Angular Velocity is a measure of how quickly something moves about a fixed point. In other words, how quickly something moves about the circumference of a circle. The speed of this movement is normally measured in radians or degrees per unit of time, most commonly seconds (though if you were measuring the movement of stars then the unit of time could be years).
In the diagram at right let’s assume that the two green lines represent the movement of a stick that is anchored at the center of the two circles. Moving the stick from its raised position downward 45° (the angle represented by Θ) is represented by the two green lines. Let us further assume this movement takes one second. So the angular velocity is forty five degrees per second. Whether speaking of the blue inner circle or the red outer circle the angular velocity is the same. It takes the same amount of time for the stick to move 45° along the circumference of both circles.
Tangential velocity is quite different. If we look at the two arrows labeled Arrow 1 and Arrow 2 we see that these two straight lines are just to the outside of their respective circles. These barely (if at all) touch the circles. They are tangent to the circle and are therefore called tangential lines. These tangential lines represent the speed and direction at which the stick is moving at the moment where it crosses the point represented by where the line and circle touch. In the diagram the stick would have moved downward roughly 22.5° before the stick would have interested the point represented by the circle and arrows.
The first thing you will notice is that Arrow 2 is larger than Arrow 1. This indicates that the stick is moving faster at any moment in time along the outer circle than it is along the inner circle. These are linear speeds, not angular speeds.
This makes sense if you think about it. Since a stick (assuming it is rigid) would move in a circular path then the end of the stick must be moving faster than the middle of the stick, otherwise the stick would bend. Both the end of the stick and the middle of the stick must both reach the bottom line at the same time. Since these two points are further from the center of the circles the distance traveled by the end of the stick is greater than the distance traveled by the middle of the stick. The only conclusion one can reach is that the end of the stick must therefore be traveling faster than the middle of the stick.
We can conclude that Arrow 1 and Arrow 2 represent vectors. They have a direction of travel and their length represents their relative speeds. If you were to move Arrow 1 closer to the center of the circles then Arrow 1 would become smaller. If you moved Arrow 2 further away from the center of the circles then it would grow larger.
If you now think of the stick as being a Jo weapon you can see that the far end of the Jo will be moving must faster tangentially than the end you are holding. To derive maximum power from the Jo you would want to strike with the very end of the weapon. This is where the most tangential speed and therefore the most energy will be focused.
If on the other hand you were defending yourself from a Jo attack (assuming something like a Shomen, or vertical downward, strike) then you would want to block the weapon as close to the opponent’s hands as possible. You do not want to block anywhere near the free end of the weapon. There will be far less energy near the end being held by your opponent. (None of this suggests that we advocate blocking a Jo weapon with your bare hands).
Now let’s consider that an attacker doesn’t have a Jo but has a club in his or her right hand instead. The person raises his or her hand, steps forward, and crashes the weapon downward in your direction. Where would you block now?
Since the entire arm and the club make one long “stick” then we would want to block as near to the center of the circle being drawn by that stick. The center of that circle would be the shoulder joint. The closer you can block to the shoulder joint the less energy you will encounter. The closer you get to the end of the club the greater the energy you will need to absorb.
This helps explain why we block with an Age Uke placed such that our forearm is near our forehead or the bridge of our nose. This is the correct elevation for blocking a descending arm with the center of rotation being the shoulder joint. We block as close to the shoulder as possible, so our block need not rise higher than this level. This is different than blocking when we must block further away from the shoulder so we use a completely different blocking strategy (Haiwan Nagashi Uke) in that case.
Angular and tangential velocities are quite important in both hand and weapon utilization. There are subtle ways to derive the most benefit from these concepts while performing hand and weapon skills. Your instructor should point these uses out for you when you are learning many different skills. You will encounter these ideas quite frequently and should become quite familiar with these concepts and their practical applications.