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Unformatted text preview: Part 1 HARMONIC OSCILLATIONS Oscillatory motion is everywhere in nature. Any object which has both inertia and a restoring force will oscillate around an equilibrium position if displaced from that equilibrium. As we will see, the descriptions of essentially all oscillating systems are very similar - and hence we will look most closely at mass-spring systems and pendula, the most common oscillators in our daily experience. Describing the motions of objects that oscillate about a point of equilibrium, as with the motion of any object, requires a solid understanding of Newton's laws. It is useful at this point to review the essential ideas contained in those laws. As we will then see, the descriptions of the motions of masses on springs, pendula, marbles oscillating about in the bottom of a bowl, and even the periodic motions of buildings or bridges or violin strings will follow from understanding the forces acting and then solving Newton's second law. So we will begin with a review of Newton's laws of motion. NEWTON'S LAWS While the kinematics equations describe the motions of objects, it is Newton's three laws that relate the motion to the causes of the motion. As simple as Newton's laws are at one level, it is difficult to overemphasize their importance. They represent the fundamental principles that govern how things move - the connection between how objects interact with each other and the changes in motion that result from that interaction. Solving Newton's law problems is often very difficult for many students. To be successful at it requires internalizing what is actually meant by Newton's laws - not just learning the statements or knowing the equation that expresses Newton's second law. When Newton's laws are understood, setting up problems to solve for the motion of some object simply becomes an exercise in identifying all the forces on the object and expressing Newton's second law in algebraic form speciallized to each object in the specific problem being described. Newton's 1st Law: The Law of Inertia In the absence of a net force - or unbalanced force, an object either remains at rest or moves in a straight line at constant speed. The first law just identifies that changes of motion occur because of forces - and so the state of motion remains unchanged either if no forces act or if all the forces that do act are balanced in such a way to add to zero. This is the basis of all equilibrium problems. The inertia of an object is its tendency to remain in its current state of motion. The object's mass is a measure of its inertia. The significance of the first law is to state that because of the inertia of an object, an unbalanced force is required to change the object's motion....
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