A weight is attached to a spring that is oscillating up and down. It takes 6 sec. for the spring to complete one cycle, and the distance from the highest to lowest point is 7 in. Write an equation that models the position of the weight at time t seconds?
Let the position of the weight at time t seconds be denoted as y(t). Since the spring is oscillating up and down, we can use a sinusoidal function to model its motion, specifically a cosine function because it starts at a maximum (the highest point).
The period of the oscillation is 6 seconds, so the angular frequency ω = 2π / 6 = π/3.
The amplitude of the oscillation is half of the distance between the highest and lowest points, so A = 7/2 inches.
We are looking for an equation in the form y(t) = A*cos(ω*t) + D, where A is the amplitude, ω is the angular frequency, and D is the vertical shift.
Since the spring oscillates between 3.5 inches above and below its equilibrium position, we have a vertical shift of D = 3.5 inches.
So the equation that models the position of the weight at time t seconds is:
y(t) = (7/2) * cos(π/3 * t) + 3.5