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15 November, 13:53

A harmonic oscillator is described by the function x (t) 5 (0.200 m) cos (0.350t). Find the oscillator's (a) maximum velocity and (b) maximum acceleration. Find the oscillator's (c) position, (d) velocity, and (e) acceleration when t 5 2.00 s

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  1. 15 November, 14:02
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    a) Maximum velocity = 0.07 m/s

    b) Maximum acceleration = 0.0245 m/s²

    c) Position at t = 2s is 0.153 m

    d) velocity at t = 2s is - 0.0451 m/s

    e) Acceleration at t = 2s is 0.0187 m/s²

    Explanation:

    x (t) = (0.200 m) cos (0.350t)

    The general expression for the displacement of a body in simple harmonic motion = A cos wt

    Comparing, A = amplitude = maximum displacement = 0.20 m

    w = angular frequency/velocity = 0.350 rad/s

    a) v = dx/dt = (d/dt) (A cos wt) = - Aw sin wt

    Maximum velocity is given by Aw = 0.2 * 0.35 = 0.07 m/s

    b) a = dv/dt = (d/dt) (-Aw sin wt) = - Aw² cos wt

    Maximum acceleration = Aw² = 0.2 * 0.35² = 0.0245 m/s²

    c) Position at t = 2,

    x = A cos wt = 0.2 cos (0.35t)

    At t = 2

    x = 0.2 cos (0.35 * 2) = 0.2 cos (0.7) = 0.153 m (Take note, wt is in radians)

    d) velocity at t=2s

    v = - Aw sin wt = - (0.2 * 0.35) sin (0.35 * 2) = - 0.07 sin (0.7) = - 0.0451 m/s

    e) Acceleration at t = 2s

    a = - Aw² cos wt = - (0.2 * 0.35²) cos (0.35 * 2) = - 0.0245 cos (0.7) = - 0.0187 m/s²
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