: What is angular velocity?
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The speed at which you complete cycles when you’re rotating around something or yourself. RPM is the most commonly known unit for it, but HZ and gHZ also count depending on where.
Angular velocity is the rate at which something turns around a central point.
Imagine a baseball batter swinging their bat. Their body becomes the central point, or axis, and the bat has an angular velocity that describes how quickly it is rotating around the player’s body.
If they swing the bat faster, it has a higher angular velocity.
The angular velocity is measured in radians or degrees / time. So you could have something with an angular velocity of 360 degrees/minute meaning it makes one full rotation and ends up back where it started every minute.
(It can also be measured in revolutions/time, like RPM with an engine)
I was curious about my own example, so I looked it up. Apparently a baseball bat swing is something like 2000°/sec.
*tl;dr: Angular velocity is the vector form of the rate of change of something’s angle. You measure the angle something is at, you work out how that angle is changing over time, and then you include the direction it is changing in.*
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If we have something that is repeating in a pattern it can be useful to describe where it is at the moment in the pattern (rather than where it is absolutely).
For example, time comes in a whole bunch of repeating patterns (seconds in a minute, minutes in an hour, hours in a day, days in a week, days in a year, months in a year and so on). Rather than counting time as an absolute number (like UNIX time – the number of seconds since 1 January 1970) we care about where we are in a particular cycle – how many minutes past the hour, how many hours into a day and so on. And with any of these we can measure where we are in a cycle in terms of the “angle” it would make if our cycle was a circle, so where 0° would be the start, 90° a quarter of the way through, 180° halfway through and 360° back to the start again (although in maths, physics and engineering we’d usually use radians rather than degrees). This is how analogue clocks work. Each hand is representing how far through the minute, hour or half-day we are as an angle.
Normal speed tells us how a distance changes over time. *Angular* speed then tells us how quickly our angle is changing – for example how many degrees per second. The second hand on a clock moves at an angular speed of 360° per minute, or 6° per second (each second “tick” is 6° from the previous one). We could also measure it in terms of the frequency; how many complete cycles we get through in a time period. A second hand moves at 1 RPM (one complete rotation per minute), or 1/60 Hz (rotating 1/60th of a circle per second).
Angular *velocity* takes this a step further, by turning quantity into a vector, not just a number. Angular velocity would be the same as the angular speed, but would also tell us which way around our thing is cycling (so on a clock, the second hand moves with an angular speed 6° per second, but an angular velocity of 6° per second *clockwise*). Mathematically encoding this information in the term – rather than having to mention it separately – makes calculations a lot easier, and lets us do all sorts of sneaky things.
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One common uses of angular velocity is when things are moving, and we care about how they are moving around a point, rather than in a particular direction. Often because the thing is moving in a circle, but sometimes just because we care about that fixed point. In that case, angular velocity directly links to the normal “linear” velocity (i.e. movement through space), but is telling us the same thing as above; how is the angle the object is at changing over time, and in which direction.
**Regular velocity is the rate at which something is moving**. 15 meters per second, 10 miles per hour, that sort of thing.
***Angular*** **velocity is the rate at which something is spinning**. A more intuitive name would be “rotational velocity”, but we use “angular” because then you can say something is rotating through an *angle* of 60 degrees every second or something like that. But you can also give an angular velocity in units like revolutions per minute (RPM), turns per second (Hertz), etc.
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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