Deriving Relation between Angular & Translational Velocities in Circular Motion

[platonic]

relationship of angular velocy and translational velocity in simple circular motion??

Like if something is going in a circle of radiues R at speed v with angular velocity w, then w=v/R. Where does this relation come from? I'm having trouble deriving it. Please help!

EDIT:

I think i got it. if period is T, then w=(2pi rad)/T and v=(2piR)/T, solve for T in each, equate them, and you get the relation.

 

[tiny-tim]

hi platonic!

(have a pi: π and an omega: ω )

it's much more fundamental than that …

draw two lines of length R from the same point, at a very small angle of θ …

then divide the distance between the endpoints by the time ​

(also works for arc-distance and for angular acceleration)

 

[platonic]

hi platonic!

(have a pi: π and an omega: ω )

it's much more fundamental than that …

draw two lines of length R from the same point, at a very small angle of θ …

then divide the distance between the endpoints by the time ​

(also works for arc-distance and for angular acceleration)

I don't get this explation at all, please clear it up for me!

 

[arildno]

Hi,.platonic!
First off, RELAX!

Your own thinking is very good, and what you have derived is the correct relationship, under the simplifying assumption that both speed&angular velocities are CONSTANTS over time.

But, how should you proceed for an angular velocity varying over time?
Essentially, that is what tiny-tim hinted at.

Note that in this case, the relationship between period and velocities you used simply do not hold.

However, by the method indicated by tiny-tim, you can prove that the relationship w=v/R still holds, it is in a sense a DEEPER relationship, since it is independently valid of any restrictions of constancy of the quantities over time.

 

[platonic]

I definitely do see why it's a deeper relationship since what I did assumed constant velocities. But I don't quite understand the method.

 

[arildno]

Well, tiny-tim can explain his method to you, I'll give you a rougher argument:

look at a tiny time interval, say from instant T to T+dT, where dT is tiny.

Let dT be sufficiently tiny so that the velocity V(t) is roughly constant int that interval, i.e V(T) is approximately equal to V(T+dT) (and every instant in between those instants). Call that velocity V.

In the same time interval, moving along a circular arc of angular width dA, the actual distance traversed is R*dA, where R is the radius.

Thus, we get:
V*dT=R*dA, or V=R*(dA/dT), where we recognize dA/dT as the angular velocity W, i.e, we have the relationship V=R*WNote that this ALSO uses (effectively) constant velocity, but justifies this by making the time interval so tiny that constancy is necessarily valid for that time period.

That is quite different from assuming constant velocity for the whole orbital period as you did, but reaches the same result.

 

[platonic]

Okay I've got it. Thanks for the clarification.