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IUPhysicsP201F2009
Assignment 7a
Due at 12:00pm on Tuesday, October 21, 2008
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The ParallelAxis Theorem
Description:
Contains several simple examples demonstrating that the parallelaxis theorem correctly predicts the
values of the moment of inertia.
Learning Goal:
To understand the parallelaxis theorem and its applications
To solve many problems about rotational motion, it is important to know the moment of inertia of each object
involved. Calculating the moments of inertia of various objects, even highly symmetrical ones, may be a lengthy and
tedious process. While it is important to be able to calculate moments of inertia from the definition (
),
in most cases it is useful simply to recall the moment of inertia of a particular type of object. The moments of inertia
of frequently occurring shapes (such as a uniform rod, a uniform or a hollow cylinder, a uniform or a hollow sphere)
are well known and readily available from any mechanics text, including your textbook. However, one must take into
account that an object has not one but an infinite number of moments of inertia. One of the distinctions between the
moment of inertia and mass (the latter being the measure of tranlsational inertia) is that the moment of inertia of a
body depends on the axis of rotation. The moments of inertia that you can find in the textbooks are usually calculated
with respect to an axis passing through the center of mass of the object. However, in many problems the axis of
rotation does not pass through the center of mass. Does that mean that one has to go through the lengthy process of
finding the moment of inertia from scratch? It turns out that in many cases, calculating the moment of inertia can be
done rather easily if one uses the parallelaxis theorem. Mathematically, it can be expressed as
,
where
is the moment of inertia about an axis passing through the center of mass,
is the total mass of the
object, and
is the moment of inertia about another axis, parallel to the one for which
is calculated and located a
distance
from the center of mass. In this problem you will show that the theorem does indeed work for at least one
object: a dumbbell of length
made of two small spheres of mass
each connected by a light rod (see the figure).
NOTE: Unless otherwise noted, all axes considered are perpendicular to the plane of the page.
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10/10/2008
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View Full DocumentPart A
Using the definition of moment of inertia, calculate
, the moment of inertia about the center of mass, for this
object.
Express your answer in terms of
and
.
Hint A.1
Location of the center of mass
The center of mass is halfway between the two spheres, at point A (see the figure).
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 Spring '09
 physics
 Physics

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