Falling Body On Rotating
Earth (3D)
General Description
This simulations illustrates the trajectory of an object dropped from
rest (relative to the rotating Earth) from a point near Earth's
surface. The user can set the initial height of the object above (or
below) the Earth's surface, as well as the rotational speed of Earth.
The
motion of the object can be viewed from an inertial reference frame,
or from a frame that rotates with the Earth. The simulation can be
made to stop when the object hits Earth's surface, or it can allow the
object to move through the Earth.
A variety of
models can be used to determine the forces on the object once it
passes into the Earth. The gravitational forces can be modeled by
treating the Earth as a uniform solid sphere, by treating the Earth as
a point mass, or by treating the Earth as the source of a constant
magnitude attractive force pointing toward Earth's center (with
magnitude equal to Earth's surface gravity). There are also options
to include a resistive force proportional to the object's speed,
proportional to the square of its speed, or no resistive force. The
user can adjust the coefficient of proportionality for the resistive
force (linear or quadratic).
In this 3D version of the simulation the object can be dropped from
any latitude. The simulation can define "up" either as directly away
from Earth's center, or as the direction of a plumb line (which
includes the centrifugal effect of the rotating Earth). The
simulation can also display plots of the object's longitude and
latitude as functions of time.
This simulations illustrates the Eastward deflection of a falling body
on the rotating Earth (now attributed to the Coriolis effect in
a non-inertial reference frame). It also shows that although the
object's motion is confined to a single plane in the inertial frame,
it is NOT confined to a single plane in the rotating frame. The
simulation also serves to illustrate
certain aspects of a historical debate between Robert Hooke and Isaac
Newton over the path of a body falling through the Earth.
Rotating Frame
This frame shows the motion of the object in a reference frame that
rotates with Earth. This frame also includes all of the controls for
the simulation.
- General Options Menu
- Show Center of Earth: select to display a black
dot at the Earth's center (both views).
- Show Trace of Object's Path: select to display the
trajectory of the object in this frame (both views).
- Show Earth's Surface: select to show a partially
transparent sphere representing the Earth.
- Pass Through Earth: select to allow the object
to pass through the Earth.
- Show View from Inertial Frame: select to show a
separate frame that shows the motion as seen by an inertial
observer.
- Use True Up/Down: select to define "up" as
directly away from Earth's center (otherwise the simulation
uses the direction of a plumb line, which includes the
centrifugal effect of Earth's rotation).
- Show Inertial Plane: show a disk defining the plane in the
inertial reference frame to which the object's motion is
confined.
- Show Plot of Latitude vs. Time: select to show
a plot of the object's latitude as a function of time.
- Show Plot of Longitude vs. Time: select to show
a plot of the object's longitude as a function of time.
- Earth Model Menu
Note: these options only affect the motion of the object if it
is passing through Earth. When the object is outside
of Earth's surface, the Earth is treated as a uniform
sphere or, equivalently, as a point mass.
- Uniform Density Sphere: select to treat the Earth
as a solid sphere of uniform density throughout.
- Point Mass: select to treat the Earth as a point
mass located at the center.
- Constant Force: select to treat the Earth as a
source of a constant magnitude force directed toward the
center. The magnitude is the same as at Earth's surface.
- Resistance Menu
Note: these options only affect the motion of the
object if it is passing through Earth. There is no
resistive force on the object when it is outside of
Earth's surface.
- None: select to include no resistive force on the
object.
- Linear: select to include a resistive force on
the object that is proportional to the object's speed
and points opposite its velocity relative to the point
of Earth through which it is passing.
- Quadratic: select to include a resistive force
on the object that is proportional to the square of
the object's speed and points opposite its velocity
relative to the point of Earth through which it is passing.
- Visual Elements
- Blue semi-transparent sphere: the Earth.
- Blue lines: lines of constant longitude/latitude.
- Green line: plumb line from the object to the
point on the Earth's surface directly below (or above)
the object. Note that "below" is defined according to
the way "up" is defined in the menu items.
- Black dot: the center of Earth.
- Red dot: the object.
- Red trail: the trace of the object's trajectory.
- Pink disk: the plane in the inertial frame to
which the object's motion is confined.
- Magenta line: a line in the inertial plane from
Earth's center to the point on Earth's surface
directly below (or above) the object.
Inertial Frame
This frame shows the motion of the object in an inertial reference
frame.
- Visual Elements
- Blue semi-transparent sphere: the Earth.
- Blue lines: lines of constant longitude/latitude.
- Green line: plumb line from the object to the
point on the Earth's surface directly below (or above)
the object. Note that "below" is defined according to
the way "up" is defined in the menu items.
- Black dot: the center of Earth.
- Red dot: the object.
- Red trail: the trace of the object's trajectory.
- Pink disk: the plane in the inertial frame to
which the object's motion is confined.
- Magenta line: a line in the inertial plane from
Earth's center to the point on Earth's surface
directly below (or above) the object.
Latitude Frame
This frame displays a plot of the object's latitude (in degrees) as a
function of time (in seconds).
- Visual Elements
- Red trail: plot of the object's latitude as a
function of time. Positive latitudes indicate
degrees North of the equator, negative
latitudes indicate degrees South of the equator.
Longitude Frame
This frame displays a plot of the object's longitude (in degrees) as a
function of time (in seconds).
- Visual Elements
- Red trail: plot of the object's longitude as a
function of time. Positive longitudes indicate
degrees East of the starting point, negative
longitudes indicate degrees West of the starting point.
Controls (in Inertial Frame)
- Buttons
- Play/Pause: to play or pause the
simulation.
- Step: to advance the simulation by one time
step.
- Reset: to reset the simulation to its
initial state.
- Erase: to erase all trails, etc.
- Initialize: to reset the simulation to
using the current parameter values.
- Sliders
- Height: sets the initial height of the
object above (positive) or below (negative) the
Earth's surface (in units of Earth radii).
- Rotations: sets the number of rotations
the Earth performs every 24 hours.
- Latitude: sets the latitude of the launch
point (the point on Earth's surface directly below, or above, the
object's initial location).
- Resistance: sets the constant of
proportionality for the resistive force (linear or
quadratic). Higher values produce stronger resistive
forces. (Units depend on the type of resistive force
used, but generally lengths are in Earth radii and times are
in seconds.)
Todd K. Timberlake (ttimberlake@berry.edu)