Problem:
Consider a right triangular lamina of areal density m, with one edge of length a along the x-axis and another edge of length b along the y-axis, as shown in the diagram.
(a) Find the center of mass (X,Y,Z) in this coordinate system.
(b) Find the components of the inertia tensor in this coordinate system.
(c) Transform the inertia tensor to principal axes, giving the angle between the
principal axes and those shown in the diagram, and find the three moments of inertia in
the principal axes system.
(d) Show that the parallel axis theorem applies to the inertia tensor, and find the
moments of inertia about the center of mass.
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Problem:
The expression for the angular momentum of a rigid body is
for a body made up of point masses, mk, situated at points rk relative to an origin of the coordinates at P.
(a) From this expression obtain the general expression for the inertia tensor
I.
(b) Show that I is symmetric.
(c) A thin disk of mass
M and radius a lies with its center at O, and
its axis in the y-z plane at 45o to the y-axis. Find the
inertia tensor in the given (x,y,z) coordinate system.
(d) If the disk is constrained to rotate about the
z-axis with an angular
velocity w, find the angular momentum vector
L in
the (x,y,z) coordinate system.
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Problem:
A top of mass M is spinning about a fixed point under gravity,
and its axis is vertical
but
the angular velocity around its axis w3 is
insufficient for stability in that position. The Lagrangian for a top is
where
q,
f, and y are the usual
Euler angles, I1 and I3 are the moments of inertia
about their respective axes, N is the line of nodes, and l is the distance
from the point of the top O to the center of mass C .
(a) Derive all the first integrals of the motion and evaluate them in terms of the given initial conditions.
(b) Show that the head will descend to an angle q
given by
(c) Show that the time dependence of this q
is
given by the solution of
You do not need to solve for
q(t).
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Problem:
A thin rectangular solid of sides 6a and 8a rotates freely in space.
(a) Find the moments of inertia along the three principal axes
going through its center of mass.
(b) Give the Euler equations for this system and use them to
describe qualitatively the subsequent rotational motion of the object in
the 3 cases when the initial angular velocity w
is almost, but not exactly, parallel to each of the principal axes.
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