0.1 4.2 rotation angle and angular velocity  (Page 2/8)

From $\text{Δ}\theta =\frac{\text{Δ}s}{r}$ we see that $\text{Δ}s=r\text{Δ}\theta$ . Substituting this into the expression for $v$ gives

We write this relationship in two different ways and gain two different insights:

The first relationship in $v=\mathrm{r\omega }\text{ or }\omega =\frac{v}{r}$ states that the linear velocity $v$ is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest $r$ ), as you might expect. We can also call this linear speed $v$ of a point on the rim the tangential speed . The second relationship in $v=\mathrm{r\omega }\text{ or }\omega =\frac{v}{r}$ can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed $v$ of the car. See [link] . So the faster the car moves, the faster the tire spins—large $v$ means a large $\omega$ , because $v=\mathrm{r\omega }$ . Similarly, a larger-radius tire rotating at the same angular velocity ( $\omega$ ) will produce a greater linear speed ( $v$ ) for the car.

Both $\omega$ and $v$ have directions (hence they are angular and linear velocities , respectively). Angular velocity has only two directions with respect to the axis of rotation—it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in [link] .

Section 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

  $\mathrm{2\pi }\text{rad}=\text{360º}\text{= }1\text{ revolution.}$
• The conversion between radians and degrees is $1\text{rad}=\text{57}\text{.}3\text{º}$ .
• 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

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

Conceptual questions

Problem exercises