Starting at its rightmost position, it takes 1 second for the pendulum of a grandfather clock to swing a horizontal distance of 12 inches from right to left, and 1 second for the pendulum to swing back from left to right. Write a cosine function, d = acos(bt), to model the distance, d, of the pendulum from the center (in inches) as a function of time t (in seconds).

For the answer to the question above, asking to write a cosine function, d = acos(bt), to model the distance, d, of the pendulum from the center (in inches) as a function of time t (in seconds).T = 2s ------> f = 1/T = 0.5 Hz -----> ω = 2πf = π radians/sec
A = 6"
d(t) = 6"cos(πt)

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bhoopendrasisodiya34 bhoopendrasisodiya34 · Answer 2 · 2022-02-01T10:58:11+01:00

Function which describe the motion of the pendulum is called the function of the pendulum. The cosine function of the pendulum is,

[tex]d(t)=6\cos (\pi t)[/tex]

What is the function of the pendulum?

Function which describe the motion of the pendulum is called the function of the pendulum.

Given information-

The given cosine function for the pendulum is,

[tex]d=a\cos (bt)[/tex]

The pendulum takes 1 second to swing horizontal distance of 12 inches from right to left.

The pendulum takes 1 second to swing horizontal distance of 12 inches from left to right.

The total time taken by the pendulum to complete one cycle is,

[tex]t=1+1\\t=2[/tex]

Thus the time taken by the pendulum to complete one cycle is 2 seconds.

As the frequency of the pendulum is inverse of the time. Thus frequency,

[tex]f=\dfrac{1}{t} \\f=\dfrac{1}{2}[/tex]

Angular speed of the pendulum can be given as,

[tex]\omega=2\pi f\\\omega=2\times\pi\times\dfrac{1}{2} \\\omega=\pi[/tex]

Thus value of angular acceleration [tex]b[/tex] is [tex]\pi[/tex] red per seconds.

The distance of the pendulum from the center is half of the total distance. Thus,

[tex]a=\dfrac{12}{2} \\a=6[/tex]

Hence the distance of the pendulum from the center is 6 inch.

Put the value in given cosine function as,

[tex]d(t)=6\cos (\pi t)[/tex]

Thus the cosine function of the pendulum is,

[tex]d(t)=6\cos (\pi t)[/tex]

Learn more about the function of pendulum here;

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