Motion inside a cone
A point mass moves on the interior surface of an upside-down cone.
Friction is negligible. The simulation allows displaying the mass motion,
as well of that of the equivalent one-dimensional problem.
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The unit mass is the particle mass m.
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The unit length is uL = (L2/m2g)1/3,
where L is the angular momentum of particle m around the
cone vertical axis.
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The unit time is uT = (L/mg2)1/3.
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In these units we have m = L = g = 1 and the
total mass is 1+M/m.
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Gravitational energy is 0 when m is in the hole.
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On the right the effective potential energy
m g r cos
α + 1/2 r2
sin2 α is displayed in red (α
is the cone half-opening angle). You may use the mouse (or the
controls below) to select the mechanical energy E and the
initial polar distance r.
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The point of view of the three-dimensional projection can be changed
with the mouse.
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Each whole image can be moved with the mouse while pressing Ctrl.
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To change the zoom in the projections, press Shift
when moving up or down the mouse pointer.
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Put the mouse pointer over an element to get the corresponding tooltip.