# Connecting Linear and Rotational Velocty

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)

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.

• very helpful thanks, One quick question thou, were do the unit vectors K^ and theta^ play into this Oct 4, 2021 at 16:40
• @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. Oct 4, 2021 at 17:16
• 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 Oct 4, 2021 at 17:50
• 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. Oct 4, 2021 at 17:54
• she does not "drop" the unit vector, but speaks of the amount of the vector $|\vec{\omega}| and |\vec{k}|=1$ Oct 4, 2021 at 21:22

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 $$|v|=r*|\omega|$$.