Question

a spring attached to a wall was displaced horizontally to model simple periodic motion. which function represents the oscillations of the spring if it had an amplitude of 5, a frequency of \\(\frac{3}{2\pi}\\), and a midline of 4?\\(f(t) = 5\sin 4t + 3\\)\\(f(t) = 3\sin 5t + 4\\)\\(f(t) = 5\sin 3t + 4\\)\\(f(t) = 4\sin 3t + 5\\)

Explanation:

Step1: Recall the general form of simple harmonic motion

The general form of a sinusoidal function for simple harmonic motion is \( f(t) = A\sin(\omega t) + k \), where:

• \( A \) is the amplitude,
• \( \omega \) is the angular frequency,
• \( k \) is the midline (vertical shift).

Step2: Determine the amplitude (\( A \))

We are given that the amplitude \( A = 5 \). So the coefficient of the sine function should be 5.

Step3: Determine the angular frequency (\( \omega \))

The frequency \( f \) is related to the angular frequency \( \omega \) by the formula \( \omega = 2\pi f \). We are given \( f=\frac{3}{2\pi} \).
Substitute \( f \) into the formula:
\( \omega = 2\pi\times\frac{3}{2\pi} = 3 \)

Step4: Determine the midline (\( k \))

We are given that the midline \( k = 4 \). So the vertical shift should be 4.

Step5: Form the function

Putting it all together, the function should be \( f(t) = 5\sin(3t) + 4 \).

Answer:

\( f(t) = 5\sin 3t + 4 \) (the third option)