Chapter 7: Angular Kinematics
Section Summary
7.1 Rotation Angle and Angular Velocity
- Uniform circular motion is motion in a circle at constant speed. The rotation angle [latex]\text{Δ}\theta[/latex] is defined as the ratio of the arc length to the radius of curvature:
[latex]\text{Δ}\theta =\frac{\text{Δ}s}{r}\text{,}[/latex]
where arc length [latex]\text{Δ}s[/latex] is distance traveled along a circular path and [latex]r[/latex] is the radius of curvature of the circular path. The quantity [latex]\text{Δ}\theta[/latex] is measured in units of radians (rad), for which
[latex]2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}=\text{360º}\text{= }1\text{ revolution.}[/latex] - The conversion between radians and degrees is [latex]1\phantom{\rule{0.25em}{0ex}}\text{rad}=\text{57}\text{.}3\text{º}[/latex].
- Angular velocity [latex]\omega[/latex] is the rate of change of an angle,
[latex]\omega =\frac{\text{Δ}\theta }{\text{Δ}t}\text{,}[/latex]
where a rotation [latex]\text{Δ}\theta[/latex] takes place in a time [latex]\text{Δ}t[/latex]. The units of angular velocity are radians per second (rad/s). Linear velocity [latex]v[/latex] and angular velocity [latex]\omega[/latex] are related by
[latex]v=\mathrm{r\omega }\text{ or }\omega =\frac{v}{r}\text{.}[/latex]
7.2 Angular Acceleration
- Uniform circular motion is the motion with a constant angular velocity [latex]\omega =\frac{\Delta \theta }{\Delta t}[/latex].
- In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is [latex]\alpha =\frac{\Delta \omega }{\Delta t}[/latex].
- Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as [latex]{a}_{\text{t}}=\frac{\Delta v}{\Delta t}[/latex].
- For circular motion, note that [latex]v=\mathrm{r\omega }[/latex], so that
[latex]{a}_{\mathrm{\text{t}}}=\frac{\text{Δ}\left(\mathrm{r\omega }\right)}{\Delta t}.[/latex]
- The radius r is constant for circular motion, and so [latex]\mathrm{\text{Δ}}\left(\mathrm{r\omega }\right)=r\Delta \omega[/latex]. Thus,
[latex]{a}_{\text{t}}=r\frac{\Delta \omega }{\Delta t}.[/latex]
- By definition, [latex]\Delta \omega /\Delta t=\alpha[/latex]. Thus,
[latex]{a}_{\text{t}}=\mathrm{r\alpha }[/latex]
or
[latex]\alpha =\frac{{a}_{\text{t}}}{r}.[/latex]
7.3 Centripetal Acceleration
- Centripetal acceleration [latex]{a}_{\text{c}}[/latex] is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity [latex]v[/latex] and has the magnitude
[latex]{a}_{\text{c}}=\frac{{v}^{2}}{r};\phantom{\rule{0.25em}{0ex}}{a}_{\text{c}}={\mathrm{r\omega }}^{2}.[/latex]
- The unit of centripetal acceleration is [latex]\text{m}/{\text{s}}^{2}[/latex].
7.4 Centripetal Force
- Centripetal force [latex]{\text{F}}_{\text{c}}[/latex] is any force causing uniform circular motion. It is a “center-seeking” force that always points toward