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**Unformatted text preview: **Lecture 16 Vibrations And Simple Harmonic Motion Periodic motion Periodic motion A motion is called periodic when the system comes back to the same situation every time interval T. Same position Same velocity Period Examples: Uniform circular motion Earth around the Sun Toy train in a circuit } π ϖ π = = = 1 Frequency 2 Angular 2 frequency f T f T Oscillations about an equilibrium position Oscillations about an equilibrium position U x When a 1D system is released near a stable equilibrium point, the motion is periodic: oscillations between two turn-around points x 1 and x 2 . E x 1 x 2 Forbidden region Forbidden region Oscillations SE Simple Harmonic Motion (SHM) Simple Harmonic Motion (SHM) SHM is the oscillatory motion that happens when the restoring force is proportional to the displacement from the equilibrium position: F ∝ –x …or when the potential energy is a quadratic function of the displacement from the equilibrium position: U ∝ x 2 (a parabola) equivalent to Example of SHM: Spring that obeys Hooke’s law = - F kx = 2 1 2 U kx Oscillations about x = 0 U x E –x x Most oscillatory systems can be approximated by SHM when the oscillations are small enough: U x E x 1 x 2 If not too far from the minimum, the curve is approximately a parabola. The SHM equation The SHM equation ϖ ϖ = = sin cos x t x t Solutions to this differential equation: = - F kx- =...

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