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Instructors_Guide_Ch14 - 14 Oscillations Recommended class...

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14-1 14 Oscillations Recommended class days: 3 Background Information Oscillations occupy a middle ground between mechanics and wave motion. Simple harmonic motion, the basic model for oscillations, is still single-particle dynamics, and it plays a critical role in all of physics and engineering. But simple harmonic motion is also the starting point for the development of harmonic waves, and much of Part V will depend on ideas introduced and devel- oped in this chapter. Adequate time spent here will help the chapters on wave motion go more smoothly. Oscillations of springs, pendula, and other objects are familiar to students, but most have never observed the motion systematically. Thinking about oscillatory motion immediately reawakens the difficulties students have with concepts of motion, especially acceleration. Although the kinematic graphs and motion diagrams of Part I will certainly have benefited students, don’t expect to find them free of difficulties with acceleration. Oscillations will provide them with another opportunity, in a new context, to develop correct mental models of motion. Students can easily become distracted by the mathematics of oscillatory motion. It is important to keep the mathematical representation connected to graphical representations and to demonstra- tions of real oscillating systems. If you set a spring or a pendulum to oscillating, most students initially find it difficult to identify the point where the velocity is zero and the acceleration positive or the range of positions where the velocity is positive. A good strategy is to have them answer questions such as these (particularly questions about acceleration and phase) while observing the motion, then show how the answers are related to graphs of the motion, and finally discuss how they are related to the mathematical functions that describe oscillations. Although all students studied trigonometry in high school, many never got beyond thinking of trigonometry as simply dealing with triangles. A significant fraction of students don’t really under- stand the idea of sines and cosines as oscillatory functions , and this becomes a severe hindrance to their understanding of oscillations and waves. A recitation hour devoted to the mathematics of sinusoidal functions is an hour well spent. Many students don’t know the term sinusoidal function and think that it refers only to the specific function sin( x ). Common student difficulties with the mathematics of oscillations include: Unfamiliarity with radians. Not knowing that the argument of an oscillatory function needs to be an angle, which has no physical units. Student statements such as sin( t ) are common. Unfamiliarity with trigonometric identities, such as sin( x ) = sin( x ) or sin( x + π /2) = cos( x ).
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14-2 Instructor’s Guide Thinking that sine and cosine are two distinct, independent functions, whereas we want them to understand that sine and cosine are the same oscillatory function with different phase constants.
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