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14 December, 17:49

A 94 g particle undergoes SHM with an amplitude of 8.3 mm, a maximum acceleration of magnitude 7.8 x 103 m/s2, and an unknown phase constant φ. What are

(a) the period of the motion,

(b) the maximum speed of the particle, and (c) the total mechanical energy of the oscillator? What is the magnitude of the force on the particle when the particle is at

(d) its maximum displacement and

(e) half its maximum displacement?

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Answers (1)
  1. 14 December, 19:57
    0
    a) T = 6.49*10^-3 s

    b) v = 8 m/s

    c) E = 3 J

    d) F = 733 N

    e) F = 366.5 J

    Explanation:

    Given

    Mass of particle, m = 94 g = 0.094 kg

    Amplitude of the particle, A = 8.3 mm = 8.3*10^-3 m

    Maximum acceleration of particle, a = 7.8*10^3 m/s²

    the equation describing Simple Harmonic Motion is given as

    x = A cos (wt + φ)

    To fond the acceleration of this relationship, we would have to integrate. Twice, the first would be a Velocity, and the second acceleration that we need.

    Velocity = dx/dt = - Aw sin (wt + φ)

    Acceleration = d²x/dt = - Aw² cos (wt + φ)

    From the question, we were given, magnitude of acceleration to be 7.8*10^3 m/s²

    Aw² = 7.8*10^3

    w² = 7.8*10^3 / A

    w² = 7.8*10^3 / 8.3*10^-3

    w² = 939759

    w = √939759

    w = 969

    Recall, T = 2π/w, so that

    T = (2 * 3.142) / 969

    T = 6.49*10^-3 s

    Maximum speed = Aw

    Maximum speed = 8.3*10^-3 * 969

    Maximum speed = 8.0 m/s

    Total mechanical energy oscillator =

    mgx + 1/2mx² =

    1/2mv (max) ² =

    1/2 * 0.094 * 8² =

    3 J

    Maximum displacement

    x = A cos (wt + φ)

    For x to be maximum here, then cos (wt + φ) Must be equal to 1

    Acceleration = d²x/dt² = - Aw²

    And force = mass * acceleration

    Force = 0.094 * 7.8*10^3

    Force = 733 N

    x = A cos (wt + φ), where cos (wt + φ) = 1/2

    d²x/dt² = - Aw² * 1/2

    d²x/dt² = 733 * 0.5

    = 366.5 N
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