Re: [Phys-L] gravity along a chord
I haven't a clue whether this has been done in AJP, though I suspect it has.
There is no need to consider mass "above" the chord at all (if by this you
mean mass outside the radius of the particular location on the chord) - it
is only necessary to consider the mass "below" the chord (inside the radius
of the particular location on the chord). You can determine by this the
magnitude of the gravitational force on the sled/train/car/object that's
travelling in the chord at any given point along the chord - you'll need to
determine the relationship between r and delta x, the displacement measured
from the center of the chord. You'll then find, through simple
trigonometry that the along-the-chord component of the force has the form
of a simple harmonic oscillator force - it looks like a negative constant
times delta x - so that delta x is the relevant independent variable.
Gravity does "assist" in this way (and interestingly the calculated period
of the motion will be the same as that for the tunnel through the center of
the Earth (and that for an orbit of a satellite at just above the Earth's
surface, too, ignoring air resistance). Naturally in all this one has to
posit a chord-tunnel that has slippery insides such that there are no or
negligible frictional losses.
If the chord is only 2 miles deep, the "gravitational assist" force along
the chord will be quite small, though - but it is calculable and finite.
Todd
On Thu, Sep 25, 2014 at 8:56 AM, Peter Schoch <pschoch@fandm.edu> wrote:
> We just did gravity in my calc-based physics class. As part of that, I did
> the example of a tunnel directly through the diameter of the planet, along
> the rotation axis.
>
> I have a student who wants to do a similar problem -- gravity through a
> 'tube' which is along a chord of the planet (assuming the planet is a
> sphere, he wants to go 2 miles deep and go point to point. He thinks the
> calculation will show that gravity will assist in the motion and make it a
> viable transportation alternative.
>
> I was almost certain I had sen this done in either the Physics Teacher or
> the AJP, but I can't find it after a morning of searching. Does anyone know
> if this has been done already?
>
> If not, am I right in being able to tell him to just find the CM of the
> portion "below" the chord, and the CM "above" the chord, and just find the
> effects due to both of them at each point along the chord?
>
> Thank you,
> Peter Schoch
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--
Todd K. Pedlar
Associate Professor of Physics
Luther College, Decorah, IA
pedlto01@luther.edu