Previous posts in this series assumed that the hole went from pole to pole, and ignored such complications as tidal effects and wobbles of the axis of rotation.  In this post we will examine the effect of moving the hole to the equatorial plane, so the ends of the hole (and anything dropped down them) have a significant velocity with respect to the centre.  The up-dated spreadsheet (including full open source code) may be downloaded from ODE-Buckle.zip.

But first, where can we place the ends of the hole?  There are actually very few suitable locations with land close to sea level at both ends.  The location I have chosen is shown below, with ends on the coast of Ecuador, and close to the West coast of Sumatra (found with the aid of antipodemap.com):

Ends of the Earth - at the Equator

The ODE to be solved in this case is shown below.  It is very similar to ODEFunc4 (for a hole along the Polar Axis) , except the position, velocity and acceleration values are now each split into X and Y components, with X being along a line through the centre of the Earth on a non-rotating axis, and Y being distance in the perpendicular direction, on the Equatorial plane. 

ODE for hole through the centre of the Earth on the Equatorial plane

  The Y offset and acceleration in the Y direction are initially set to zero.  In order to check the output the Y velocity was set to orbital velocity.  The output results show a path following the surface of the Earth exactly, indicating that the ODE solution to the problem is providing accurate results:

Input and Output for a body released at the surface with orbital velocity

Plot of orbital velocity results

Finally the initial Y velocity was set to the surface of the Earth (463 m/s), modelling the situation when an object is dropped into a vertical hole at the Equator, with no horizontal velocity, relative to the Earth’s surface.  The results are shown below.  The blue line shows the path that would be followed if the object was unconstrained, and the green line the path followed if the object was constrained by the frictionless sides of a rotating hole.

Path for object dropped with zero velocity relative to the Earth's surface.