rad/s to Hz Converter
Welcome to the comprehensive radians per second to Hertz converter designed to help engineers, physicists, students, and professionals working with rotational motion, oscillations, and wave mechanics perform accurate conversions between rad/s and Hz with instant calculations and detailed mathematical formulas.
Angular Frequency Converter Tool
Radians per Second
Hertz (Hz)
6.283 rad/s = 2π rad/s = 1 Hz (1 revolution per second)
rad/s to Hz Conversion Formula
Angular Frequency to Frequency Formula
\[ f = \frac{\omega}{2\pi} \]
Where f is frequency (Hz) and ω is angular frequency (rad/s)
Numerical Conversion
\[ \text{Hz} = \frac{\text{rad/s}}{2\pi} \]
Divide rad/s by 2π (approximately 6.28318) to get Hz
Understanding the Units
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 ordinary 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. The unit rad/s directly expresses how many radians of rotation occur per second.
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) or cycles per second (cps). Hz counts complete cycles, while rad/s measures angular displacement per second.
The Relationship Between rad/s and Hz
The conversion between rad/s and Hz 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 (approximately 6.283 rad/s). This relationship is fundamental in converting between angular frequency (radians/second) and linear frequency (cycles/second). Understanding this conversion is essential in physics, engineering, signal processing, and analyzing rotational or oscillatory motion.
Step-by-Step Conversion Process
Example 1: Convert 377 rad/s to Hz (AC Power)
Using the formula:
f = ω/(2π)
f = 377/(2π)
f = 377/6.28318
f ≈ 60
Result: 377 rad/s = 60 Hz
Standard AC power frequency (60 Hz in North America)
Example 2: Convert 6283.2 rad/s to Hz (Audio Tone)
Using the formula:
f = 6283.2/(2π)
f = 6283.2/6.28318
f ≈ 1000
Result: 6283.2 rad/s = 1000 Hz
1 kHz audio reference tone
Common Angular Frequency Conversions
| Radians/Second (rad/s) | Hertz (Hz) | Application |
|---|---|---|
| 3.14 rad/s (π rad/s) | 0.5 Hz | Slow pendulum swing |
| 6.28 rad/s (2π rad/s) | 1 Hz | 1 revolution per second |
| 314 rad/s (100π rad/s) | 50 Hz | AC power (Europe, Asia) |
| 377 rad/s (120π rad/s) | 60 Hz | AC power (North America) |
| 2765 rad/s | 440 Hz | Musical note A4 |
| 6283 rad/s | 1000 Hz | 1 kHz reference tone |
| 22,619 rad/s | 3600 Hz | 3600 RPM motor |
Hz to rad/s Conversion (Reverse)
Reverse Conversion Formula
\[ \omega = 2\pi f \]
Multiply Hz by 2π to get rad/s
Why Convert Between rad/s and Hz?
Practical Reasons for Conversion
- Intuitive Understanding: Hz is more intuitive for counting cycles or rotations
- Motor Speeds: RPM converts to Hz, then to rad/s for calculations
- AC Power Systems: Power frequency is specified in Hz (50/60 Hz)
- Communication: Hz is more commonly understood than rad/s
- Measurement Devices: Many instruments display frequency in Hz
- Practical Reference: Easier to visualize cycles per second than radians per second
When to Use Each Unit
- Use rad/s: Physics equations, angular motion analysis, mathematical derivations
- Use Hz: Practical measurements, motor speeds, audio frequencies, power systems
- Engineering: Convert between units as needed for calculations vs. specifications
- Rotational motion: rad/s for angular velocity, Hz for rotation rate
Practical Applications
Rotational Mechanics
In rotational motion, angular velocity is naturally expressed in radians per second. A motor running at 3600 RPM rotates at 60 Hz (3600/60) or 377 rad/s (60 × 2π). While rad/s is essential for physics calculations involving torque, angular momentum, and rotational kinetic energy, Hz provides an intuitive measure of rotation rate. Converting from rad/s to Hz helps communicate rotation speed in more understandable terms—saying "60 revolutions per second" is clearer than "377 radians per second" for most audiences.
Oscillatory Systems and Vibration Analysis
Simple harmonic oscillators—pendulums, mass-spring systems, LC circuits—have natural frequencies expressed in both rad/s and Hz. The differential equation for simple harmonic motion uses angular frequency (ω in rad/s), but practical measurements and specifications use Hz. A vibrating structure might have a natural frequency of 314 rad/s, which converts to 50 Hz—a more intuitive value for engineers assessing resonance risks. Converting rad/s to Hz helps in vibration monitoring, structural analysis, and predictive maintenance.
