Hertz to Radian per Second (rad/s) Calculator

Enter the frequency in hertz:

Converting Hertz (Hz) to radians per second (rad/s) is essential for analyzing rotational motion, angular velocity, and AC circuits in physics and engineering. This guide explains the conversion with formulas, examples, and practical applications.

---

Key Concepts

1. Hertz (Hz)

• Definition: Frequency in cycles per second (1 Hz = 1 cycle/second)

• Applications:

            a. AC power (50/60 Hz)
            b. Audio signals (20 Hz – 20 kHz)
            c. Digital clock speeds

2. Radians per Second (rad/s)

• Definition: Angular velocity (1 rad/s = 1 radian of rotation per second)

• Applications:

         a. Rotational mechanics (e.g., motors, turbines)
         b. AC circuit analysis (angular frequency)
         c. Wave propagation (phase velocity)

---

Conversion Formula

The relationship between Hz and rad/s is:

$$\boxed{{\displaystyle \omega =2\pi f}}$$

Where:

• $\omega$ = angular frequency (rad/s)
• $f$ = frequency in Hertz (Hz)
• $2\pi$ ≈ 6.28318 (radians in one full cycle)

Examples:

1. 50 Hz to rad/s:

   $$\omega =2\pi \times 50=314.16\text{ rad/s}$$

2. 1 kHz to rad/s:

   $$\omega =2\pi \times 1000=6283.18\text{ rad/s}$$

Reverse Conversion (rad/s → Hz):

$$f=\frac{\omega }{2\pi }$$

Example:

• 100 rad/s → Hz:

$$f=\frac{100}{2\pi }\approx 15.92\text{ Hz}$$

---

Conversion Table

Hertz (Hz)	Radians per Second (rad/s)
1 Hz	6.283 rad/s
10 Hz	62.832 rad/s
50 Hz	314.159 rad/s
60 Hz	376.991 rad/s
100 Hz	628.318 rad/s
1 kHz	6,283.185 rad/s
10 kHz	62,831.853 rad/s

---

Practical Applications

1. Electrical Engineering

• AC Circuits:

           a. Angular frequency ($\omega$) determines reactance ($X_L=\omega L$, $X_C=\frac{1}{\omega C}$)

           b. Example:

           c. A 60 Hz AC signal → $\omega =377\text{ rad/s}$

2. Mechanical Systems

• Rotational Speed:

            a. A motor running at 30 Hz → $\omega =188.5\text{ rad/s}$
            b. Used in torque calculations ($\tau =I\alpha$)

3. Signal Processing

• Digital Filters:

          a. Cutoff frequencies are often specified in rad/s
          b. Example:

•  A 1 kHz filter → ωc​=6,283 rad/s

${\omega }_c=6,283\text{ rad/s}$