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# Angular motion variables

David explains the meaning of angular displacement, angular velocity, and angular acceleration. Created by David SantoPietro.

## Want to join the conversation?

• would you mind explaining how pi radians equals 180 degrees? i'm not sure i understand that part.
• A radian is the angle subtended by an arc when the arc is equal to the radius. So the circumference is the arc subtended when the angle goes 360 round right? That means ...
so 2 pi radians is the same as 360 degrees....
divide both sides by 2....
pi radians is the same as 180 degrees.
Hope this helped!
(Pls.upvote)
• At when Dave says that if you were to revolve around the circle twice, the displacement wouldn't be two revolutions. Why would it not be zero because it ended at the same place it started?
• Angular displacement and displacement is different. One talks about minimum distance while the other the minimum angle
• Wait. Why do we use radians instead of degrees when dealing with angular variables?
• Radians end up being more helpful if you end up ever using calculus. For example, angular acceleration is the derivative of angular velocity, but derivatives for trig functions can only be calculated in radians, which is why using radians is better.
• the shape that the tennis ball took was a circle, what would happen if it was a oval? would the equations stay the same or would they change? thanks.
• It would be the same. Even there was no ball there, just the string, still equations will be perfect. Note that we not dealing with the shape of the object. We just care about angle, speed, radius etc.,. Nothing about the ball. Hope I didn't confuse you.
• Since displacement means the shortest path length,then why was the angular displacement of the tennis ball 720 degrees and not 0 as the initial and final positions are same?
• Since you did not ask why angular displacement is not a vector, let me give a simpler answer to what you asked, without going into commutative vector laws. Angular displacement unlike linear displacement isn't defined as the shortest path traveled between to points. When you talk about shortest path between two position coordinates in space, you, in principle, invoke vector laws to find the change in the position vectors which in turn tells you the displacement or the shortest distance you traveled from one position coordinate to the other. In circular motions you move in arcs not straight lines and the concept of shortest distance as defined in angles, is moot. An arc between any two point is always greater than a line between them. However, you can argue that we can define something like shortest angular path for angular motions, analogous to linear displacement. This is where the properties of vectors come forth.
Although defined in a similar fashion like displacement, that is, as the difference between the final (angular) position - initial (angular) position, angular displacement is not a vector.

NOTE: A rigorous mathematical answer can be given regarding why angular displacement is not like linear vector displacement. For that you need to understand commutative vector law.
• In angular displacement, if the tennis ball rotates a full circle, would the angular displacement be 0 or 2pi radians.If the angular displacement is 2pi radians, then why is it called angular "displacement" since in displacement the actual value would be 0.
• Good question. But as per my knowledge there is no angular distance...
• can angular velocity be negative?
• Yes, if you define clockwise as positive rotation and something is rotating counter-clockwise it will have a negative angular velocity.
• Why is counter-clockwise positive?

If we're going off which way is rightward, counter-clockwise is only rightward on the bottom. In the popular saying righty-tighty, lefty-loosey, direction is judged from the top. Why not here as well?
• If you take a unit circle on an XY plane and use the definition of the Y position as Sine of an angle and the X position as the Cosine of the same angle. As the angle increases the point moves around the unit circle in a counter-clockwise direction.
• can we calculate angular acceleration using average angular velocity? Of course in this case velocity is not constant
• In general the answer is no, but there are special cases in which the answer is yes. For instance, suppose the initial angular velocity is zero and the angular acceleration is constant. In that case the ang. acceleration is equal to the av. ang. acceleration and that av. ang. acceleration is equal to the change in ang. velocity divided by the time taken and the change in ang. velocity is twice the av. ang. velocity. Consequently, we have,

'alpha' = av 'alpha' = change in 'omega' / change in time = 2 (av 'omega') / change in time.

There are some other examples also, but, in general there is no simple relation between ang. acceleration and av. ang. velocity.