46.1: Introduction to the Coriolis Force
- Page ID
- 92308
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Imagine you're on a rotating merry-go-round, and you throw a ball to another person who's on the opposite side of the merry-go-round. If you aim directly at the other person, you'll miss them - the ball will travel in a straight line relative to the ground, but the merry-go-round will have rotated during the time the ball is in the air. Relative to the merry-go-round, the ball will appear to move along a curved path. You can attribute this curvature to a "fictitious force" called the Coriolis force. The Coriolis force is not a real force-it's just an artifact of viewing the ball's motion in a rotating reference frame. The ball really moves in a straight line relative to the ground.
So in the rotating reference frame of the merry-go-round, you'll see the ball move in a curved path, which can't happen unless there is a "force" present. We can compute the magnitude of this Coriolis force by considering the following situation. Suppose you're at the center of the merry-go-round, and throw a ball outward with velocity \(v\) while the merry-go-round is rotating with an angular velocity \(\Omega\). After a time \(t\), the ball will have moved a radial distance \(r=v t\). At time \(t\), a point on the merry-go-round a distance \(r\) from the center will have moved through an arc length
\[s =r \theta \]
\[ =r(\Omega t) \]
\[ =(v t) \Omega t \]
\[ =\Omega v t^{2} .\]
But under a constant acceleration \(a_{c}\), we know
\[s=\frac{1}{2} a_{c} t^{2} .\]
Comparing Eq. \(\PageIndex{4}\) with Eq. \(\PageIndex{5}\), we deduce that the Coriolis acceleration \(a_{c}\) is given by
\[a_{c}=2 \Omega v \text {. }\]
More generally, in terms of vectors, the Coriolis acceleration vector \(\mathbf{a}_{c}\) is given by
\[\mathbf{a}_{c}=-2(\boldsymbol{\Omega} \times \mathbf{v})\]
From Newton's second law, the corresponding Coriolis force \(\mathbf{F}_{c}\) on a body of mass \(m\) is then
\[\mathbf{F}_{c}=-2 m(\boldsymbol{\Omega} \times \mathbf{v})\]