Chapter Summary
5.1 Rotation Angle and Angular Velocity
- Uniform circular motion is motion in a circle at constant speed. The rotation angle [latex]\Delta \theta[/latex] is defined as the ratio of the arc length to the radius of curvature:
[latex]\Delta \theta = \frac{\Delta s}{r} ,[/latex]
where arc length [latex]\Delta 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]\Delta \theta[/latex] is measured in units of radians (rad), for which [latex]2π \text{rad} = \text{360}º =\textrm{ } 1 \textrm{ }\text{revolution}.[/latex]
- The conversion between radians and degrees is [latex]1 \text{rad} = \text{57} . 3 º[/latex].
- Angular velocity [latex]\omega[/latex] is the rate of change of an angle, [latex]\omega = \frac{\Delta \theta}{\Delta t} ,[/latex] where a rotation [latex]\Delta \theta[/latex] takes place in a time [latex]\Delta 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 = rω \textrm{ }\text{or}\textrm{ } \omega = \frac{v}{r} .[/latex]
5.2 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} ; a_{\text{c}} = rω^{2} .[/latex]
- The unit of centripetal acceleration is [latex]\text{m} / \text{s}^{2}[/latex].
5.3 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]F_{\text{c}} = \text{ma}_{\text{c}} ,[/latex] which can also be expressed as
[latex]\left\begin{matrix} F_{\text{c}} = m \frac{v^{2}}{r} \\ \begin{matrix}\text{or} \\ F_{\text{c}} = \text{mr} \omega^{2}\end{matrix} , \end{matrix}\right}[/latex]
5.4 Newton’s Universal Law of Gravitation
- Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form, this is [latex]F = G \frac{\text{mM}}{r^{2}} ,[/latex] where F is the magnitude of the gravitational force. [latex]G[/latex] is the gravitational constant, given by [latex]G = 6 . \text{674} \times \text{10}^{–\text{11}} \text{N} \cdot \text{m}^{2} /\text{kg}^{2}[/latex].
- Newton’s law of gravitation applies universally.
“Section Summary” from College Physics 2e by OpenStax is licensed under a Creative Commons Attribution 4.0 International License.
