9716

Planetary Tunnels with Coriolis Effect

If you were to dig a tunnel into a planet, what shape would the tunnel have to be so that when you drop an object into it, the object's fall is unobstructed? The object will not necessarily fall in a straight line because of Coriolis forces. This Demonstration shows the path needed for the tunnel under various planetary conditions, including planet radius, surface gravity, rotational period, and initial starting latitude.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The following equation gives the general form for the relative motion of a particle between an inertial reference frame and a rotating one:
.
Here is the acceleration of the particle with respect to the inertial frame, is the acceleration of the rotating frame's origin with respect to the inertial frame, is the angular velocity of the rotating frame about its origin, and is the position vector of the particle in the rotating frame.
Using Gauss's law applied to gravity, it can be shown that for , where is the gravitational acceleration at the surface of the planet and is the radius of the planet. In detail, , where , the effective attractive mass of the planet inside a sphere of radius (). Also, note that . Given that position is a function of time, is zero, and is constant, the appropriate substitutions can be made into the above equation to give
.
Solving this differential equation for gives the path for the falling object with respect to the rotating reference frame of the planet.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+