Hz to rad/s Converter
Welcome to the comprehensive Hertz to radians per second converter designed to help engineers, physicists, students, and professionals working with rotational motion, oscillations, and wave mechanics perform accurate conversions between Hz and rad/s with instant calculations and detailed mathematical formulas.
Angular Frequency Converter Tool
Hertz (Hz)
Radians per Second
1 Hz = 2π rad/s ≈ 6.283 rad/s
Hz to rad/s Conversion Formula
Angular Frequency Formula
\[ \omega = 2\pi f \]
Where ω is angular frequency (rad/s) and f is frequency (Hz)
Numerical Conversion
\[ \text{rad/s} = \text{Hz} \times 2\pi \]
Multiply Hertz by 2π (approximately 6.28318) to get rad/s
Understanding the Units
What is Hertz (Hz)?
Hertz (Hz) is the SI unit of frequency, representing the number of complete cycles per second. Named after Heinrich Hertz, one Hertz means one complete oscillation or revolution occurs every second. Frequency describes how often a periodic event repeats—from sound waves and electromagnetic radiation to mechanical vibrations and rotational motion. In rotational systems, Hz represents revolutions per second (rps).
What is Radians per Second (rad/s)?
Radians per second (rad/s) is the SI unit of angular frequency or angular velocity, representing the rate of change of angular position. Since one complete revolution equals 2π radians (approximately 6.28318 radians), angular frequency relates to frequency by the factor 2π. Angular frequency is fundamental in physics and engineering for describing rotational motion, oscillations, waves, and periodic phenomena using angular measurements rather than cycle counts.
The Relationship Between Hz and rad/s
The conversion between Hz and rad/s involves the constant 2π because one complete cycle (1 Hz) corresponds to rotating through 2π radians. If something rotates at 1 Hz (one revolution per second), it sweeps through 2π radians per second. This relationship is fundamental in converting between linear frequency (cycles/second) and angular frequency (radians/second). Understanding this conversion is essential in physics, engineering, signal processing, and anywhere rotational or oscillatory motion is analyzed.
Step-by-Step Conversion Process
Example 1: Convert 60 Hz to rad/s (Motor Speed)
Using the formula:
ω = 2πf
ω = 2π × 60
ω = 120π rad/s
ω ≈ 377 rad/s
Result: 60 Hz = 377 rad/s
Standard AC power frequency (60 Hz in North America)
Example 2: Convert 1000 Hz to rad/s (Audio Frequency)
Using the formula:
ω = 2π × 1000
ω = 2000π rad/s
ω ≈ 6283.2 rad/s
Result: 1000 Hz = 6283.2 rad/s
1 kHz audio tone angular frequency
Common Frequency Conversions
| Hertz (Hz) | Radians/Second (rad/s) | Application |
|---|---|---|
| 0.5 Hz | 3.14 rad/s (π rad/s) | Slow pendulum swing |
| 1 Hz | 6.28 rad/s (2π rad/s) | 1 revolution per second |
| 50 Hz | 314 rad/s (100π rad/s) | AC power (Europe, Asia) |
| 60 Hz | 377 rad/s (120π rad/s) | AC power (North America) |
| 440 Hz | 2765 rad/s | Musical note A4 |
| 1000 Hz | 6283 rad/s | 1 kHz reference tone |
| 3600 Hz | 22,619 rad/s | 3600 RPM motor |
rad/s to Hz Conversion (Reverse)
Reverse Conversion Formula
\[ f = \frac{\omega}{2\pi} \]
Divide rad/s by 2π to get Hertz
Why Convert Between Hz and rad/s?
Practical Reasons for Conversion
- Physics Equations: Many formulas use angular frequency (ω) rather than frequency (f)
- Rotational Mechanics: Angular velocity naturally uses radians per second
- Wave Analysis: Wave equations often express frequencies in rad/s
- Control Systems: Transfer functions and Bode plots use rad/s
- Signal Processing: Digital filters and Fourier analysis use angular frequency
- Oscillatory Motion: Pendulums, springs, and LC circuits use ω notation
When to Use Each Unit
- Use Hz: When counting cycles, revolutions, or oscillations per second
- Use rad/s: When dealing with angular motion, phase angles, or mathematical analysis
- Engineering: Control systems and frequency response typically use rad/s
- Audio/RF: Sound and radio frequencies typically use Hz
- Motors/Rotation: RPM (revolutions per minute) converts to Hz, then to rad/s for analysis
Practical Applications
Rotational Mechanics
In rotational motion, angular velocity is naturally expressed in radians per second. A spinning wheel, motor shaft, or rotating machinery has an angular velocity ω (rad/s) related to its rotational frequency f (Hz or revolutions per second) by ω = 2πf. For example, a motor running at 3600 RPM rotates at 60 Hz (3600/60) or 377 rad/s (60 × 2π). Understanding this conversion is essential for analyzing torque, angular momentum, and rotational kinetic energy.
