Let's define the cosine function as follows:
h(t) = A cos(ωt + φ) + C
Where:
- A is the amplitude (half the difference between the maximum and minimum height)
- ω is the angular frequency (related to the period of the function)
- φ is the phase shift
- C is the average resting position of the spring
Given:
- A = (7 feet 4 inches - 6 feet 2 inches) / 2 = 1 foot 2 inches = 14 inches
- Period = 2 seconds
Since the function should complete one cycle (from minimum to maximum and back) in 2 seconds, the angular frequency ω can be calculated as:
ω = 2π / Period = 2π / 2 = π
The phase shift is typically 0 for these kinds of models because the spring starts at its minimum length, assuming t = 0.
Therefore, the cosine function that models the spring's length in inches above and below its average resting position is:
h(t) = 14 cos(πt) + 6 feet 8 inches
This function shows the position of the spring at any given time t in seconds above or below its average resting position of 6 feet 8 inches.
The motion of a simple spring hanging from the ceiling can be modeled with a cosine function. The bottom of the spring has a maximum height of 7 feet 4 inches and a minimum height of 6 feet 2 inches from the floor. It takes 2 seconds for the spring to expand from its minimum length to its maximum length. What is a cosine function that models the spring’s length in inches above and below its average, resting position? Express the model as a function of time in seconds.
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