An Introduction to Modern Astrophysics · Problem 19.22

The Planetary System — Carroll & Ostlie

Physics — Astrophysics · coriolis-force-components

Problem

Suppose that a ball of mass

m

is thrown with a velocity

\mathbf{v} = v_x \hat{\mathbf{i}} + v_y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}},

where

\hat{\mathbf{i}}

\hat{\mathbf{j}}

, and

\hat{\mathbf{k}}

are unit vectors pointing directly east, north, and upward, respectively, at the point where the ball is thrown. The latitude of the ball is

L

when it is thrown. (a) Show that the components of the Coriolis force on the ball are given by

\mathbf{F}_C = -2m\omega[(v_z \cos L - v_y \sin L)\hat{\mathbf{i}} + v_x \sin L\, \hat{\mathbf{j}} - v_x \cos L\, \hat{\mathbf{k}}].

Hint: Be sure to represent the components of the vector

\boldsymbol{\omega}

in terms of the coordinate system on the surface of Earth defined by

\hat{\mathbf{i}}

\hat{\mathbf{j}}

, and

\hat{\mathbf{k}}

, with the origin of the system at the position where the ball was thrown. (b) What is the value of

\omega

for Earth? (c) If the ball is thrown eastward with an initial velocity vector

\mathbf{v} = 30

m s

^{-1}\hat{\mathbf{i}}

on the surface of Earth at a latitude of

40^\circ

, what are the components of the acceleration vector that are due to the Coriolis force? (d) If the ball is thrown northward with

\mathbf{v} = 30

m s

^{-1}\hat{\mathbf{j}}

, what are the components of the acceleration vector? (e) If the ball is thrown straight up with

\mathbf{v} = 30

m s

^{-1}\hat{\mathbf{k}}

, what are the components of the acceleration vector? Give a simple physical explanation for the result.