How do you convert Hertz to Radian per second?

Multiply the value in Hertz by the conversion factor 6.283185307179586 to obtain the result in Radian per second.

Convert Hertz to Radian per second (Hz to rad/s)

Reference ID: EC-FREQ-HZRA | EngConverter Calculation Guide

Convert From (Hertz):

Result To (Radian per second):

rad/s

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In process engineering, rotating machinery analysis, and industrial automation, characterizing cyclical phenomena requires distinguishing between cyclic frequency and angular frequency. Hertz (Hz) is the SI unit of cyclic frequency, defined as one complete cycle or oscillation per second ($1\text{ Hz} = 1\text{ s}^{-1}$). Formally adopted by the General Conference on Weights and Measures (CGPM) in 1960 in honor of Heinrich Hertz, it measures repetitive occurrences per unit time, such as AC grid oscillations, vibration harmonics, or shaft revolutions.

Conversely, radian per second (rad/s) is the coherent SI unit of angular velocity and angular frequency (often symbolized by $\omega$). A radian is the plane angle subtended by a circular arc equal in length to the radius of the circle, making it a dimensionless geometric ratio ($\text{m}/\text{m}$). Because a single cycle corresponds to a rotation through a full circle of $2\pi$ radians, the analytical transformation between cyclic frequency $f$ and angular frequency $\omega$ is governed by the exact relationship:

$$\omega = 2\pi f$$

Thus, multiplying a value in Hertz by $2\pi$ (approximately $6.283185307179586$) yields the equivalent angular rate in radians per second.

Engineering Applications & Technical Considerations

Accurate translation between $\text{Hz}$ and $\text{rad/s}$ is fundamental across several core disciplines in industrial engineering:

Turbomachinery & Rotor Dynamics: Vibration analysis via Campbell diagrams relies on determining whether excitation frequencies (in Hz) coincide with system natural frequencies. However, governing differential equations for shaft bending, gyroscopic moments, and bearing stiffness utilize angular velocity $\omega$ ($\text{rad/s}$). Conflating the two introduces an erroneous factor of $2\pi$ (a $528\%$ magnitude distortion), which can lead to miscalculated critical speeds and catastrophic mechanical resonance.
Shaft Power and Torque Transduction: In pump, blower, and compressor sizing, mechanical power $P$ is evaluated via torque $\tau$ and angular velocity: $P = \tau \cdot \omega$. Substituting cyclic frequency $f$ (in Hz) directly into this relation instead of $\omega$ (in rad/s) severely underestimates dynamic power requirements, leading to undersized drives and electrical trips.
Variable Frequency Drives (VFDs) & Electric Machines: Digital motor controllers operate on inverter output frequency measured in Hz. However, calculating the mechanical angular velocity of the rotor requires accounting for both the $2\pi$ conversion factor and the motor's internal pole-pair count $p$: $\omega_{\text{mech}} = \frac{2\pi f_{\text{elec}}}{p}$.
Signal Processing & Control Systems: In distributed control systems (DCS) and digital sensor acquisition, filter design (such as Butterworth or Chebyshev low-pass filters) defines cutoff points in radians per second ($\omega_c$), while data sheets frequently specify filter passbands in Hertz. Failure to execute proper conversion results in improper aliasing mitigation and signal attenuation.

Engineers must maintain explicit dimensional tracking in numerical integration environments (e.g., MATLAB, Python, or PLC code) to avoid implicit unit mismatches, ensuring that constants retain sufficient double-precision resolution (at least 15 significant digits for $2\pi$) to prevent cumulative drift in control loops.

Hertz to Radian per second Conversion Reference Table

Hertz (Hz)	Radian per second (rad/s)
0.1	0.6283
0.5	3.1416
1.0	6.2832
2.0	12.5664
5.0	31.4159
10.0	62.8319
20.0	125.6637
50.0	314.1593
100.0	628.3185
500.0	3141.5927
1000.0	6283.1853

To convert an operating frequency of $10\text{ Hz}$ (such as a large agitator rotating at 10 cycles per second) to angular velocity in radians per second, apply the angular frequency formula:

$$\omega = 2\pi \cdot f$$

Substitute $f = 10\text{ Hz}$ into the equation:

$$\omega = 2 \times 3.141592653589793 \times 10\text{ Hz} = 62.83185307179586\text{ rad/s}$$

Rounded to four decimal places for typical instrumentation display, $10\text{ Hz}$ corresponds to $62.8319\text{ rad/s}$.

In the International System of Units (SI), mechanical power is defined as rate of work: $P = \tau \cdot \omega$, where power $P$ is in Watts (Joules/second), torque $\tau$ is in Newton-meters ($\text{N}\cdot\text{m}$), and angular velocity $\omega$ is in radians per second ($\text{rad/s}$). Because the radian is a dimensionless unit ratio (arc length divided by radius), the dimensional equation evaluates directly to $(\text{N}\cdot\text{m})/\text{s} = \text{J}/\text{s} = \text{W}$. If frequency in Hertz ($\text{cycles}/\text{s}$) or rotational speed in RPM is used without converting to radians per second, the missing geometric factor of $2\pi$ will produce an inaccurate power value.

The electrical frequency $f_{\text{elec}}$ (in Hz) generated by an AC drive dictates synchronous magnetic field rotation. The synchronous mechanical angular speed $\omega_{\text{sync}}$ (in rad/s) depends on the magnetic pole pairs $p$ of the stator winding: $\omega_{\text{sync}} = \frac{2\pi f_{\text{elec}}}{p}$. In an induction motor, the actual shaft angular velocity $\omega_{\text{shaft}}$ is slightly lower due to rotor slip $s$: $\omega_{\text{shaft}} = (1 - s) \cdot \frac{2\pi f_{\text{elec}}}{p}$. Consequently, direct conversion from inverter Hz to shaft rad/s requires knowing both the pole count and operational slip.

"On fait la science avec des faits, comme on fait une maison avec des pierres ; mais une accumulation de faits n'est pas plus une science qu'un tas de pierres n'est une maison."
"Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house."
— Henri Poincaré (French Mathematician, Theoretical Physicist & Mining Engineer)