radians per second to megahertz conversion

How to convert radians per second to megahertz?

Sure! Converting radians per second (rad/s) to megahertz (MHz) involves understanding the relationship between radians per second and hertz (Hz), as well as the relationship between hertz and megahertz. Here we go step-by-step through the conversion:

1. Understanding the Definitions:

   • Radians per second (rad/s) is a unit of angular frequency.
   • Hertz (Hz) is a unit of frequency representing one cycle per second.
   • Megahertz (MHz) is one million hertz.

2. Conversion from Radians per Second to Hertz:

   • One cycle corresponds to $2\pi$ radians.
   • Therefore, 1 Hz is equivalent to $2\pi$ radians per second.
   • To convert radians per second to hertz, you divide by $2\pi$.

   $f (\text{Hz}) = \frac{\omega (\text{rad/s})}{2\pi}$

3. Conversion from Hertz to Megahertz:

   • 1 MHz is $10^6$ Hz.
   • To convert hertz to megahertz, you divide by $10^6$.

   $f (\text{MHz}) = \frac{f (\text{Hz})}{10^6}$

4. Combining the Steps:

   $f (\text{MHz}) = \frac{\omega (\text{rad/s})}{2\pi \times 10^6}$

For 1 rad/s:

$f (\text{MHz}) = \frac{1}{2\pi \times 10^6}$
$f (\text{MHz}) = \frac{1}{6.2832 \times 10^6}$
$f (\text{MHz}) \approx 1.5915 \times 10^{-7} \text{ MHz}$

So, 1 rad/s is approximately $1.5915 \times 10^{-7}$ MHz.

Real-World Examples for Other Quantities of Radians per Second:

1. Electric Power Grid Frequency:

   • Standard electrical power grid frequency in many countries is 50 Hz or 60 Hz.
   • In radians per second, this is:
     • For 50 Hz: $\omega = 2\pi \times 50 \approx 314.16$ rad/s.
     • For 60 Hz: $\omega = 2\pi \times 60 \approx 376.99$ rad/s.

2. Audio Frequencies:

   • Middle C on a piano is approximately 261.63 Hz.
   • In radians per second, this is:
     • $\omega = 2\pi \times 261.63 \approx 1644.69$ rad/s.

3. Microwave Oven Frequency:

   • Standard microwaves operate at 2.45 GHz (gigahertz).
   • In radians per second, this is:
     • $\omega = 2\pi \times 2.45 \times 10^9$
     • $\omega \approx 15.392 \times 10^9$ rad/s.

4. Rotational Speed of a CD:

   • A typical CD spins at about 200 to 500 RPM (revolutions per minute).
   • At 200 RPM, this is:
     • $200 \text{ RPM} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}}$
     • $\omega \approx 20.94$ rad/s.
   • At 500 RPM, this is:
     • $\omega \approx 52.36$ rad/s.

Hopefully, this gives you a clear understanding of converting radians per second to megahertz and some context with real-world examples of different angular frequencies.