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}{2pi}), and a midline of 4? (f(t) = 5 sin 4t + 3) (f(t) = 5 sin 3t + 4) (f(t) = 4 sin 3t + 5) (f(t) = 3 sin 5t + 4)

Explanation:

Step1: Recall the general form of a sinusoidal function for simple harmonic motion, which is \( f(t) = A\sin(Bt) + C \), where \( A \) is the amplitude, \( B \) is related to the frequency, and \( C \) is the midline.

Step2: Determine the amplitude \( A \). Given the amplitude is 5, so \( A = 5 \).

Step3: Calculate the angular frequency \( B \). The frequency \( f \) is given by \( f=\frac{3}{2\pi} \), and the relationship between frequency and angular frequency is \( B = 2\pi f \). Substituting \( f=\frac{3}{2\pi} \), we get \( B = 2\pi\times\frac{3}{2\pi}=3 \).

Step4: Determine the midline \( C \). Given the midline is 4, so \( C = 4 \).

Step5: Substitute \( A = 5 \), \( B = 3 \), and \( C = 4 \) into the general form, we get \( f(t)=5\sin(3t)+4 \).

Answer:

\( f(t) = 5\sin 3t + 4 \)