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6 October, 20:26

Starting at its rightmost position, it takes 2 seconds for the pendulum of a grandfather clock to swing a horizontal distance of 18 inches from right to left and 2 seconds for the pendulum to swing back from left to right. Which of the following equations models d, the horizontal distance in inches of the pendulum from the center as a function of time, t, in seconds? Assume that right of center is a positive distance and left of center is a negative distance.

a) d=9cos (pi/4 t)

b) d=9cos (pi/2 t)

c) d=18cos (pi/4 t)

d) d=18cos (pi/2 t)

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Answers (2)
  1. 6 October, 23:59
    0
    The correct answer is b, x = 9 cos (pi / 2 t)

    Explanation:

    The equation that describes a simple pendulum is

    θ = θ₀ cos (wt + φ)

    The angle is measured is radians

    θ = x / L

    We replace

    d / L = x₀ / L cos (wt + φ)

    x₀ = 9 in

    We replace

    d = 9 cos (wt + φ)

    Angular velocity is related to frequency and period.

    w = 2π f = 2π / T

    The period is the time of a complete oscillation T = 4 s

    w = 2π / 4

    w = π / 2

    Let's replace

    x = 9 cos (π/2 t + φ)

    As the system is released from the root x = x₀ for t = 0 s

    x₀ = x₀ cos φ

    Cos φ = 1

    φ = 0°

    The final equation is

    x = 9 cos (pi / 2 t)

    The correct answer is b
  2. 7 October, 00:21
    0
    b) d=9cos (pi/2 t)

    Explanation:

    This is a cosine function in the such as: y = a cos bt ...

    a = (maximum distance - minimum distance) / 2:

    a = (max - min) / 2

    maximum distance = 18 inches minimum distance = 0

    a = (18 - 0) / 2 = 18/2 = 9

    That is a = 9

    To solve for b, similar to the period:

    The period in radians:

    P = 2pi/b, which is the amount of time it takes to revolve one full cycle ...

    multiply that time by time from minimum to maximum, 2 to give the period as 2 * 2 = 4.

    Thus to find b ...

    4 = 2pi/b

    4b = 2pi/b * b

    4b = 2pi

    or

    b = pi/2

    Also

    a = 9

    b = pi/2

    So our cosine function is:

    d = 9cos ((pi/2) t)

    Hence the equation that models d
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