What is the relationship between instantaneous and average angular velocity?

[Aristotle]

Can somebody explain to me the reason why? Thanks!

 

[elegysix]

Chances are your velocity was changing. The instantaneous = the average only when it is absolutely constant.

 

[Aristotle]

Chances are your velocity was changing. The instantaneous = the average only when it is absolutely constant.

Say there was an object that was experiencing a varying angular acceleration decreasing over time to zero. Would that mean its velocity continues to increase/change over time?

 

[haruspex]

Can somebody explain to me the reason why? Thanks!

The instantaneous is the average over an arbitrarily short time interval. In general, an average velocity can be over a substantial interval. In notation, d is used for an infinitesimal change, $\Delta$ for a general change. So $\frac{\Delta x}{\Delta t}$ is an average velocity over time $\Delta t$, while $\frac{dx}{dt}$ is the instantaneous velocity.

 

[elegysix]

Say there was an object that was experiencing a varying angular acceleration decreasing over time to zero. Would that mean its velocity continues to increase/change over time?

If the acceleration is nonzero, then velocity is changing, regardless of whether the acceleration varies. It is only when acceleration = 0 that velocity is constant.

 

[Aristotle]

If the acceleration is nonzero, then velocity is changing, regardless of whether the acceleration varies. It is only when acceleration = 0 that velocity is constant.

Ah I see. So because the velocity is changing over time, we get a curved position vs time graph (representing velocity) to show that its increasing. and so if we take the instant angular velocity at a single time it wouldn't equal with the average angular velocity (between two time intervals) because like you said, that velocity changes for every time, correct? Just wanted to make sure I follow what you're telling me.

Thanks!

 

[elegysix]

It sounds like you've got it right.
I'll provide an example for you.
suppose you have velocities v=(1,2,3,4) at times t=(1,2,3,4)
the average velocity over the whole time is 2.5.
The average velocity over the first 3 seconds is 2.
The instantaneous velocities are only defined at a point, so for instance at t=4, the instantaneous velocity is 4.

make sense?

 

[Aristotle]

It sounds like you've got it right.
I'll provide an example for you.
suppose you have velocities v=(1,2,3,4) at times t=(1,2,3,4)
the average velocity over the whole time is 2.5.
The average velocity over the first 3 seconds is 2.
The instantaneous velocities are only defined at a point, so for instance at t=4, the instantaneous velocity is 4.

make sense?

Thanks you're the best!

 

[haruspex]

if we take the instant angular velocity at a single time it wouldn't equal with the average angular velocity (between two time intervals)

Right, except there is certain to be some instant in the interval at which the instantaneous velocity equals the average over the interval (mean value theorem).