To determine the correct function that models the position of the weight in simple harmonic motion, we need to analyze the given data and apply the principles of oscillatory motion.
1. Amplitude (A):
- The maximum displacement from the equilibrium position is given to be 6 inches. Therefore, [tex]\( A = 6 \)[/tex].
2. Period (T):
- The time it takes for one complete oscillation is 4 seconds. Therefore, [tex]\( T = 4 \)[/tex] seconds.
3. Angular Frequency ([tex]\(\omega\)[/tex]):
- The angular frequency is given by [tex]\( \omega = \frac{2\pi}{T} \)[/tex].
- Substituting [tex]\( T = 4 \)[/tex] seconds: [tex]\( \omega = \frac{2\pi}{4} = \frac{\pi}{2} \)[/tex].
4. Form of the Displacement Function:
- In simple harmonic motion, the displacement [tex]\( s(t) \)[/tex] from the equilibrium position can be modeled by a cosine function when the object starts from its maximum displacement. Hence, the general form is:
[tex]\[
s(t) = A \cos(\omega t) \quad \text{or} \quad s(t) = -A \cos(\omega t)
\][/tex]
- Since the weight is pulled down to the maximum displacement and then released, we use the negative cosine function:
[tex]\[
s(t) = -A \cos(\omega t)
\][/tex]
- Substituting [tex]\( A = 6 \)[/tex] and [tex]\( \omega = \frac{\pi}{2} \)[/tex]:
[tex]\[
s(t) = -6 \cos\left(\frac{\pi}{2} t\right)
\][/tex]
5. Best Fit Function:
- Considering the given options, the function that best fits this form is:
- A. [tex]\( s(t) = -6 \cos\left(\frac{\pi}{2} t\right) \)[/tex]
Thus, the function that best models the position of the weight is given by:
[tex]\[
s(t) = -6 \cos \left(\frac{\pi}{2} t\right)
\][/tex]
Therefore, the correct answer is:
A. [tex]\( s(t) = -6 \cos \left(\frac{\pi}{2} t\right) \)[/tex]
