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 the center of rotation. It is perpendicular to linear velocity [latex]v[/latex] and has magnitude [latex]\phantom{\rule{0.25em}{0ex}}{F}_{\text{c}}={\text{ma}}_{\text{c}}\text{,}[/latex]
which can also be expressed as
[latex]\begin{array}{c}{F}_{\text{c}}=m\frac{{v}^{2}}{r}\\ \begin{array}{}\text{or}\\ {F}_{\text{c}}=\text{mr}{\omega }^{2}\end{array}\end{array},[/latex]
7.5 Relating Angular and Translational Quantities
- The linear kinematic equations have their rotational counterparts such that there is a mapping [latex]x\to \theta ,v\to \omega ,a\to \alpha[/latex].
- A system undergoing uniform circular motion has a constant angular velocity, but points at a distance r from the rotation axis have a linear centripetal acceleration.
- A system undergoing nonuniform circular motion has an angular acceleration and therefore has both a linear centripetal and linear tangential acceleration at a point a distance r from the axis of rotation.
- The total linear acceleration is the vector sum of the centripetal acceleration vector and the tangential acceleration vector. Since the centripetal and tangential acceleration vectors are perpendicular to each other for circular motion, the magnitude of the total linear acceleration is [latex]|\overset{\to }{a}|=\sqrt{{a}_{\text{c}}^{2}+{a}_{\text{t}}^{2}}[/latex].
7.6 Fictitious Forces and Non-inertial Frames: The Coriolis Force
- Rotating and accelerated frames of reference are non-inertial.
- Fictitious forces, such as the Coriolis force, are needed to explain motion in such frames.
7.7 Kinematics of Rotational Motion
- Kinematics is the description of motion.
- The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
- Starting with the four kinematic equations we developed in Chapter 2, we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in the Rotational Kinematic Equations table.
- In these equations, the subscript 0 denotes initial values ([latex]{x}_{0}[/latex] and [latex]{t}_{0}[/latex] are initial values), and the average angular velocity [latex]\stackrel{-}{\omega }[/latex] and average velocity [latex]\stackrel{-}{v}[/latex] are defined as follows:
[latex]\overline{\omega }=\frac{{\omega }_{0}+\omega }{2}\text{ and }\overline{v}=\frac{{v}_{0}+v}{2}.[/latex]