Question

margaret pulls a slinky a distance of 22 inches from its equilibrium position and then releases it. the time for one oscillation is 2 seconds.
step 2 of 2 : find the function in terms of (t) for the displacement of the slinky. assume that the slinky is at its maximum displacement at (t = 0). also assume that the function includes no horizontal or vertical shifts.

Explanation:

Step1: Determine the form of the function

Since the slinky is at its maximum displacement at $t = 0$ and has no horizontal or vertical shifts, we use the cosine - function form $y = A\cos(\omega t)$. The amplitude $A$ is the maximum displacement from the equilibrium position. Here, $A = 22$ inches.

Step2: Calculate the angular frequency $\omega$

The period $T$ is the time for one oscillation. Given $T=2$ seconds. The formula for the angular frequency is $\omega=\frac{2\pi}{T}$. Substituting $T = 2$ into the formula, we get $\omega=\frac{2\pi}{2}=\pi$.

Step3: Write the function

Substitute $A = 22$ and $\omega=\pi$ into the function $y = A\cos(\omega t)$. The displacement function is $y = 22\cos(\pi t)$.

Answer:

$y = 22\cos(\pi t)$