconds (time for one complete cycle)
• 2π ≈ 6.2832 radians in one complete revolution

Angular frequency represents the rate of change of phase angle in oscillatory motion. In simple harmonic motion, a mass on a spring oscillates with angular frequency ω = √(k/m). In AC electrical circuits, voltage and current oscillate at angular frequency ω = 2πf where f is typically 50 Hz or 60 Hz. Wave phenomena including sound, light, and electromagnetic waves all have characteristic angular frequencies. Understanding the relationship between ω, f, and T is fundamental to wave mechanics, quantum mechanics, signal processing, and electrical engineering.

• AC Power (60 Hz): f = 60 Hz, ω ≈ 377 rad/s, T ≈ 16.67 ms
• AC Power (50 Hz): f = 50 Hz, ω ≈ 314 rad/s, T = 20 ms
• Pendulum (1 Hz): f = 1 Hz, ω ≈ 6.28 rad/s, T = 1 s
• Radio Wave (100 MHz): f = 100 MHz, ω ≈ 6.28×10⁸ rad/s, T = 10 ns
• Rotating machinery (3600 rpm): f = 60 Hz, ω ≈ 377 rad/s

In simple harmonic motion (SHM), the angular frequency ω determines how rapidly the system oscillates. The displacement can be written as x(t) = A cos(ωt + φ), where A is amplitude and φ is phase. The velocity is v(t) = -Aω sin(ωt + φ) and acceleration is a(t) = -Aω² cos(ωt + φ). Higher angular frequency means faster oscillation and greater maximum velocity and acceleration for the same amplitude. This applies to pendulums, springs, vibrating strings, and molecular vibrations.

Angular frequency is related to several other physics concepts. The Angular Velocity Calculator deals with rotational motion of rigid bodies, while angular frequency describes oscillations. The Frequency Calculator helps with basic frequency conversions. For wave applications, check the Wavelength Calculator to relate frequency to wavelength. In AC circuits, angular frequency connects to the Capacitive Reactance Calculator via Xc = 1/(ωC).