# Hz units vs RPM units

I have seen Hz defined in different ways, if $$\Omega = 1 \ \text{Hz} = 1\frac{\text{cycle}}{\text{s}}$$ Here $$1 \ \text{cycle} =$$ a full turn of the rotor $$= 2\pi$$

If I wish to express this a rpm:

By definition $$1 \ \text{rev} =$$ a full turn of the rotor = $$2\pi$$

So, $$1 \ \text{Hz} = 1\frac{\text{cycle}}{\text{s}} = 1\frac{\text{rev}}{\text{s}} = 1\frac{\text{rev}}{\text{s}} \cdot \frac{60 \ \text{s}}{\text{min}} = 60 \ \text{rpm}$$ I wonder if this is true because I saw a different transformation from Hz to rpm: $$\omega = 24 \ \text{Hz} = 24 \ \text{Hz} \left( \frac{1}{s}\frac{\text{rev}}{2\pi}\frac{60 \ \text{s}}{\text{min}} \right) = 231 \ \text{rpm}$$ He has that "random" pi factor. In here, omega is the angular speed of the rotor, so 24 turns per second.

• The Engineering book I'm reading has a different convention: $$\omega = 24 Hz * (\frac{1}{s} \frac{rev}{2\pi}\frac{60s}{min} = 231rpm$$ He has that "random" pi factor.
– RSM
Feb 4 at 21:07
• That's just incorrect useage of Hz. The correct unit would be rad/s. Which book is this?
– noah
Feb 4 at 21:46
• I thought I wrote the name of the book earlier, my bad. The name is: alternative energy systems and applications, 2nd ed, bk hodge
– RSM
Feb 5 at 2:29
• @RSM What page in that book, please?
– nasu
Feb 5 at 15:56

What is done by the formula is not a conversion of units per se but a conversion of qantities. He calculates the angular velocity (or angular frequency) when the frequency (in Hz or rpm) is given. The relationship is $$\omega =2\pi f$$ where $$\omega$$ is the angular frequency (or velocity) in $$s^{-1}$$ or $$rad/s$$ and f is the frequency in Hz or rpm. So this is where the pi comes from, it is not random, of course. The rest is just converting between frequency in Hz to frequency in rpm.

• Yeah I think he refers to Hz as rad/s, I was confused as I made the connection from Hz = cycles/s, so Hz is equivalent to rpm if 1 cycle was a full turn (2Pi) on the rotor. But like you explained, the usual convention is that Hz = rad/s ; thus Hz (rev/2Pi*rad)(60s/min)
– RSM
Feb 6 at 18:07
• I never said that Hz is radian/s. Hz is simply number of something per second. The something can be rotations or oscillations or beats or falling drops etc. But it's the number (no units) per second. It has nothing to do with radians. Angular frequency though is given by some authors in radians/s but but never in Hz.
– nasu
Feb 6 at 22:48
• I understand you didn't say that explicitly. Since Hz = something/s, then something = 1 rad. Which i think is what the author had in mind. Otherwise what he did wouldn't make sense.
– RSM
Feb 6 at 23:25
• No, is not 1 rad/s. If anything, for circular motion 1 Hz means 2pi radians per second or about 6.28 rad/s. This value, 6.28 rad/s, is the angular velocity of a circular motion with the frequency of 1 Hz. An angular velocity of 1 rad/s corresponds to a frequency of 1/(2pi) Hz or about 0.16 Hz. What page in the book is the example in the book? Or what chapter, at least. I don't see any reference to Hz so far. He uses rpm for frequency and rad/s for angular velocity.
– nasu
Feb 7 at 3:22