Problem GPS 4.20 (2
nd
ed. 49)
In general, we would need six generalized coordinates to describe the motion
of the sphere. Three of these would be for the center of mass (
X
,
Y
,
Z
) and three
for the orientation of a body
fi
xed frame rotating with the sphere (the three Euler
angles). Because the sphere is con
fi
ned to a plane, one COM coordinate, say
Z
,
is always a constant.
So our task comes down to
fi
nding suitable relationships
among (
X
,
Y
,
ψ
,
θ
,
φ
) to govern the constraint of rolling without slipping.
Let’s start by imagining that we are sitting at the origin of a space
fi
xed
Cartesian frame (SFF) watching the sphere rolling in the
x
−
y
plane located at
z
= 0
.
The COM vector
R
runs from the origin of the SFF to the center of
the sphere.
Another vector
r
runs from the center of the sphere to its surface.
This vector is
fi
xed in the sphere, so it rotates as the sphere rolls.
Finally, a
third vector
r
0
connects the SFF origin with the end of vector
r
on the sphere’s
surface.
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 Fall '10
 Wilemski
 mechanics, Derivative, Center Of Mass, Mass, Ω, Eqs., sin θ cos

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