$$  $$A buoy floating in the sea is bobbing in simple harmonic motion with period 2 seconds and amplitude 8 in. Its displacement d from sea level at time t = 0 seconds is 0 in, and initially it moves downward. (Note that downward is the negative direction.) Give the equation modeling the displacement d as a function of time t.

To model the displacement $ d $ of the buoy as a function of time $ t $, we will use the equation for simple harmonic motion. The general form of the equation is:

\[
d(t) = A \cos(\omega t + \phi)
\]

where:
- $ A $ is the amplitude,
- $ \omega $ is the angular frequency,
- $ \phi $ is the phase shift.

Given:
- The amplitude $ A $ is 8 inches.
- The period $ T $ is 2 seconds.
- At $ t = 0 $, the displacement $ d $ is 0 inches.
- Initially, the buoy moves downward.

### Step-by-Step Solution

1. **Determine the angular frequency $ \omega $**:
   The angular frequency $ \omega $ is related to the period $ T $ by the formula:
   \[
   \omega = \frac{2\pi}{T}
   \]
   Substituting $ T = 2 $ seconds:
   \[
   \omega = \frac{2\pi}{2} = \pi \text{ radians per second}
   \]

2. **Determine the phase shift $ \phi $**:
   At $ t = 0 $, $ d(0) = 0 $. We need to find $ \phi $ such that $ \cos(\phi) = 0 $.

   The cosine function is zero at $ \phi = \frac{\pi}{2} + k\pi $ for any integer $ k $. To satisfy the condition that initially the buoy moves downward (negative direction), we choose $ \phi = \frac{\pi}{2} $ because the derivative of $ \cos $ at $ \frac{\pi}{2} $ is negative, indicating a downward motion.

   Thus, $ \phi = \frac{\pi}{2} $.

3. **Write the displacement equation**:
   Substituting $ A = 8 $, $ \omega = \pi $, and $ \phi = \frac{\pi}{2} $ into the general form:
   \[
   d(t) = 8 \cos(\pi t + \frac{\pi}{2})
   \]

4. **Simplify the equation**:
   Using the trigonometric identity $ \cos(x + \frac{\pi}{2}) = -\sin(x) $, we can simplify the equation:
   \[
   d(t) = 8 \cos(\pi t + \frac{\pi}{2}) = 8(-\sin(\pi t)) = -8 \sin(\pi t)
   \]

### Final Equation
\[
d(t) = -8 \sin(\pi t)
\]

This equation models the displacement $ d $ of the buoy from sea level as a function of time $ t $.

Would you like more details or have any questions?

Here are 5 more questions you might be interested in:
1. How do you derive the period of a simple harmonic oscillator from its angular frequency?
2. Can you explain the significance of phase shift in simple harmonic motion?
3. What are the differences between sine and cosine functions in modeling periodic motion?
4. How can you determine the velocity of an object in simple harmonic motion?
5. How does the amplitude affect the energy in simple harmonic motion?

**Tip:** When dealing with harmonic motion, always check the initial conditions to correctly determine the phase shift.

Learn how to model the displacement of a buoy floating in the sea using simple harmonic motion. The problem involves determining the equation of motion based on given parameters such as amplitude, period, and initial conditions. Suitable for advanced high school students.

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