A buoy floating in the sea is bobbing in simple harmonic motion with amplitude 13 in and period 0.2 seconds. Its displacement *d* from sea level at time *t =* 0 seconds is 0 in, and initially it moves upward. (Note that upward is the positive direction.) Give the equation modeling the displacement *d* as a function of time *t*.

Given that a buoy is bobbing in simple harmonic motion, we can model its displacement using the equation for simple harmonic motion:

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

where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase shift,
- \( d(t) \) is the displacement at time \( t \).

From the problem, we have:
- Amplitude \( A = 13 \) inches,
- Period \( T = 0.2 \) seconds.

### Step-by-Step Solution:

1. **Determine the angular frequency (\(\omega\)):**
   \[ \omega = \frac{2\pi}{T} \]
   \[ \omega = \frac{2\pi}{0.2} = 10\pi \text{ rad/s} \]

2. **Initial conditions to find the phase shift (\(\phi\)):**
   At \( t = 0 \), \( d(0) = 0 \) inches, and initially, the buoy moves upward. This means at \( t = 0 \), \( \cos(\phi) = 0 \).

   The cosine function is zero at \(\phi = \frac{\pi}{2}\) and \(\phi = \frac{3\pi}{2}\). Given the buoy moves upward initially, we use:
   \[ \phi = \frac{\pi}{2} \]

3. **Construct the displacement equation:**
   \[ d(t) = 13 \cos(10\pi t + \frac{\pi}{2}) \]

4. **Simplify using trigonometric identities:**
   \[ \cos(10\pi t + \frac{\pi}{2}) = -\sin(10\pi t) \]

   Thus, the equation becomes:
   \[ d(t) = -13 \sin(10\pi t) \]

### Final Equation:

\[ d(t) = -13 \sin(10\pi t) \]

This equation models the displacement \( d \) of the buoy as a function of time \( t \).

Would you like further details or have any questions about this solution?

### Relative Questions:
1. How is the amplitude related to the maximum displacement in simple harmonic motion?
2. What is the significance of the angular frequency in the context of harmonic motion?
3. How do you determine the phase shift when given initial conditions?
4. What other forms can the equation of simple harmonic motion take?
5. How would the equation change if the buoy started at the maximum displacement?
6. What is the relationship between period and frequency in harmonic motion?
7. How can the displacement equation be verified experimentally?
8. What are some real-life examples of simple harmonic motion other than a buoy in the sea?

### Tip:
Always remember that the phase shift determines where the oscillation starts in its cycle, which is crucial for accurately modeling the motion.

Explore how to model the displacement of a buoy in the sea using simple harmonic motion. Learn about the amplitude, angular frequency, and phase shift involved in the equation d(t) = 13 * cos(10πt + π/2). Suitable for high school students studying physics.

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