Environmental Engineering Reference
In-Depth Information
10.4 A rectangular plate of mass
M
has length 2
a
and width
a
. Determine the
moment of inertia tensor for rotations about a corner. Use a Cartesian
co-ordinate system in which the
x
1
axis is aligned with the long axis of
the plate, the
x
3
axis is perpendicular to the plane of the plate and the
x
2
axis is parallel to the edge that has length
a
.
In the same co-ordinate system the plate rotates at constant angular velocity
ω
√
2
ω
=
(
1
,
1
,
0
).
Determine the magnitude of the angular momentum of the plate, and hence
the torque needed to maintain the rotation. Calculate the rotational kinetic
energy.
10.5 Determine the principal axes and principal moments of inertia for rotations
about a corner of the solid rectangular plate in the previous problem.
10.6 A solid uniform cylinder of density
ρ
has radius
R
and height
h
. Use sym-
metry to deduce the principal axes for rotations about the centre-of-mass.
Calculate the corresponding principal moments of inertia.
A cylindrical artillery shell of radius 0.1 m and length 0.4 m is fired from
a gun into the air. Estimate the rate of precession of the symmetry axis if
the barrel of the gun imparts a spin of 50 revolutions per second about the
symmetry axis of the shell.
10.7 A rotating thin circular disc, moving through a fluid, is subject to a damping
torque about its centre of mass that is given by
τ
1
=−
κω
1
,
τ
2
=−
κω
2
,
τ
3
=
0
,
where
e
1
and
e
2
are the principal axes that lie in the plane of the disc,
e
3
is
the symmetry axis and
ω
is the angular velocity. Use Euler's equations with
iω
2
to determine the time-dependence of
ω
1
+
ω
2
.
the substitution
η
=
ω
1
+
Describe the motion of the disc in the lab frame.
10.8 Show that for a general rigid body the rate of change of rotational kinetic
energy can be expressed as
d
T
d
t
=
ω
·
τ
.
10.9 Show that the total angular momentum
L
of a system of particles may be
written as
×
R
L
=
M
R
+
L
c
,
where
R
is the position vector of the centre-of-mass,
M
is the total mass of
the system and
L
c
is the angular momentum of the system relative to the
centre-of-mass.
