Find an equation

Question

A spring is attached to the ceiling and pulled 9 cm down from equilibrium and released. The amplitude decreases by 16% each second. The spring oscillates 13 times each second. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t.

The population of rabbits oscillates 33 above and below average during the year, hitting the lowest value in January (t = 0). The average population starts at 700 rabbits and increases by 170 each year. Find an equation for the population, P, in terms of the months since January, t.

Answer 1

An undamped spring would oscillate like D(t) = -9cos(26πt) assuming the equilibrium point is "0" and below the equilibrium point is negative. But damping the spring adds an "e^λt" damping factor. To achieve the 16% decrease each second λ = -0.1744 so the equation becomes:

D(t) = -(e^-0.1744t)(9cos(26πt))

For the rabbits, without oscillation the average is modeled by P = 170t + 700 (t in years) but we are asked for an equation in months so P = (170/12)t + 700.

The oscillation is required to initially be negative (January is a minimum) so we would need to add a negative cosine wave with a period of 12 months. That makes the equation:

P(t) = (170/12)t + 700 - 33cos((2π/12)t) or (simplifying)

P(t) = (85/6)t + 700 - 33cos((π/6)t)