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Unformatted text preview: Notes for Day : Â§ . : Unforced Mechanica Vibrations Hookeâ€™s Law Imagine a spring at rest, suspended vertically from a ceiling. A downward force is exerted on the spring, causing the spring to stretch. As the spring stretches, the spring itself exerts a force, called a restoring force which pulls the spring back to its resting position. Hookeâ€™s Law says that when the distance by which the spring is originally displaced, Î” y , is small, the restoring force F R is proportional to that distance, but in the opposite direction; in other words, F R = - k Î” y . e constant k is called the spring constant and represents the â€śsti nessâ€ť of the spring. e Spring-Mass System: Undamped Motion Suppose a spring with natural length l is suspended from the ceiling, and an object having mass m is attached to the bottom of the spring. e weight of the object will pull the spring downwards. e spring will bounce a few times, and eventually come to rest at a new equilibrium position ( Î” y = Y ) in which the length of the spring is greater than l . At the equilibrium position, the downward force due to gravity acting on the object ( mg ) balances out the upward restoring force due to the spring pulling itself back in (- k Î” y ) : mg- kY = rough a little symbolic manipulation (p. ), we can represent the motion of the spring with the di erential...
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- Spring '06