Oscillatory Systems
Simple harmonic oscillators—pendulums, mass-spring systems, LC circuits—have natural frequencies expressed in both Hz and rad/s. The differential equation for simple harmonic motion uses angular frequency: d²x/dt² + ω²x = 0. For a pendulum with period T = 2 seconds (frequency f = 0.5 Hz), the angular frequency is ω = π rad/s. This angular frequency directly relates to the restoring force and energy of the oscillating system.
Electrical Engineering
AC power systems operate at 50 Hz or 60 Hz, corresponding to angular frequencies of 314 rad/s or 377 rad/s. When analyzing AC circuits using phasors and complex impedance, engineers use ω = 2πf in calculations. Inductance reactance is XL = ωL, and capacitance reactance is XC = 1/(ωC), where ω is in rad/s. Bode plots for frequency response analysis use logarithmic scales with rad/s on the horizontal axis.
Mathematical Relationships
Key Formulas Involving Angular Frequency
Period and Frequency:
\[ T = \frac{1}{f} = \frac{2\pi}{\omega} \]
Angular Velocity:
\[ \omega = \frac{d\theta}{dt} = 2\pi f \]
Simple Harmonic Motion:
\[ x(t) = A\cos(\omega t + \phi) \]
Wave Equation:
\[ y(x,t) = A\sin(kx - \omega t) \]
Common Questions
Why multiply by 2π instead of just π?
One complete cycle corresponds to rotating through 360 degrees, which equals 2π radians (not π radians). Since frequency in Hz counts complete cycles per second, and each cycle is 2π radians, the angular frequency is ω = 2πf. The factor of 2 comes from the full circle (360°), while π converts degrees to radians (180° = π radians, 360° = 2π radians). This is not arbitrary—it's the mathematical relationship between circular motion and periodic oscillations.
Is rad/s the same as angular velocity?
Yes and no. Technically, angular frequency (ω) and angular velocity have the same units (rad/s) and are often used interchangeably, but they can have subtle differences in meaning. Angular velocity typically refers to the rate of rotation of a physical object (like a spinning wheel), while angular frequency refers to the rate of oscillation or phase change in a wave or oscillating system. In practice, both use rad/s and follow the same conversion: ω = 2πf.
How do I convert RPM to rad/s?
First convert RPM (revolutions per minute) to Hz (revolutions per second) by dividing by 60, then multiply by 2π to get rad/s. Formula: ω (rad/s) = RPM × 2π/60 = RPM × π/30. For example, 3600 RPM = 3600/60 = 60 Hz = 60 × 2π ≈ 377 rad/s. Alternatively, 3600 × π/30 = 120π ≈ 377 rad/s. This conversion is common in motor and machinery analysis.
Why does physics prefer rad/s over Hz?
Radians are the natural mathematical unit for angles because they simplify calculus and trigonometry. In the equation ω = dθ/dt (angular velocity equals rate of change of angle), using radians makes the math cleaner—derivatives and integrals of trigonometric functions work elegantly with radians. Additionally, many physics formulas (centripetal acceleration: a = ω²r, rotational kinetic energy: KE = ½Iω²) naturally use angular frequency in rad/s. While Hz is more intuitive for counting cycles, rad/s is more fundamental mathematically.
What's the exact value of 2π?
2π is an irrational number approximately equal to 6.283185307179586... It never terminates or repeats. For most practical calculations, using 6.283 or 6.28 provides sufficient accuracy. In exact mathematical work, it's best to leave the answer in terms of π (e.g., 120π rad/s) rather than converting to decimals. Many calculators and programming languages have a built-in π constant for precise calculations.
Quick Reference Guide
Conversion Shortcuts
- Hz to rad/s: Multiply by 2π (≈ 6.283)
- rad/s to Hz: Divide by 2π (≈ 6.283)
- Quick estimate: Hz × 6.28 ≈ rad/s (or × 6.3 for mental math)
- 1 Hz: Exactly 2π rad/s
- Common multiple: 60 Hz = 120π ≈ 377 rad/s (AC power)
Common Conversion Mistakes to Avoid
- Using π instead of 2π: Must multiply by 2π, not just π
- Forgetting units: Always specify rad/s vs Hz vs RPM
- Rounding too early: Use full precision of π in calculations
- Confusing f and ω: Remember ω = 2πf, they're not the same
- Degree/radian confusion: Always use radians, never degrees in rad/s
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Our converter combines mathematical precision with instant calculations and comprehensive explanations to help students, engineers, physicists, and professionals understand and apply frequency and angular frequency conversions in rotational mechanics, oscillatory motion, signal processing, and wave analysis.
Note: This Hz to rad/s converter uses the fundamental relationship ω = 2πf, where ω is angular frequency in radians per second and f is frequency in Hertz. The conversion factor 2π (approximately 6.283185) represents one complete revolution in radians. To convert Hz to rad/s, multiply by 2π. To convert rad/s to Hz, divide by 2π. This conversion is essential in physics, engineering, rotational mechanics, oscillatory motion, wave analysis, and control systems. Angular frequency (rad/s) is preferred in mathematical analysis and physics equations, while frequency (Hz) is more intuitive for counting cycles or rotations.
