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11 November, 05:53

A 1.15-kg mass oscillates according to the equation where x is in meters and in seconds. Determine (a) the amplitude, (b) the frequency, (c) the total energy, and (d) the kinetic energy and potential energy when

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  1. 11 November, 06:08
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    The complete question is;

    A 1.15-kg mass oscillates according to the equation x = 0.650 cos (8.40t) where x is in meters and t in seconds. Determine (a) the amplitude, (b) the frequency, (c) the total energy, and (d) the kinetic energy and potential energy when x = 0.360 m.

    Answer:

    A) Amplitude; A = 0.650 m

    B) Frequency; f = 1.337 Hz

    C) total energy = 17.142 J

    D) Kinetic energy = 11.884 J

    Potential Energy = 5.258 J

    Explanation:

    We are given;

    Mass; m = 1.15 kg

    Equation; x = 0.650 cos (8.40t)

    (a) The standard form of a wave function is in the form y (x, t) = Asin (kx-ωt+ϕ)

    So, comparing terms in our equation in the question to this, the amplitude is;

    A = 0.650 m

    (b) we know that formula for frequency is;

    f = ω/2π

    Again, comparing terms in the standard equation and our question, we can see that ω = 8.4

    Thus;

    f = 8.4 / (2π)

    f = 1.337 Hz

    (c) Formula for the total energy is given by;

    E = m•ω²•A²/2

    Plugging in the relevant values, we have;

    E = (1.15) (8.40) ² (0.650) ²/2

    E = 17.142 J

    (d) we want to find the kinetic energy and potential energy when x = 0.360 m.

    The formula for kinetic energy in this case is given by;

    K = (1/2) •m•ω²• (A² - x²)

    Thus;

    K = (1/2) * (1.15) * (8.40) ² * ((0.650) ² - (0.360) ²)

    K = 11.884 J

    Also, the formula for the potential energy in this case is given by;

    U = (1/2) •m•ω²•x²

    Thus;

    U = (1/2) * (1.15) * (8.40) ² * (0.360) ²

    U = 5.258 J
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