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Lambda, Sigma, and Rho
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by David Castro
Charles A. Dana Center
University of Texas
Austin, Texas
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Every year, the AP Physics C instructor must decide what aspects of calculus to include in the mechanics portion of the syllabus. A cursory review of previously released AP Physics C Exam questions underscores the need for a calculus-based understanding of motion (both as derivatives and integrals). Other topics that readily come to mind are work as 
and impulse as 
. However, there remains a significant calculus-based aspect of mechanics that many instructors choose to omit -- the calculation of moment of inertia using 
.
This may seem prudent, since during the last 20 years of Physics C free-response questions, students have been required to perform only one such integral. In 1996, they were asked to prove that the inertia of a rod pivoted about its center is 
. Although students who took the 2004 AP Physics C Exam might have benefited from being able to integrate r2dm, they could have used the parallel axis theorem to bypass the integral.
The real advantage of teaching 
during mechanics is to introduce integration of density to find a second quantity. This type of integral comes up regularly in the Electricity and Magnetism portion of the AP Physics C Exam. Examples include:
Total charge of a nonconducting rod:
Total charge enclosed by a Gaussian surface:
Total current enclosed by an Amperian loop:
Total magnetic flux in a loop of wire:
A detailed study of moment of inertia is invaluable in laying the foundation for the second semester, especially since many students feel more comfortable visualizing mechanical systems. It also provides an opportunity to reinforce one of the underlying principles of physics: the application of simple mathematical relationships can explain a wide variety of phenomena.
The question below asks students to use calculus to find the moment of inertia of a variety of objects and provides an opportunity to introduce students to density problems of three types. Note that these values are usually presented in textbooks without proof or explanation.
Using the relationship 
and basic calculus, prove each of the following statements. You may not use the parallel axis theorem.
(A) The moment of inertia of a rod pivoted about its end is
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(B) The moment of inertia of a rod pivoted about its center is
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(C) The moment of inertia of a thin hoop pivoted about its center is
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(D) The moment of inertia of a disk pivoted about its center is
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(E) The moment of inertia of a thick hoop of inner radius a and outer radius b is
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(F) The moment of inertia of a solid sphere pivoted about its center is
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(G) The moment of inertia of a hollow sphere about its center is
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| Click here to view the answers and commentary!
David Castro taught AP Physics (B and C), AP Calculus (AB and BC), and AP U.S. and European History in a teaching career spanning 14 years, including 5 years as a master AP Physics teacher. In 1997, he received a Special Recognition Teaching Award, and in 2002 his combined AP Physics and AP Calculus syllabus was published in the AP Physics Teacher's Guide. Active as an AP Physics consultant in the Southwest Region since 1995, his areas of expertise include Pre-AP middle school science, AP Vertical Teams, as well as interdisciplinary physics/calculus. He also serves as a Reader for AP Physics. Mr. Castro recently joined the Charles A. Dana Center at the University of Texas, where he continues to focus on providing support for science educators.
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