: What is angular velocity?
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Say you took a dinner plate and put it on the ground, and you started to take baby steps to walk in a tight loop around it. Even with such small steps you’d make it around the dinner plate reasonably quickly.
Now say while you did this, you extended your arm on the side of you opposite the plate straight out. And in your hand you held a meter stick pointing straight out even further from the dinner plate. You instruct your friend to grab the other end of the stick and walk around the plate with you. You also tell them that they have to keep up with you so that the stick always stays pointing straight out away from the plate. If your friend does that, they’ll probably find that they’ll have to walk quite quickly to keep pace with you. Possibly even start jogging, depending on the size of step you’re taking.
Think about how weird that is. The two of you are clearly moving at two completely different speeds (you are taking baby steps, while your friend is near a full-out jog), but you are still managing to complete the same activity in the same amount of time (walking around the dinner plate exactly once).
In a straight-line motion scenario, if you take a person’s speed and multiply it by the time they traveled, you get their distance. So if two of you were walking at different speeds for the same amount of time, you’d have two different distances. But in this scenario where you’re walking around a dinner plate, the two of you were moving at two different speeds for the same amount of time, but you ended up at the same exact rotational place. What’s going on here?
The only difference between you two besides your speeds are how far away from the dinner plate the two of you are. If you happen to take each of your speeds and divide them by each of your respective distances to the dinner plate, you’ll both arrive at the same value. Just like how you arrived at the same rotational place. So it suggests that this is how you should calculate your “speed” when you’re rotating around something. Your “rotational speed”. How long it takes you to walk around something, regardless of how far away you are from it.
Typical straight-line speed has units distance per unit time (like meters/second). When we modified it to measure rotational speed, though, we divided your straight-line velocity by a measurement of distance. The distance units cancel and you’re left with units of just “per unit time” (like “per second”). Literally just a measurement of how long it takes some some *thing* to occur. The SI unit for this is the hertz (Hz). So rotational speed is measured in hertz in the SI system.
You may have noticed that I kept using the word “speed” instead of “velocity”. Velocity is essentially the same thing as speed, but also incorporating information about direction of travel. If I started walking east at 1 m/s and you started walking west at 1 m/s, we’d both be walking with the same speed, but since we’re walking in different directions, we’d have different velocities.
For rotation, rotational velocity will be the same thing as rotational speed plus information about which way the thing in question is actually rotating. In 2D, we’d usually say “clockwise” or “counterclockwise”.
It’s harder to figure out exactly what to say in 3D, since if you looked at something spinning and said, “It’s spinning clockwise!”, then walked around to look at the opposite side, it would look like it’s spinning counterclockwise. You can’t tell the two apart. Instead, the convention in physics to solve this problem is to use something called the “right hand rule”. It works like this: Imagine the spinning object was shrunk down so it fits in the palm of your right hand. Grab the object into a tight fist in such a way that your fingers curl around it pointing in the direction the object is spinning. Then, stick your thumb straight out in a “thumbs-up” gesture. The direction your thumb points is considered to be the way the rotation is “pointing”. In diagrams it will typically be shown as an arrow pointing straight out of the object. If the object was spinning the other direction, you’d have to flip your hand over to grab it according to our finger-curling rules, meaning your thumb would also point in the opposite direction. Thus, with the right hand rule, you can uniquely describe how something spins in 3D space in a way that gives the same answer no matter what direction you look at it from.
Why the right hand specifically, and not the left hand? No particular reason. At least, no reason as far as physics are concerned. It was just an arbitrary choice we made so we could all agree on something. Most people are right-handed, so it’s probably why we picked that one, but if it were a “left hand rule” instead, it would work just as well.
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