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I'm was watching the MIT open courseware classical mechanics and don't understand why the presenter did what she did to equate liner speed to rotational speed. I was wondering if anyone would be able to explain it. I get everything up to 3:42 and get lost when she starts to rearrange (w)

https://www.youtube.com/watch?v=q785KV5ZIN0&list=PLUl4u3cNGP61qDex7XslwNJ-xxxEFzMNV&index=57

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Unless I missed it, I didn't hear the lecturer say they are the same.

But you can connect the linear speed (magnitude of tangential velocity, $v$) to rotational speed (magnitude of angular velocity $\omega$) as follows:

$$\omega=\frac{d\theta}{dt}$$

$$v=r\frac{d\theta}{dt}$$

The differential tangential displacement, $ds$, is

$$ds=rd\theta$$ Or $$r=\frac{ds}{d\theta}$$

Substituting for $r$ in the second equation gives us

$$v=\frac{ds}{dt}$$

Hope this helps.

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  • $\begingroup$ very helpful thanks, One quick question thou, were do the unit vectors K^ and theta^ play into this $\endgroup$
    – CatsOnAir
    Oct 4, 2021 at 16:40
  • $\begingroup$ @CatsOnAir They come into play to define the rotational and tangential velocities as vectors. The $k$ unit vector is perpendicular to the plane of rotation because that's defined as the direction of the angular velocity. The $\theta$ unit vector is a unit angular vector that is in the plane of rotation, because that's the direction of the tangential velocity. $\endgroup$
    – Bob D
    Oct 4, 2021 at 17:16
  • $\begingroup$ but why don't the carry over to the mathematics?, like at 5:18 when she equates angler velocity to velocity, she drops the unit vectors and that's why I'm confused $\endgroup$
    – CatsOnAir
    Oct 4, 2021 at 17:50
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    $\begingroup$ I don't have the time to keep viewing the lecture, but the magnitude and direction of a vector are independent of one another. So you can treat the angular and tangential speeds without consideration of the unit vectors. $\endgroup$
    – Bob D
    Oct 4, 2021 at 17:54
  • $\begingroup$ she does not "drop" the unit vector, but speaks of the amount of the vector $|\vec{\omega}| and |\vec{k}|=1$ $\endgroup$
    – trula
    Oct 4, 2021 at 21:22
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She did not equate linear speed and rotational speed, she defined angular velocity as $\vec{\omega}=\frac{d\Theta}{dt}*\vec{k}$ and before defined the unit vector k by the right hand rool. with this definition you have the velocity as $|v|=r*|\omega|$.

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