To find the equation of the simple harmonic motion, let's first recall the general form for simple harmonic motion (SHM):

$s(t) = A \sin\left(\omega t + \phi\right)$ or $s(t) = A \cos\left(\omega t + \phi\right)$ Where:

• $A$ is the amplitude (the maximum displacement from equilibrium),
• $\omega$ is the angular frequency (in radians per second),
• $\phi$ is the phase constant (which determines the initial conditions).

Step 1: Identify Given Information

• The amplitude $A$ is the maximum displacement from the equilibrium position. The object is pulled down 3 inches from equilibrium, so: $A = 3 \, \text{inches}$
• The time for one complete oscillation (the period $T$) is 7 seconds: $T = 7 \, \text{seconds}$
• The angular frequency $\omega$ is related to the period by the formula: $\omega = \frac{2\pi}{T}$ Substituting the given value for $T$: $\omega = \frac{2\pi}{7}$

Step 2: Analyze the Initial Conditions

The problem states that the object is initially pulled downward (so the displacement is negative) and then released. This suggests that at $t = 0$, the displacement $s(0) = -3$ inches. This is a typical starting condition for a cosine function, which starts at its maximum or minimum value.

Thus, the equation should be in the cosine form, with a negative amplitude because the initial displacement is downward.

Step 3: Write the Equation

The equation for simple harmonic motion, given that the amplitude is 3 inches, the angular frequency is $\frac{2\pi}{7}$, and the initial displacement is negative, is:

$s(t) = -3 \cos\left(\frac{2\pi}{7} t \right)$

This matches the form of the equation in option C.

Final Answer:

C.) $s(t) = -3 \cos\left(\frac{2\pi}{7} t \right)$

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Would you like further clarification on how I derived the solution, or is there another concept you'd like to explore? Here are a few related questions to deepen the understanding:

1. What would the phase constant $\phi$ be if the displacement were $+3$ inches instead of $-3$?
2. How do you calculate the velocity and acceleration in simple harmonic motion?
3. What is the relationship between angular frequency $\omega$ and frequency $f$?
4. How do you find the period of a mass-spring system using Hooke's Law?
5. What happens to the motion if the object is displaced more than the amplitude?

Tip: The cosine function is typically used when an object starts at its maximum displacement, while the sine function is used when the object starts from the equilibrium position moving in the positive direction.