Radians per Second (rad/s) - Unit Information & Conversion
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What is a Radians per Second?
Radians per second (rad/s or rad·s⁻¹) is a unit of angular frequency measuring the rate of angular displacement in radians per unit time. Related to hertz by ω = 2πf. Used in physics, engineering, and rotational dynamics.
History of the Radians per Second
Angular frequency in radians per second derives from calculus and rotational kinematics. Essential for describing oscillatory motion, AC circuits, and rotational dynamics. The relationship ω = 2πf connects angular frequency to cyclic frequency (Hz).
Quick Answer
What is Radians per Second? Radians per second (rad/s) measures angular frequency - how fast an angle changes. Related to hertz (Hz) by: ω = 2πf or f = ω/(2π). 1 rad/s ≈ 0.159 Hz. Used in physics for oscillations, AC circuits, rotating machinery, and wave motion. 2π rad/s = 1 Hz = 1 cycle/second. Use our frequency converter for instant conversions.
Definition
1 rad/s = 1 radian per second = 1/(2π) Hz ≈ 0.159155 Hz
Relationship to Hz:
- ω (rad/s) = 2πf (Hz)
- f (Hz) = ω/(2π) (rad/s)
Key relationship: 2π rad/s = 1 Hz (one complete cycle)
Angular Frequency vs Cyclic Frequency
Cyclic Frequency (f in Hz):
- Measures complete cycles per second
- f = 1 Hz = 1 full rotation per second
Angular Frequency (ω in rad/s):
- Measures angular displacement per second
- ω = 2π rad/s = one full rotation per second (360° per second)
Why the 2π factor?
- One complete cycle = 2π radians = 360°
- So ω = 2πf converts cycles/sec to radians/sec
Common Uses
Physics: Simple harmonic motion, pendulums, springs, wave equations. Electrical Engineering: AC circuit analysis, phasors, impedance. Mechanical Engineering: Rotational dynamics, angular velocity, vibration analysis. Control Systems: Transfer functions, frequency response, stability analysis.
Real-World Applications
Oscillations & Vibrations
Simple Harmonic Motion:
- Mass on spring: ω = √(k/m)
- Pendulum: ω = √(g/L)
Example:
- Pendulum with ω = 3.14 rad/s
- Period T = 2π/ω = 2 seconds
- Frequency f = ω/(2π) = 0.5 Hz
AC Electrical Circuits
North America (60 Hz power):
- f = 60 Hz
- ω = 2πf = 377 rad/s
Europe/Asia (50 Hz power):
- f = 50 Hz
- ω = 2πf = 314 rad/s
AC voltage: V = V₀ sin(ωt) AC current: I = I₀ sin(ωt + φ)
Rotational Motion
Angular velocity (same units, different context):
- 1 rad/s = 1 radian per second rotational speed
- 1 rad/s = 60/(2π) RPM ≈ 9.55 RPM
Example:
- Wheel rotating at 10 rad/s
- Equivalent to 10/(2π) ≈ 1.59 Hz
- Equivalent to 10 × (60/2π) ≈ 95.5 RPM
Control Systems
Transfer functions: H(s) where s = jω Frequency response: Magnitude and phase vs ω Bode plots: X-axis in rad/s Stability: Gain and phase margins at specific ω
Radians per Second Conversion Formulas
To Hertz:
To Millihertz:
To Kilohertz:
To Megahertz:
To Gigahertz:
To Terahertz:
To Revolutions per Minute:
To Revolutions per Second:
To Beats per Minute:
To Cycles per Second:
Frequently Asked Questions
Formula: Hz = rad/s ÷ (2π) Examples:
- 2π rad/s = 1 Hz
- 6.28 rad/s ≈ 1 Hz
- 377 rad/s = 60 Hz (US power frequency)
- 314 rad/s = 50 Hz (European power frequency) Reverse: rad/s = Hz × 2π rad/s to Hz converter →
Convert Radians per Second
Need to convert Radians per Second to other frequency units? Use our conversion tool.
Radians per Second Quick Info
Related Frequency Units
Popular Conversions
- Radians per Second to HertzConvert →1 rad/s = 0.159155 Hz
- Radians per Second to MillihertzConvert →1 rad/s = 159.154943 mHz
- Radians per Second to KilohertzConvert →1 rad/s = 0.000159 kHz
- Radians per Second to MegahertzConvert →1 rad/s = 1.5915e-7 MHz
- Radians per Second to GigahertzConvert →1 rad/s = 1.5915e-10 GHz
- Radians per Second to TerahertzConvert →1 rad/s = 1.5915e-13 THz