12: Non-inertial Reference Frames
- Page ID
- 9627
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This chapter will analyze the behavior of dynamical systems in accelerated frames of reference, especially rotating frames such as on the surface of the Earth. Newtonian mechanics, as well as the Lagrangian and Hamiltonian approaches, will be used to handle motion in non-inertial reference frames by introducing extra inertial forces that correct for the fact that the motion is being treated with respect to a non-inertial reference frame. These inertial forces are often called fictitious even though they appear real in the non-inertial frame. The underlying reasons for each of the inertial forces will be discussed followed by a presentation of important applications.
- 12.1: Introduction to Non-inertial Reference Frames
- Inertial frames of reference make it possible to use either Newton’s laws of motion, or Lagrangian, or Hamiltonian mechanics, to develop the necessary equations of motion. There are certain situations where it is much more convenient to treat the motion in an acceleratory non-inertial frame of reference.
- 12.2: Translational acceleration of a reference frame
- Reaction to translational acceleration.
- 12.3: Rotating Reference Frame
- Rotating non-inertial reference frames are used extensively to describe motion on Earth and other rotating bodies.
- 12.4: Reference Frame Undergoing Rotation Plus Translation
- Reaction to translation plus rotation.
- 12.5: Newton’s Law of Motion in a Non-Inertial Frame
- Derivation of the equations of motion in a rotating frame.
- 12.6: Lagrangian Mechanics in a Non-Inertial Frame
- The above derivation of the equations of motion in the rotating frame is based on Newtonian mechanics. Lagrangian mechanics provides another derivation of these equations of motion for a rotating frame of reference by exploiting the fact that the Lagrangian is a scalar which is frame independent, that is, it is invariant to rotation of the frame of reference.
- 12.7: Centrifugal Force
- The centrifugal force is experienced when riding in a car driven rapidly around a bend. The passenger experiences an apparent centrifugal force that thrusts them to the outside of the bend relative to the inside of the turning car. In reality, relative to the fixed inertial frame, i.e. the road, the friction between the car tires and the road is changing the direction of the car towards the inside of the bend and the car seat is causing the centripetal acceleration of the passenger.
- 12.8: Coriolis Force
- An important non-inertial force in a rotating frame.
- 12.9: Routhian Reduction for Rotating Systems
- The Routhian reduction technique is a hybrid of Lagrangian and Hamiltonian mechanics that exploits the advantages of both approaches for solving problems involving cyclic variables.
- 12.11: Free Motion on the Earth
- The calculation of trajectories for objects as they move near the surface of the earth is required for many applications. Such calculations require inclusion of the non-inertial Coriolis force.
- 12.12: Weather systems
- Weather patterns in rotating reference frame.
- 12.13: Foucault pendulum
- Spherical pendulum in rotating frame.
- 12.S: Non-inertial reference frames (Summary)
- This chapter has focussed on describing motion in non-inertial frames of reference. It has been shown that the force and acceleration in non-inertial frames can be related using either Newtonian or Lagrangian mechanics by introducing additional inertial forces in the non-inertial reference frame.
Thumbnail: This low-pressure system over Iceland spins counterclockwise due to balance between the Coriolis force and the pressure gradient force. (Public Domain; NASA’s Aqua/MODIS satellite).