Answered

Connect with knowledgeable individuals and find the best answers at IDNLearn.com. Get thorough and trustworthy answers to your queries from our extensive network of knowledgeable professionals.

A pendulum is swinging next to a wall. The distance from the bob of the swinging pendulum to the wall varies in a periodic way that can be modeled by a trigonometric function.

The function has a period of 0.8 seconds, amplitude of 6 cm, and midline [tex]$H=15$[/tex] cm. At time [tex]$t=0.5$[/tex] seconds, the bob is at its midline, moving towards the wall.

Find the formula of the trigonometric function that models the distance [tex]$H$[/tex] from the pendulum's bob to the wall after [tex]$t$[/tex] seconds. Define the function using radians.

[tex]\[ H(t) = \square \][/tex]

Sagot :

GinnyAnsweridn
To find the trigonometric function that models the distance [tex]\( H \)[/tex] from the pendulum's bob to the wall after [tex]\( t \)[/tex] seconds, given the information:

- Period [tex]\( T = 0.8 \)[/tex] seconds
- Amplitude [tex]\( A = 6 \)[/tex] cm
- Midline [tex]\( H = 15 \)[/tex] cm
- At [tex]\( t = 0.5 \)[/tex] seconds, the bob is at its midline and moving towards the wall.

We can proceed as follows:

### Step 1: Determine the Angular Frequency
The angular frequency [tex]\( \omega \)[/tex] is calculated using the formula:
[tex]\[ \omega = \frac{2\pi}{T} \][/tex]
Given the period [tex]\( T = 0.8 \)[/tex] seconds:
[tex]\[ \omega = \frac{2\pi}{0.8} = 7.853981633974483 \][/tex]

### Step 2: Formulate the Function Based on Trigonometric Characteristics
The displacement of the pendulum from the midline can be represented by a cosine function because at [tex]\( t = 0.5 \)[/tex] seconds, the bob is at the midline and moving towards the wall, which aligns well with the behavior of the cosine function offset by [tex]\(\pi/2\)[/tex] phase shift:
[tex]\[ H(t) = A \cos(\omega t + \phi) + \text{midline} \][/tex]

### Step 3: Determine the Phase Shift [tex]\(\phi\)[/tex]
Since the bob is at the midline at [tex]\( t = 0.5 \)[/tex] seconds and moving towards the wall, let's set:
[tex]\[ \cos(\omega \cdot t + \phi) = 0 \][/tex]
At [tex]\( t = 0.5 \)[/tex] seconds:
[tex]\[ \cos(7.853981633974483 \cdot 0.5 + \phi) = 0 \][/tex]
Solving for [tex]\(\phi\)[/tex]:
[tex]\[ 7.853981633974483 \cdot 0.5 + \phi = \frac{\pi}{2} \][/tex]
[tex]\[ \phi = \frac{\pi}{2} - 3.9269908169872415 = -1.5707963267948966 \][/tex]

### Step 4: Write the Final Equation
Now we combine all the values into the cosine function to represent the distance [tex]\( H(t) \)[/tex].

Thus, the distance [tex]\( H(t) \)[/tex] from the pendulum's bob to the wall is:
[tex]\[ H(t) = 6 \cos(7.853981633974483 \cdot t - 1.5707963267948966) + 15 \][/tex]

Therefore, the formula for [tex]\( H(t) \)[/tex] is:
[tex]\[ H(t) = 6 \cos(7.853981633974483 t - 1.5707963267948966) + 15 \][/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.