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What is the relationship between linear speed and angular speed?

1 Answer

Jul 17, 2015

$v = ω R$ $v=omegaR$

Explanation:

Linear velocity $v$ $v$ is equal to the angular speed $ω$ $omega$ times the radius from the center of motion $R$ $R$ .

We can derive this relationship from the arclength equation
$S = θ R$ $S=thetaR$ where $θ$ $theta$ is measured in radians.

Start with
$S = θ R$ $S=thetaR$

Take a derivative with respect to time on both sides
$d \frac{S}{dt} = d \frac{θ}{dt} R$ $d S/"dt"=d theta/"dt"R$

$d \frac{S}{dt}$ $d S/"dt"$ is linear velocity and $d \frac{θ}{dt}$ $d theta/"dt"$ is angular velocity

So we're left with:
$v = ω R$ $v=omegaR$

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