# 3.7 - Math 334 Lecture#18 3.7 Mechanical Vibrations...

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Math 334 Lecture #18 § 3.7: Mechanical Vibrations Mass-Spring Systems. A spring is attached to a ﬁxed support, an object is suspended on the spring, and a dashpot is placed about the object. [Sketch picture of this conﬁguration.] Let u ( t ) be the position of the center of the object at time t , with u = 0 being the position of the object when it is at rest, and u > 0 is the downward direction. The principle governing the motion of the mass is Newton’s second law: ma = sum of forces acting on the mass , where m is the mass of the object and a is its acceleration. The relationship between a and u is a = u 00 . [We will consider four forces that act on the object.] Force due to Gravity . The weight of the object is F g = mg , where g is the gravitation constant. Force due to the Spring . Attaching the object to the spring stretches the spring by a distance L , called the elongation . [Sketch picture of spring before and after the object is suspended on the spring.] Hooke’s Law states that the force exerted by the spring is negatively proportional to the elongation: - kL where k > 0. When the object is at rest (i.e. is in equilibrium), the weight of the object and the force

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3.7 - Math 334 Lecture#18 3.7 Mechanical Vibrations...

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