radians per second to hertz conversion

How to convert radians per second to hertz?

To convert radians per second (rad/s) to hertz (Hz), you need to understand the relationship between these two units.

1. Formula: $\text{Frequency (Hz)} = \frac{\text{Angular Velocity (rad/s)}}{2\pi}$

   This is because 1 Hz represents one cycle per second, and one complete cycle is $2\pi$ radians.

2. Conversion Example: $1 \, \text{rad/s} = \frac{1 \, \text{rad/s}}{2\pi} \approx 0.159 \, \text{Hz}$

   Hence, 1 radian per second is approximately 0.159 Hz.

3. Real-World Examples:

   • Electrical Engineering: An alternating current (AC) electrical signal in the USA typically has a frequency of 60 Hz. In radians per second, this is: $60 \, \text{Hz} \times 2\pi \approx 377 \, \text{rad/s}$

   • Mechanical Engineering: A car engine running at 3000 revolutions per minute (rpm) can be converted to radians per second: $3000 \, \text{rpm} = 3000 \times \frac{2\pi}{60} \approx 314 \, \text{rad/s}$

   • Physics (Circular Motion): The Earth's rotation speed can also be expressed in radians per second. Since the Earth makes one complete rotation every 24 hours: $1 \, \text{day} = 24 \, \text{hours} \times 3600 \, \text{seconds/hour} = 86400 \, \text{seconds}$ $\Omega_{\text{Earth}} = \frac{2\pi \, \text{radians}}{86400 \, \text{seconds}} \approx 7.27 \times 10^{-5} \, \text{rad/s}$

By understanding these concepts and using the given formula, you can easily convert between radians per second and hertz for various applications.