radians per second to kilohertz conversion

How to convert radians per second to kilohertz?

Sure, I'd be happy to explain how to convert radians per second to kilohertz and provide some real-world examples of different quantities of radians per second.

Conversion from Radians per Second to Kilohertz

1 radian per second (rad/s) is a measure of angular velocity, while kilohertz (kHz) is a measure of frequency in cycles per second.

To convert from radians per second to hertz (Hz), and then to kilohertz (kHz), we need to understand the relationship between these units.

\text{Frequency (Hz) = Angular Velocity (rad/s) / (2\pi) }

1 Hz is equal to one cycle per second, and 1 kHz is equal to 1,000 cycles per second.

1. Convert radians per second to hertz:

$1 \text{ rad/s} = \frac{1}{2\pi} \text{ cycles/second} \approx 0.1592 \text{ Hz}$

2. Convert hertz to kilohertz:

$0.1592 \text{ Hz} = 0.1592 \times 10^{-3} \text{ kHz} = 0.0001592 \text{ kHz}$

Therefore, 1 rad/s approximately equals 0.0001592 kHz.

Real-World Examples of Radians per Second

1. Electric Motor Speed:

   A typical electric motor operating at a speed of 3,600 revolutions per minute (RPM) can be converted to radians per second as follows:

   $3,600 \text{ RPM} \times \frac{2\pi \text{ rad}}{60 \text{ s}} \approx 377 \text{ rad/s}$

2. Earth's Rotation:

   The Earth's rotation speed in radians per second can be calculated knowing that it makes one full rotation every 24 hours (86,400 seconds):

   $1 \text{ rotation/day} \times \frac{2\pi \text{ rad}}{86,400 \text{ s}} \approx 7.272 \times 10^{-5} \text{ rad/s}$

3. Sound Frequency:

   Consider a sound wave with a frequency of 440 Hz (A note above middle C):

   $440 \text{ Hz} \times 2\pi \text{ rad} \approx 2,764 \text{ rad/s}$

These examples illustrate a range of angular velocities from very slow (Earth's rotation) to high frequencies (sound waves), showing how radian per second is a versatile unit in many fields, including mechanical engineering and acoustics.