Radian per Second to Hertz Converter

Enter the angular velocity in rad/s:

A Radian per Second to Hertz (rad/s to Hz) converter is a tool that translates angular velocity (measured in radians per second) to frequency (measured in hertz). This conversion is essential in fields like rotational mechanics, signal processing, and wave physics where both angular and cyclic frequencies are used.

Understanding the Units

• Radian per Second (rad/s):

  • Measures angular velocity (how fast an object rotates)

  • 1 rad/s = angle of 1 radian covered per second

  • Used in rotational dynamics, AC circuits, and oscillation analysis

• Hertz (Hz):

  • Measures cyclic frequency (cycles per second)

  • 1 Hz = 1 complete oscillation per second

  • Standard unit for waves, vibrations, and alternating currents

Conversion Formula

The conversion between these units relies on the relationship between angular and cyclic frequency:

$$\text{Frequency in Hertz (Hz)}=\frac{\text{Angular frequency in rad/s}}{2\pi }$$

Derivation:

• 1 complete cycle = 2π radians

• Therefore, frequency (cycles/sec) = angular frequency (radians/sec) ÷ 2π

Conversion Steps

1. Take the value in rad/s
   Example: 62.83 rad/s

2. Divide by 2π (≈6.2832)

   $$\frac{62.83}{6.2832}\approx 10$$

3. Result:
   62.83 rad/s ≈ 10 Hz

Conversion Table

rad/s	Hz (approx)
6.2832	1
12.5664	2
31.416	5
62.832	10
125.664	20
314.159	50

Practical Applications

• Electrical Engineering: Converting angular frequency of AC signals to standard frequency

• Mechanical Systems: Relating rotational speed to vibration frequency

• Wave Physics: Translating between angular wave number and temporal frequency

• Control Systems: Analyzing system responses in different frequency terms

Calculation Examples

Example 1: Convert 15.708 rad/s to Hz

$$\frac{15.708}{6.2832}\approx 2.5\text{ Hz}$$

Example 2: A motor rotates at 376.99 rad/s. What's the frequency in Hz?

$$\frac{376.99}{6.2832}\approx 60\text{ Hz (standard AC power frequency)}$$

Reverse Conversion (Hz to rad/s)

To convert back:

$$\text{rad/s}=\text{Hz}\times 2\pi$$

Example: 50 Hz → 50 × 6.2832 ≈ 314.16 rad/s