21: Central Forces and Equivalent Potential
- Page ID
- 8537
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 21.1: Introduction to Central Forces
- This page explains the dynamics of a particle in a co-rotating reference frame, highlighting how it experiences a centrifugal force, contrasting it with the centripetal acceleration felt in inertial frames. The centrifugal force is conservative and related to a potential energy function.
- 21.2: Motion Under a Central Force
- This page covers the two-dimensional motion of a particle under a conservative central force in polar coordinates. It derives the motion equations relating radial acceleration and centripetal forces, highlighting constant angular momentum due to the lack of transverse forces. By using a co-rotating frame, the dynamics are simplified, leading to an effective force and potential energy function, \(V'\), which integrates the original potential with an angular momentum-related term.
- 21.3: Inverse Square Attractive Force
- This page explores celestial motion, focusing on gravitational forces and orbital dynamics. It outlines the relationship between force, potential energy, and kinetic energy, emphasizing the conservation of total energy. Orbits are classified based on total energy: negative energy leads to elliptic orbits with specific perihelion and aphelion distances, and positive energy indicates hyperbolic orbits with a lower limit.
- 21.4: Hooke’s Law
- This page explores the oscillatory motion of a particle attached to a spring, highlighting Hooke's law and the effective potential energy of the system. It illustrates motion through Lissajous ellipses and examines various force laws, including the inverse square law. This discussion aims to enhance understanding of complex natural interactions by investigating hypothetical forces.
- 21.5: Inverse Fourth Power Attractive Force
- This page explores the interplay between force, potential energy, and total energy in a central force field, detailing how negative total energy restricts the motion of a particle and may lead to collapse at the center. It contrasts this with positive total energy scenarios, where the motion can be either bounded or unbounded based on initial conditions, and highlights how the total energy influences the range of allowable distances, making some distances inaccessible.
- 21.6: A General Central Force
- This page explores the motion of a particle under a central force, detailing radial and transverse equations of motion. It defines key concepts like radial force per unit mass and angular momentum per unit mass, which are crucial for understanding the particle's path. By eliminating time from the equations, it establishes a relationship between radial distance and angle.
- 21.7: Inverse Cube Attractive Force
- This page explores the motion of a particle under an attractive force inversely related to the cube of the distance from a center. It derives a differential equation for the particle's path and examines how angular momentum affects motion, leading to circular, oscillatory, or diverging paths. Additionally, it discusses potential energy concepts and provides solutions to the motion equations for different levels of angular momentum.
Thumbnail: Gravitational field lines around the Earth are an example of a central force. (Public Domain; Sjlegg).