To model the distance [tex]\( H(t) \)[/tex] of the pendulum's bob from the wall using a trigonometric function, we need to consider the given parameters and information. Here’s a detailed step-by-step solution:
1. Identify Given Information:
- Period ([tex]\( T \)[/tex]) = 0.8 seconds
- Amplitude ([tex]\( A \)[/tex]) = 6 cm
- Midline = 15 cm
- At [tex]\( t = 0.2 \)[/tex] seconds, the bob is at its midline moving towards the wall.
2. Determine the Angular Frequency ([tex]\( \omega \)[/tex]):
The angular frequency [tex]\(\omega\)[/tex] is given by:
[tex]\[
\omega = \frac{2\pi}{T}
\][/tex]
Substituting the given period:
[tex]\[
\omega = \frac{2\pi}{0.8} = 7.853981633974483 \text{ rad/s}
\][/tex]
3. Identify Phase Shift:
- When [tex]\( t = 0.2 \)[/tex] seconds, the distance [tex]\( H(t) \)[/tex] is at the midline and moving towards the wall. This suggests that in the trigonometric function, [tex]\( \sin(\theta) \)[/tex] should ideally be at [tex]\(\sin(-\frac{\pi}{2}) = -1\)[/tex]. Since the bob is at the midline (15 cm) and moving towards the wall, the appropriate sine value would make [tex]\( \theta = -\frac{\pi}{2} \)[/tex].
- Determine the total phase shift for [tex]\( t = 0.2 \)[/tex]:
[tex]\[
\text{Phase shift} = \omega \cdot t - \frac{\pi}{2}
\][/tex]
- Substituting the values:
[tex]\[
\text{Phase shift} = 7.853981633974483 \times 0.2 - \frac{\pi}{2} = 0
\][/tex]
- Hence, the phase shift is [tex]\( 0 \)[/tex].
4. Formulate the Function [tex]\( H(t) \)[/tex]:
Given the amplitude, midline, angular frequency, and determined phase shift, the distance [tex]\( H(t) \)[/tex] can be modeled as:
[tex]\[
H(t) = A \sin(\omega t - \text{phase shift}) + \text{midline}
\][/tex]
Since the phase shift is 0:
[tex]\[
H(t) = 6 \sin(7.853981633974483 \cdot t) + 15
\][/tex]
Therefore, the formula for the trigonometric function that models the distance [tex]\( H \)[/tex] between the bob and the wall after [tex]\( t \)[/tex] seconds is:
[tex]\[
H(t) = 6 \sin(7.853981633974483 \cdot t) + 15
\][/tex]
