Lect 30 - Oscillations - Oscillatory Motion Chapter 15...

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Unformatted text preview: Oscillatory Motion Chapter 15 Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A special kind of periodic motion occurs in mechanical systems when the force acting on the object is proportional to the position of the object relative to some equilibrium position If the force is always directed toward the equilibrium position, the motion is called simple harmonic motion Motion of a Spring-Mass System A block of mass m is attached to a spring, the block is free to move on a frictionless horizontal surface When the spring is neither stretched nor compressed, the block is at the equilibrium position x = 0 Hookes Law Hookes Law states F s = - kx F s is the restoring force It is always directed toward the equilibrium position Therefore, it is always opposite the displacement from equilibrium k is the force (spring) constant x is the displacement Acceleration The force described by Hookes Law is the net force in Newtons Second Law F Hooke = - = = - The acceleration is proportional to the displacement of the block The direction of the acceleration is opposite the direction of the displacement from equilibrium An object moves with simple harmonic motion whenever its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium Acceleration The acceleration is not constant Therefore, the kinematic equations cannot be applied If the block is released from some position x = A , then the initial acceleration is kA / m When the block passes through the equilibrium position, a = 0 The block continues to x = - A where its acceleration is + kA / m Motion of the Block The block continues to oscillate between A and + A These are turning points of the motion The force is conservative In the absence of friction, the motion will continue forever Real systems are generally subject to friction, so they do not actually oscillate forever Orientation of the Spring When the block is hung from a vertical spring, its weight will cause the spring to stretch If the resting position of the spring is defined as x = 0, the same analysis as was done with the horizontal spring will apply to the vertical spring-mass system Simple Harmonic Motion Mathematical Representation Model the block as a particle Choose...
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Lect 30 - Oscillations - Oscillatory Motion Chapter 15...

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