I need to find the equation. I got them wrong for the first time. So i need help!!!

Question

A spring is attached to the ceiling and pulled 14 cm down from equilibrium and released. After 4 seconds the amplitude has decreased to 13 cm. The spring oscillates 20 times each second. Assume that the amplitude is decreasing exponentially. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t.

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.

A 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

A spring is attached to the ceiling and pulled 14 cm down from equilibrium and released. After 4 seconds the amplitude has decreased to 13 cm. The spring oscillates 20 times each second. Assume that the amplitude is decreasing exponentially. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t.

Given A₀ = 14 cm, T= 1/20 = 0.05 sec + Time period of Oscillator

13 = 14 e^{-Kt ,} when t= 4 sec ,

w = 2pi/T = 125.6

14/13 = e ^Kt

ln(14/13 ) = Kt = 4K , K = 0.01852

K = (1/4) *0.07410 = 0.018

Equation of motion is

D = 14 ( e ^{- 0.018t} Sin ( 126t +φ) ) cm

As equation of damped harmonic oscillator is Y = A₀ e-^Kt Sin ( wt +φ)

Here φ is initial phase angle ( at t= 0 sec)

If you find it OK, I will go to next

Any question?