There are many different ways to measure the speed something is turning, but you’ll need to understand what they actually mean if you need to convert from one to another. For instance, if the crankshaft of a car engine is turning at 4,000 revolutions per minute (RPM), how big of an angle does it turn in a single second?
This question might strike you as very specific, but in reality, if you learn how to do this conversion, other conversions such as radians per second to RPM (the opposite process) and many others will start to make sense.
RPM and Radians Defined
Understanding both of the measures you’re considering is the first step to performing a valid conversion. Revolutions per minute (RPM) is just what it sounds like: The number of complete turns (full revolutions) the engine or wheel makes in a one-minute time period. You don’t need to go much further than your intuitive understanding here, but remember that this is only per minute, not per second like many other measures of rotation.
A radian is a measure of an angle, like degrees, but defined in terms of π to make calculations easier in math and science in particular. There are 2π radians (rad) in a complete revolution, so π radians is half a circle and so on. You can relate this to degrees by noting that 360 degrees = 2 π rad, so 1 radian = 57.3 degrees.
RPM to Rad/s
Converting RPM to radians per second (i.e. RPM to rad/s or rev/min to rad/s) just involves applying this knowledge to your situation. The two main steps in the conversion are converting RPM to revolutions per second, then converting total revolutions to the angle covered in radians. The first step is simple: Divide the number in RPM by 60 to find the number of revolutions per second as shown:
So for 4,000 RPM you get 4,000 RPM ÷ 60 sec/min = 66.667 revolutions per second. Now, you take this value and convert to radians by multiplying by 2π. In the example, the result comes out to 418.9 rad/s. The full formula can be written as follows:
Radians Per Second to RPM
The reverse calculation (rad/s to RPM) can also be completed pretty easily once you understand what you’re doing. In fact, you can probably work it out based on the equation above or simply going through the opposite of the process described in the previous section. First, convert from radians to complete revolutions by dividing the figure by 2π, then multiply by 60 to convert from seconds to minutes. The complete formula is given below:
Of course, you can also use what you’ve learned so far to calculate the angular velocity in other units such as degrees. All you have to do is replace the 2π rad in the formula above with 360 degrees, and then you can use the same approach to convert between RPM and degrees per second or vice-versa. Additionally, because there are two clear stages to the calculation (converting revolutions to an angle and converting minutes to seconds), you can also easily find a value in radians or degrees per minute if you like.
It should go without saying that you don’t have to perform the calculation manually if you don’t want to. There are numerous online calculators (see Resources) that can handle the calculation for you, so you can convert from rad/s to RPM or vice-versa by just entering the value that you know in the appropriate field. However, if you understand the basics of the calculation, it’s easy to do on a cell phone calculator or whatever you have on hand.