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29 December, 08:09

The motion of a particle is described by x = 10 sin (πt + π/3), where x is in meters and t is in seconds. At what time in seconds is the potential energy equal to the kinetic energy?

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  1. 29 December, 08:31
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    5/12

    Explanation:

    Given

    x = 10 sin (πt + π/3)

    v = distance/time

    So, v = dx/dt

    Differentiating x with respect to t

    v = 10π cos (πt + π/3)

    Also,

    ½kx² = ½mv²

    Substituting values for x and v in the above equation

    ½k (10sin (πt + π/3)) ² = ½m (10πcos (πt + π/3)) ²

    Divide through by ½

    k (10sin (πt + π/3)) ² = m (10πcos (πt + π/3)) ²

    Open both bracket

    100ksin² (πt + π/3) = 100mπ²cos² (πt + π/3)

    Divide through by 100

    ksin² (πt + π/3) = mπ²cos² (πt + π/3)

    Divide through by kcos² (πt + π/3)

    ksin² (πt + π/3) : kcos² (πt + π/3) = mπ²cos² (πt + π/3) : kcos² (πt + π/3)

    tan² (πt + π/3) = mπ²/k

    tan² (πt + π/3) = (m/k) π²

    But w² = k/m and w = 2π/T

    (2π/T) ² = k/m

    (2π) ²/T² = k/m

    1/T² = k/m : (2π) ²

    1/T² = k/m * (2π) ²

    T² = m (2π) ²/k

    From the Question, T is when πt = 2π or T = 2

    Substitute 2 for T in the above equation

    2² = m (2π) ²/k

    4 = m (2π) ²/k

    4 = 4π²m/k

    m/k = 1/π²

    (m/k) π² = 1

    Remember that tan² (πt + π/3) = (m/k) π²

    So, tan² (πt + π/3) = 1

    This gives

    πt + π/3 = 45° = π/4

    πt + π/3 = π/4

    Divide through by π

    t + ⅓ = ¼

    t = ¼ - ⅓

    t = - 1/12 - - - Negative

    Using the second quadrant

    πt + π/3 = 3π/4

    Divide through by π

    t + ⅓ = ¾

    t = ¾ - ⅓

    t = 5/12
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