Hi,
I am researching the problem of determining mathematically when a sphere
rolling down a slope of elliptical cross-section will fall off.
I have solved the problem for a sphere rolling off a slope of circular
cross-section. It is also discussed in many textbooks.
I have made some headway with the elliptical case, but it is a much tougher
problem.
Does anyone know of any journal article(s) or texts that investigate this
question?
Any help much appreciated :-)
--
Cheers,
Brad
Give me a fast ship... for I intend to go in harm's way.

A little more information is needed here... what exactly do you mean
by "fall off"? I'm assuming you're talking about a dynamic case where
the sphere picks up enough horizontal momentum to no longer have
contact with the elipse despite gravitational acceleration? Or are
you talking about some other scenario?
If the former is the case, it's going to be a function of the
geometery of the elipse... specifically the derivative of the slope of
the contact surface (2nd derivative of the ellipse surface itself).
The point at which gravitational acceleration downward is lesser in
magnitude than the derivative of the slope of the ellipse (in relation
to the horizontal speed of the sphere) is where the sphere will being
free-fall.... (if I'm making the correct assumptions about your
problem).
Dave

A little more information is needed here... what exactly do you mean
by "fall off"? I'm assuming you're talking about a dynamic case where
the sphere picks up enough horizontal momentum to no longer have
contact with the elipse despite gravitational acceleration? Or are
you talking about some other scenario?
If the former is the case, it's going to be a function of the
geometery of the elipse... specifically the derivative of the slope of
the contact surface (2nd derivative of the ellipse surface itself).
The point at which gravitational acceleration downward is lesser in
magnitude than the derivative of the slope of the ellipse (in relation
to the horizontal speed of the sphere) is where the sphere will being
free-fall.... (if I'm making the correct assumptions about your
problem).
Dave

Hi Dave,
Thank you for taking the trouble to think about the problem.
I didn't have another scenario in mind - your interpretation is correct.
By falling off, I mean the sphere loses contact with the ellipse.
The traditional way to think about the problem of a sphere rolling off a
slope of circular cross-section is to resolve the forces normal to the
point of contact between the spheres and surface.
You get something along the lines...
mg sin(theta) - N = m v**2 /(R+r)
where R and r are the radii of the surface and sphere, m is the mass
of the sphere, N is the reaction force exerted by the surface on the
sphere etc.
Conservation of energy is used to find v in terms of theta and there
is a bit of geometry related to non-slippage of the rolling sphere.
Then it is just a matter of letting N=0, which is the condition for
the sphere to part company with the surface. It works out well.
In the case where a sphere is falling off an elliptical surface I was
using the same approach as for falling off a circular surface.
The R+r term above gets replaced by the radius of curvature
of the elliptical path taken by the center of the sphere.
This is a rather complicated formula.
The condition for non-slippage is complicated also.
Your idea of using the fall-off rate of the slope (2nd derivative) as
compared to g is intriguing. I will see where that leads.
Thanks Dave.
--
Cheers,
Brad
Give me a fast ship... for I intend to go in harm's way.

No problem!
I just figured an easier way would be to determine at what point the
velocity vector has a slope lesser in magnitude than the slope of the
ellipse surface. (ex. the ball is travelling at 3i - 4j m/s = slope
of -4/3... if the slope of the ellipse at that point is -5/3, then the
ball will loose contact).
Dave

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