Rotational KE and Moment of Inertia

29.4 Parallel Axis Theorem
29.4 Parallel Axis Theorem

A man said to the Universe: “Sir, I exist!”. “However,” replied the Universe, “the fact has not created in me a sense of obligation”.

Stephen Crane

For rotation about a single axis the kinetic energy of a system of particles can be written

But for all the
particles we can
write ; where ri
is the perpendicular distance of each particle from the axis and
all
particles
have the same angular velocity.
Therefore,

where we define the

Notice that I plays the same role as m (mass) in translational motion.

However,
unlike
mass, I does not have a simple fixed
value for rigid bodies. Its
value depends upon the axis of rotation (through the
ri).

If the system of particles is such that it may be considered continuous the summation definition of I becomes an integral,

where dm is an element of mass.

“The moment of inertia about any axis is equal to the moment of inertia about a parallel axis through the centre of mass plus the mass of the object times the square of the distance between the axes.”

The expressions for moment of inertia about axes through the centre of mass of many common objects are well known (see table in text). Application of the parallel axis theorem allows a determination of the moment of inertia about many other axes.

“A wizard is never late, nor is he early. He arrives precisely when he means to.”

Gandalf – The Lord of the Rings

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu

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