Summarizing Rotation Angle and Angular Velocity

Summary

• Uniform circular motion is motion in a circle at constant speed. The rotation angle \(\text{Δ}\theta \) is defined as the ratio of the arc length to the radius of curvature:
  \(\text{Δ}\theta =\frac{\text{Δ}s}{r}\text{,}\)

  where arc length \(\text{Δ}s\) is distance traveled along a circular path and \(r\) is the radius of curvature of the circular path. The quantity \(\text{Δ}\theta \) is measured in units of radians (rad), for which

  \(2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}=\text{360º}\text{= }1\text{ revolution.}\)
• The conversion between radians and degrees is \(1\phantom{\rule{0.25em}{0ex}}\text{rad}=\text{57}\text{.}3\text{º}\).
• Angular velocity \(\omega \) is the rate of change of an angle,
  \(\omega =\frac{\text{Δ}\theta }{\text{Δ}t}\text{,}\)

  where a rotation \(\text{Δ}\theta \) takes place in a time \(\text{Δ}t\). The units of angular velocity are radians per second (rad/s). Linear velocity \(v\) and angular velocity \(\omega \) are related by

  \(v=\mathrm{r\omega }\text{ or }\omega =\frac{v}{r}\text{.}\)

Glossary

arc length

\(\text{Δ}s\), the distance traveled by an object along a circular path

pit

a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD

rotation angle

the ratio of the arc length to the radius of curvature on a circular path:

\(\text{Δ}\theta =\frac{\text{Δ}s}{r}\)

radius of curvature

radius of a circular path

radians

a unit of angle measurement

angular velocity

\(\omega \), the rate of change of the angle with which an object