Hertz to Radians per Second

How to Convert Hz to rad/s?

One hertz equals 2π radians per second (approximately 6.2832 rad/s). To convert Hz to rad/s, multiply the Hz value by 2π. This conversion is ubiquitous in physics, electrical engineering, and signal processing. Frequency in Hz tells you how many complete cycles per second; angular frequency in rad/s tells you the rate of change in radians per second. Every wave equation, oscillation formula, and AC circuit analysis in physics uses angular frequency (ω = 2πf). If you encounter frequency in Hz and need to use it in a differential equation, Fourier transform, or LC circuit formula, multiplying by 2π is the essential first step. It also helps connect physical intuition to math-heavy equations. A familiar 60 Hz mains signal becomes 377 rad/s in circuit formulas, and a 1 kHz tone becomes 6,283 rad/s when written as angular frequency inside sinusoidal or complex-exponential analysis. Using the right form saves repeated 2π factors later and makes phase-based equations much cleaner to read. That is why textbooks switch to ω almost immediately. That consistency matters.

How to Convert Hertz to Radian per Second

1. Start with your frequency in hertz (Hz).
2. Multiply the Hz value by 2π (≈ 6.2832) to get radians per second.
3. The result is your angular frequency in rad/s.
4. The formula is: ω = 2πf, where f is frequency in Hz.
5. For example, 60 Hz x 2π = 376.99 rad/s, 1 Hz x 2π = 6.283 rad/s.

Real-World Examples

History of Hertz and Radian per Second

Angular frequency (ω) became central to physics with the development of wave mechanics and AC circuit theory in the 19th century. While ordinary frequency (f in Hz) counts complete cycles, angular frequency tracks the phase angle in radians, making it the natural variable for sinusoidal functions: sin(ωt) rather than sin(2πft). This simplification was recognized by physicists like Joseph Fourier (Fourier transforms use ω = 2πf) and adopted universally in physics and engineering. The relationship ω = 2πf is one of the most frequently used equations in all of science.

Common Mistakes to Avoid

• Forgetting to multiply by 2π. Using Hz directly in equations that expect rad/s gives results off by a factor of 6.28. In AC circuits, impedance Z = ωL is wrong if you substitute f for ω — you must use Z = 2πfL.
• Confusing ω (angular frequency) with f (ordinary frequency). They differ by 2π. A 60 Hz signal has f = 60 Hz but ω = 377 rad/s. Using the wrong one in a formula gives answers off by 6.28x.
• Dividing by 2π instead of multiplying. This converts rad/s to Hz (the opposite direction). If your result is smaller than the Hz value, you went the wrong way — rad/s should always be about 6.28x larger than Hz.
• Feeding rad/s values into software or datasheets that expect Hz. A value of 377 rad/s corresponds to 60 Hz, so entering 377 where the tool expects Hz creates a 6.28x error immediately.

Frequently Asked Questions