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24 May, 20:00

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.

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  1. 24 May, 20:04
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    (a) A = 0.650 m

    (b) f = 1.3368 Hz

    (c) E = 17.1416 J

    (d) K = 11.8835 J

    U = 5.2581 J

    Explanation:

    Given

    m = 1.15 kg

    x = 0.650 cos (8.40t)

    (a) the amplitude,

    A = 0.650 m

    (b) the frequency,

    if we know that

    ω = 2πf = 8.40 ⇒ f = 8.40 / (2π)

    ⇒ f = 1.3368 Hz

    (c) the total energy,

    we use the formula

    E = m*ω²*A² / 2

    ⇒ E = (1.15) (8.40) ² (0.650) ² / 2

    ⇒ E = 17.1416 J

    (d) the kinetic energy and potential energy when x = 0.360 m.

    We use the formulas

    K = (1/2) * m*ω² * (A² - x²) (the kinetic energy)

    and

    U = (1/2) * m*ω²*x² (the potential energy)

    then

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

    ⇒ K = 11.8835 J

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

    ⇒ U = 5.2581 J
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