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)

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

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