Electrical Engineering and AC Systems
AC power systems operate at 50 Hz or 60 Hz, corresponding to angular frequencies of 314 rad/s or 377 rad/s (ω = 2πf). While power engineers specify frequency in Hz, circuit analysis requires rad/s for calculating reactance and impedance. Inductance reactance is XL = ωL, capacitance reactance is XC = 1/(ωC), where ω must be in rad/s. Converting from rad/s back to Hz helps engineers verify they're analyzing the correct power frequency and communicate results in standard units.
Mathematical Relationships
Key Formulas Involving Angular Frequency
Frequency and Period:
\[ f = \frac{\omega}{2\pi} = \frac{1}{T} \]
Angular Velocity:
\[ \omega = \frac{d\theta}{dt} = 2\pi f \]
Simple Harmonic Motion:
\[ x(t) = A\cos(\omega t + \phi) \]
Relationship:
\[ \omega = 2\pi f = \frac{2\pi}{T} \]
Common Questions
Why divide 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 conversion from rad/s to Hz is f = ω/(2π). The factor of 2 comes from the full circle (360° = 2π radians), while π alone represents only half a circle (180°). This fundamental mathematical relationship connects circular motion to periodic oscillations.
Is rad/s the same as angular velocity?
Technically yes—both angular frequency (ω) and angular velocity have the same units (rad/s) and are often used interchangeably. However, there can be 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 to Hz: f = ω/(2π).
How do I convert RPM to Hz?
First convert RPM (revolutions per minute) to Hz (revolutions per second) by dividing by 60: Hz = RPM/60. Then if you need rad/s, multiply Hz by 2π: rad/s = (RPM/60) × 2π. For the reverse (rad/s to RPM): first convert to Hz by dividing by 2π, then multiply by 60. For example, 377 rad/s = 60 Hz = 3600 RPM. The rad/s to Hz conversion is the intermediate step.
Why does physics prefer rad/s over Hz?
Radians are the natural mathematical unit for angles because they simplify calculus and trigonometry. In physics equations like ω = dθ/dt, using radians makes derivatives and integrals of trigonometric functions work elegantly without extra constants. Many physics formulas (centripetal acceleration: a = ω²r, rotational kinetic energy: KE = ½Iω²) naturally use angular frequency in rad/s. However, Hz is more intuitive for practical applications, so converting rad/s to Hz improves communication and understanding outside of mathematical analysis.
What's the exact value of 2π?
2π is an irrational number approximately equal to 6.283185307179586... It never terminates or repeats. For practical calculations, using 6.283 or 6.28 provides sufficient accuracy. In exact mathematical work, it's best to leave answers in terms of π (e.g., 120π rad/s converts to exactly 60 Hz) rather than decimal approximations. Most calculators and programming languages have a built-in π constant for precise calculations when converting between rad/s and Hz.
Quick Reference Guide
Conversion Shortcuts
- rad/s to Hz: Divide by 2π (≈ 6.283)
- Hz to rad/s: Multiply by 2π (≈ 6.283)
- Quick estimate: rad/s ÷ 6.28 ≈ Hz (or ÷ 6.3 for mental math)
- Exact values: 2π rad/s = 1 Hz, 100π rad/s = 50 Hz
- Common multiple: 377 rad/s = 120π ≈ 60 Hz (AC power)
Common Conversion Mistakes to Avoid
- Using π instead of 2π: Must divide 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 f = ω/(2π), they're related but different
- Degree/radian confusion: Always use radians, never degrees in rad/s
Why Choose RevisionTown Resources?
RevisionTown is committed to providing accurate, user-friendly calculators and educational resources across diverse topics. While we specialize in mathematics education for curricula like IB, AP, GCSE, and IGCSE, we also create practical tools for scientific and technical applications like this rad/s to Hz converter.
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 rad/s to Hz converter uses the fundamental relationship f = ω/(2π), where f is frequency in Hertz and ω is angular frequency in radians per second. The conversion factor 2π (approximately 6.283185) represents one complete revolution in radians. To convert rad/s to Hz, divide by 2π. To convert Hz to rad/s, multiply by 2π. This conversion is essential in physics, engineering, rotational mechanics, oscillatory motion, wave analysis, and control systems. Frequency (Hz) is more intuitive for counting cycles or rotations, while angular frequency (rad/s) is mathematically natural for physics equations and analysis.
