Derivation of angular velocity using the unit circle

AI Thread Summary
The discussion focuses on deriving the angular velocity formula $$ \omega = \frac{Δα}{Δt} $$ using the unit circle and its relationship with sine and cosine functions. Angular velocity is defined as $$ \omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} $$, which can also be expressed in terms of polar coordinates. The connection to uniform circular motion is illustrated with the equations $$ x = R\cos(\omega t) $$ and $$ y = R\sin(\omega t) $$. There is a debate about whether a definition can be derived, but the emphasis is on understanding the formula's origins through the unit circle. The inquiry seeks clarity on how angular velocity relates to the unit circle's trigonometric functions.
jnuz73hbn
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Homework Statement
I am looking for a good derivation of $$ \ ω = \frac{Δα}{Δt} \ $$
, starting from the unit circle. My approach would be to first construct a right-angled triangle (Pythagorean theorem), then express $$ cos(α) $$ for the ankathete and $$ sin(α) $$ as the anticathete. Then I have a point on the arc of the circle (r=1). How do I get a suitable derivation for the initial formula of the angular velocity?
Relevant Equations
$$ \ Δα = α_2 - α_1 \ $$
$$ \ ω = \frac{Δα}{Δt} \ $$
 
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jnuz73hbn said:
Homework Statement: I am looking for a good derivation of $$ \ ω = \frac{Δα}{Δt} \ $$
That's essentiallly the definition, although usually ##\theta## or ##\phi## is used as the polar angle. The definition of angular velocity is $$\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$$See, for example:

https://en.wikipedia.org/wiki/Angular_velocity
 
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PeroK said:
That's essentiallly the definition, although usually ##\theta## or ##\phi## is used as the polar angle. The definition of angular velocity is $$\lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}$$See, for example:

https://en.wikipedia.org/wiki/Angular_velocity
however, i wanted to go via the unit circle with sinus and cosine to derive exactly this definition
 
jnuz73hbn said:
however, i wanted to go via the unit circle with sinus and cosine to derive exactly this definition
By definition, you can't derive a definition.
 
PeroK said:
By definition, you can't derive a definition.
I just want to know where the formula comes from using the unit circle or how it relates to sin cos in the unit circle
 
jnuz73hbn said:
I just want to know where the formula comes from using the unit circle or how it relates to sin cos in the unit circle
In plane polar coordinates, angular velocity ##\omega## is defined as ##\omega = \frac{d\theta}{dt}##. This means that, for example, uniform circular motion about the origin is given by:
$$x = R\cos(\omega t), \ y = R\sin(\omega t)$$
 
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