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Measuring Angles

<aside> 💡 Radian: circle with radius $r$, and arc length $r$, enclosed angle $\theta$ is 1 radian.

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Since the circumference of a circle is $2\pi r$, there are $2\pi$ radii around the circle, hence $2\pi$ radians in a circle.

Equations of Angular Motion

Just as linear motion has equations, so has angular motion. These can be derived by noticing the following correspondences:

Angular Motion Quantities

Angular and Tangential Motion

Angular Displacement

Tangential displacement is related to angular displacement by:

$$ s = r\theta $$

where:

• $s$ is the tangential displacement, in $m$.
• $r$ is the radius, in $m$.
• $\theta$ is the angular displacement, in $radians$.

Angular Velocity

The angular velocity is defined as the rate of change of angular displacement, given by:

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

where:

• $\omega$ is the angular velocity, in $rad\,s^{-1}$.
• $\theta$ is the angular displacement, in $radians$.
• $t$ is the time, in $s$.

Tangential velocity is related to angular velocity by:

$$ v = r\omega $$

where:

• $v$ is the tangential velocity, in $ms^{-1}$.
• $r$ is the radius, in $m$.
• $\omega$ is the angular velocity, in $rad\,s^{-1}$.

Angular Acceleration