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Moment of Inertia
A measure of the effort required to change that object's rotational velocity about a specified axis of rotation.
It is clear that some obejcts are more difficult to set into rotation or to stop from rotating than others. Consider four very different objects that are often rotated: a CD, a bicycle wheel, a merry-go-round in a park, and a carousel at an amusement park. Rotating a CD about its natural axis is trivial (simply brush it with your finger), and stopping its rotation is similarly trivial. Rotating a bicycle wheel is fairly easy (a push with your hand) and stopping its rotation is similarly straightforward. Rotating a park merry-go-round requires some effort (a full push with your legs) and stopping it takes some thought if you wish to avoid injury. Starting an amusement park carousel requires a large motor and stopping it requires sturdy brakes. These objects have distinctly different moments of inertia. Of course, they also have very different masses. Thus, mass is one factor that plays into moment of inertia.
Moment of inertia is not the same as mass, however, as can be seen in a straightforward experiment. Find a desk chair that swivels fairly easily and grab a pair of dumbbells or other objects with significant mass. Sit on the chair holding the dumbbells at your chest and swivel back and forth a few times. Get a sense of the effort your feet exert to start and stop your motion. Next, hold the dumbbells out to your sides at your full arms' length. Repeat the experiment and note the effort required in the new configuration. Note that your mass (plus the chair and dumbbells) has not changed in this exercise, only the position of the mass has changed.
CD |
Bike Wheel |
Merry-Go-Round |
Carousel |
![](/confluence/download/attachments/30015818/CD.jpg?version=1&modificationDate=1226865267000&api=v2)
|
![](/confluence/download/attachments/30015818/bikewheel.jpg?version=1&modificationDate=1226865278000&api=v2)
|
![](/confluence/download/attachments/30015818/merrygoround.jpg?version=1&modificationDate=1226865316000&api=v2)
|
![](/confluence/download/attachments/30015818/carousel.jpg?version=1&modificationDate=1226867937000&api=v2)
|
Photo courtesy Wikimedia Commons, by user Ubern00b. |
Photo courtesy Wikimedia Commons, by user Herr Kriss. |
Photo by Eric Hart, courtesy Flickr. |
Photo courtesy Wikimedia Commons, by user KMJ. |
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The parallel axis theorem states that if the moment of inertia of a rigid body about an axis passing through the body's center of mass is Icm then the moment of inertia of the body about any parallel axis can be found by evaluating the sum:
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where d is the (shortest) distance between the original center of mass axis and the new parallel axis.
The principle utility of the parallel axis theorem is in quickly finding the moment of inertia of complicated objects. For example, suppose we were asked to find the moment of inertia of an object created by screwing two hollow spheres of radius R and mass Ms to the end of a thin rod of length L and mass Mr. If the object is rotated about the center of the rod, then the total moment of inertia is found by adding the contributions from the rod to that from the spheres. From the table above, we can see that the rod contributes:
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Since the centers of the spheres are a distance L/2+R away from the axis of rotation of the composite object, they each contribute:
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so the total moment of inertia is:
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