For linear motion, Newton's second law relates the acceleration of a
particle of mass *m * to the force *F * applied to it. In
symbols,

*F = m a *.

From the equation (and common sense) it is clear that a larger force is required to accelerate a more massive particle. Intuitively, the mass of the particle resists acceleration. We say that the mass gives the particle linear inertia.

For angular (rotational) motion, the angular acceleration depends not
only on the applied force, but also on where the force is applied in
relation to the axis of rotation (or "pivot point"). For this reason,
as we observed in the last section, it is the *torque * a force
generates which is most naturally related to the angular acceleration
of the particle.

Torque plays the same role in rotating systems that force plays in
linear motion. However, the resistance to angular acceleration
again depends on both the mass of the particle and its distance from the
axis of rotation. Therefore we introduce a new quantity called the
*moment of inertia * to measure resistance to angular
acceleration. The moment of inertia of a rotating system is analogous
to the mass of a linearly accelerating system.

Using Newton's second law, one can show
that the relationship between the
torque, *T *, and the angular acceleration, *A *, of a
particle is given by

*T = (m r *^{2}*) A *.

Thus, we define the moment of inertia to be the quantity
*mr *^{2}
of the particle. The moment of inertia is typically denoted by *
I *.

In general, computing the moment of inertia can be quite difficult, requiring the more sophisticated techniques of iterated integrals from multivariable calculus. However, when an object is sufficiently symmetrical, we can compute the moment of inertia with a standard one-variable integral.

Now consider a planar object as in
Figure 4, and imagine rotating it about the
*x *-axis. (Note it is rotating out of the *xy *-plane into
the third dimension perpendicular to your computer screen.)
This might represent, for example, a model of a rotating turbine blade
as it rotates around its central spindle.

In this case, note that the entire strip indicated in the figure
remains at a constant distance from the axis of rotation. Therefore,
this strip has the same moment of inertia as a particle of the same
mass lying at the same distance above the *x *-axis.

If we assume that the object has been cut out of some thin material of
uniform density *p *, the mass of the strip is simply *p
w(y) dy *. Consequently, it's moment of inertia is

*p y *^{2}* w(y) dy *.

It follows that the moment of inertia for the whole object is given by the formula

where *a * and *b * are the minimum and maximum
values of the *y * variable.

- Find the moment of inertia for a rectangle of length
*l*and width*w*when the axis of rotation is- the centerline of the rectangle, and
- one of the edges.

- In order to make a building wheelchair-accessible, an engineer is told to double the width of a revolving door. By how much will this increase the moment of inertia? What practical issues might the engineer need to worry about if the door width is doubled?

Answer the following questions by interacting with the Exploring Moments of Inertia page.

- Experimentally determine the moment of inertia of a C-beam around
various horizontal axes of rotation. Record your
data in a table.
- Where does the minimum moment of inertia occur? Conjecture a
relationship between this point and other geometric properties of the
object.
- If you graph your data, it should look like a familiar function:
a quadratic function! Try to fit your data with
a quadratic function of the form
*M(y-C)*^{2}+*I_min*. - Try to give physical interpretations to the constants
*M*and*C*for your model function. In particular, we would expect from physical considerations that the moment of inertia depends on the object's mass and on its shape. How do these quantities relate to the model function you found?

The Geometry Center Calculus Development Team

Copyright 1996 by The Geometry Center. Last modified: Fri Apr 12 15:53:58 1996