Trigonometry question

Question

Suppose an object attached to a coiled spring is pulled down a distance of 18 in from its rest position and then released. If the time for one oscillation is 11 sec, develop a model that relates the displacement d(t) of the object from its rest position after time t (in seconds). Assume no friction.

d(t)=

Answer 1

d(t) = Asin(B(t+C)) + D

Since pulling down, position at time, t=0 is -18. This is the lowest point.

Period is given as 11 seconds, B = 2π/Period = 2π/11.

Midpoint = D = (-18+0)/2 = -9

The range of sin(x) = ±1.

A(-1) -9 = -18 --> A = 9

When sin(x) is at position -18, sin(x) = -1, x = -π/2 = 2π/11(0+C) --> C = -11/4

d(t) = 9sin((2π/11)(t-11/4)) - 9

d(0) = 9(-1)-9 = -18

d(11/2) = 9sin((2π/11)(11/2-11/4)-9 = 9sin(π/2)-9 = 0 Rest Point

d(11/4) = 9sin((2π/11)(11/4-11/4)-9 = 9sin(0)-9 = -9 Midpoint