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Factoring out like terms
Note that then entire wheel rotates at the same angular velocity, so we can (and did) factor the (1/2) and the w^2 out of the summation equation given above.
Integral calculus is the key
Regardless of how difficult it may seem to you to perform the summation given above, when you complete a course in integral calculus, you will have learnedhow to do that sort of thing for a variety of geometric shapes such as solid cylinders, hollow cylinders, solid spheres, hollow spheres, squares, rectangles,rods, etc. As a result, engineering and physics handbooks contain tables with this sort of information for a variety of common geometrical shapes.
(There is also such a table at (External Link) , but if you are a blind student, your accessibility equipment may not be able to read itreliably.)
Getting back to the earlier equation for rotational kinetic energy , the quantity in parentheses cannot change for a given geometric shape. The distance between each mass particle and the axis of rotation staysthe same for a rigid body, and the mass of each mass particle doesn't change. It is conventional to give the quantity in parentheses the symbol I (upper-case"I") and refer to it as either the rotational inertia , or the moment of inertia .
Therefore, using the terminology from the earlier equation for rotational kinetic energy ,
Facts worth remembering -- Rotational Inertia
I = sum from i=0 to i=N(mi*ri^2)
where
Translational versus rotational kinetic energy
Given that information, let's form an analogy between translational kinetic energy from an earlier module and rotational kinetic energy from this module.
Facts worth remembering -- Translational and Rotational Kinetic Energy
Kt = (1/2)*m*v^2
Kr = (1/2)*I*w^2
where
Comparing the terms
When we compare the terms in the two expressions, we see that angular velocity in one case is analogous to translational velocity in the othercase.
We also see that the rotational inertia in one expression is analogous to the mass in the other expression.
As I explained earlier, mass is an absolute in translational terms. The translational inertia depends directly on the amount ofmass.
However, mass is not an absolute in rotational terms. The rotational inertia for rotation depends not only upon the amount of massinvolved, but also on how that mass is geometrically distributed within the object relative to the axis of rotation.
A measure of inertia
In translational terms, mass is a measure of the inertia of an object, or how difficult it is to cause the object to change its translational velocity.
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