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Unformatted text preview: 16.6 Simple harmonic oscillations with springs Goal: Be able to solve problems involving the simple harmonic oscillations of a spring system. Remember for simple harmonic motion we must have a force that is linear in x, always toward equilibrium: F = m a = -m ω 2 x So, always for SHM: a (t) =- ω 2 x (t) Comparing the force equation for a spring given by Hooke’s law (F = − kx) with the equation of motion of a simple harmonic oscillator (F = − m ω 2 x), we find − kx= − m ω 2 which gives the angular frequency ω of a spring’s oscillation as: ω = (k/m) 1/2 The position of the spring load as a function of time is given by the equation for the position of a simple harmonic oscillator: x(t) = Asin θ (t) = Asin( ω t + φ ) The amplitude A and phase constant φ are determined by the problem’s initial conditions. Example: A strong spring, with a .5kg object attached to the end, is suspended from the ceiling and stretches 1.0cm. The object is then lifted 2.0cm from this new equilibrium position and released from stretches 1....
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