Hertz (Hz) to Radian per Second (rad/s)

Frequency in physics and engineering disciplines, such as optics, acoustics, and radio, describes how often a repeating event occurs. The unit Hertz (Hz) counts the number of complete cycles that happen every second. For instance, if a pendulum moves back and forth thrice each second, the frequency is 3 Hz.

In many science fields—electrical circuits, mechanical vibrations, wave mechanics, etc.—the frequency is expressed as angular velocity (in rad/s) instead of frequency. Radians are a way to measure the angle by which an object has rotated or is oscillating, and a full rotation equals 2π radians.

The Conversion Concept

To transform Hertz to radian per second, we apply the simple formula:

ω=2πf

Where:

• ω is angular velocity in rad/s

• f is the frequency in Hz

One complete circle measures 2π radians.

In other words, an object passes through 2π radians for each cycle per second. Multiplying 2π by the frequency gives the angular velocity.

Example

Picture a futuristic rotating light sculpture in a museum that completes seven full rotations every second. The rotation occurs 7 times each second. To see how fast the sculpture’s lights are rotating (angular velocity), we convert this to radians per second:

ω = 2π × 7 = 14π rad/s ≈ 43.98 rad/s

This means the lights are sweeping about 44 radians per second, providing a precise picture of how far those lights have rotated, making it easier to think about syncing them with music or visual effects.

Applicable Importance

In the mechanical field, angular velocity (rad/s) is commonly specified for motors, gears, and turbines.

In wave physics, radians per second is a convenient measure of rotation or angular speed and describes oscillation or phase offset as the time rate of phase change.