—
1
—
Problem.
Moment of inertia
The purpose of this project is to calculate the angular momentum of the Earth. We will model
the Earth as a solid uniform sphere with a mass density of
ρ
= 5500 kg
/
m
3
that rotates about its
north–south axis once per day. The average radius of the Earth is
a
= 6378 km
.
The angular momentum of a rotating object is
Iω
where
I
is the moment of inertia about the
rotation axis and
ω
is the angular frequency in
rad
/
s
. Computing
ω
is straightforward and involves
no calculus.
The moment of inertia is computed relative to the axis of rotation: it measures how difficult it is
to change the angular frequency of the object. The moment of inertia of a hollow cylinder, having
mass
M
and radius
r
, and rotating about its axis of symmetry, is
Mr
2
. (We assume the cylinder is
empty, with no top or bottom, and with its thickness very small relative to
r
.)
If an object lies within a distance
R
from the axis, and the differential
dM
describes the distribution
of the object’s mass with respect to distance from the axis, then the moment of inertia (of the object
about the axis, to be precise about what we’re measuring) is
I
=
R
0
r
2
dM.
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 Fall '07
 BERMAN
 Angular Momentum, Rotation, 5 rad, 106 m, 3 15 g, 8 5500 kg

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