Beams, Bending, and Boundary Conditions: Moments of Inertia (2024)

Angular vs. Linear Motion

For linear motion, Newton's second law relates the acceleration of aparticle of mass m to the force F applied to it. Insymbols,

F = m a .

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

For angular (rotational) motion, the angular acceleration depends notonly on the applied force, but also on where the force is applied inrelation to the axis of rotation (or "pivot point"). For this reason,as we observed in the last section, it is the torque a forcegenerates which is most naturally related to the angular accelerationof the particle.

Torque plays the same role in rotating systems that force plays inlinear motion. However, the resistance to angular accelerationagain depends on both the mass of the particle and its distance from theaxis of rotation. Therefore we introduce a new quantity called themoment of inertia to measure resistance to angularacceleration. The moment of inertia of a rotating system is analogousto the mass of a linearly accelerating system.

Formulas for Systems and Continuous Objects

For a rigid configuration of particles, the moment of inertia issimply the sum of all the individual moments. For a continuousdistribution of mass, just as with the center of mass, we proceed bychopping the object into tiny elements of mass, and, for each element,add up the moment of inertia due to that mass. As before, theseapproximations to the moment of inertia converge to the truemoment.

In general, computing the moment of inertia can be quite difficult,requiring the more sophisticated techniques of iterated integrals frommultivariable calculus. However, when an object is sufficientlysymmetrical, we can compute the moment of inertia with a standardone-variable integral.

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

Figure 4: Slicing an irregular object into strips.

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

If we assume that the object has been cut out of some thin material ofuniform density p , the mass of the strip is simply pw(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 bythe formula

where a and b are the minimum and maximumvalues of the y variable.

Question 3

Find the moment of inertia for a rectangle of length l and width w when the axis of rotation is
1. the centerline of the rectangle, and
2. 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?

Question 4

Because computing moments of inertia directly can be quite laborious, peoplehave worked out indirect ways of computing unknown moments of inertiafrom known moments. In particular, if we know the moment of inertiaof an object around one axis of rotation, it turns out thatwe can find the moment of inertia for the same object about an axisthat is parallel to the first axis without completely re-doing the problem! This useful result is knownas the Parallel Axis Theorem.

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

Experimentally determine the moment of inertia of a C-beam aroundvarious horizontal axes of rotation. Record yourdata in a table.
Where does the minimum moment of inertia occur? Conjecture arelationship between this point and other geometric properties of theobject.
If you graph your data, it should look like a familiar function:a quadratic function! Try to fit your data witha quadratic function of the formM(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 thatthe moment of inertia depends on the object's mass and on its shape.How do these quantities relate to the model functionyou found?

Next: Modeling Deflections in Beams
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Up: Outline

The Geometry Center Calculus Development Team