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Moment of Inertia
A measure of the tendency of an object to maintain its rotational velocity about a specified axis of rotation. The moment of inertia depends linearly on the mass and quadratically on the distance of that mass from the axis of rotation. It plays the same role for rotational motion as mass plays for translational motion, being both the ratio of angular momentum to angular velocity and the ratio of torque to resultant angular acceleration, whereas mass is the ratio of (linear) momentum to velocity and the ratio of force to resultant linear acceleration.
It is clear that some objects 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, because it depends quadratically on size as well. This can be seen in a straightforward experiment. Find two boards that have the same weight but different lengths - for example a 1" X 2" board that is 12' long and a 2" by 4" board that is 3' long. Grab each by the center in each hand and rotate them. It will require dramatically more effort to rotate the longer board - 16 times as much, in fact. Note that the mass has not changed in this exercise, only the distance between the mass and the axis of rotation.
CD |
Bike Wheel |
Merry-Go-Round |
Carousel |
 |
 |
 |
 |
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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