Question 1179768: https://gyazo.com/78c37f667cf2af63efadea4314e5642c
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step to model the height of the pendulum using a cosine function.
**1. Visualize the Situation:**
* Imagine a ceiling 4 meters high.
* A rope 3 meters long hangs from the ceiling.
* A pendulum is attached to the end of the rope.
* The pendulum swings, with its widest angle being π/3 (60 degrees) from the vertical.
**2. Determine Key Values:**
* **Rope Length (L):** 3 meters
* **Ceiling Height (C):** 4 meters
* **Amplitude (A):** The horizontal distance the pendulum swings. We'll need to find this.
* **Period (T):** The time for a complete swing (out and back). Since it swings out in 2 seconds, the total period is 4 seconds.
* **Vertical Shift (D):** The vertical distance from the ground to the midline of the pendulum's swing.
* **Horizontal Shift (C):** Since the pendulum is at its vertical position at t=0, there is no horizontal shift.
**3. Calculate Amplitude (A):**
* The horizontal displacement of the pendulum at its widest swing is L * sin(π/3).
* A = 3 * sin(π/3) = 3 * (√3 / 2) = (3√3) / 2 meters.
**4. Calculate Vertical Shift (D):**
* The vertical distance from the ceiling to the pendulum at its lowest point is L * cos(π/3).
* Vertical distance = 3 * cos(π/3) = 3 * (1/2) = 1.5 meters.
* The height from the ground to the lowest point is C - 1.5 = 4 - 1.5 = 2.5 meters.
* The midline of the swing is the average of the highest and lowest points. The highest point is 4 meters, and the lowest is 2.5.
* D = (4 + 2.5) / 2 = 6.5 / 2 = 3.25 meters.
**5. Calculate the Angular Frequency (B):**
* The period (T) is 4 seconds.
* B = 2π / T = 2π / 4 = π/2
**6. Write the Cosine Function:**
* The general form is: y = A * cos(Bt) + D
* Since the pendulum starts at its highest point (vertical position), we use a standard cosine function (no horizontal shift).
* y = ((3√3) / 2) * cos((π/2)t) + 3.25
**7. Convert to Height from Ground:**
* The height of the pendulum from the ground is given by the cosine function.
**Final Answer:**
The height of the pendulum from the ground as a function of time (t) is:
y = ((3√3) / 2) * cos((π/2)t) + 3.25
Where:
* y is the height from the ground in meters.
* t is the time in seconds.
Answer by ikleyn(52908)  (Show Source):
You can put this solution on YOUR website! .
The solution in the post by @CPhill is irrelevant to the problem
and is TOTALLY and FATALLY wrong.
The problem asks to model the sinusoidal horizontal displacements of the pendulum
from vertical line x= 0.
Instead, @CPhill models VERTICAL displacement along vertical axis,
i.e. vertical position of the pendulum versus time.
So, @CPhill incorrectly reads/interprets the problem.
Interesting to note, that @CPhill correctly determines the amplitude of the horizontal displacement,
but then apply it to model vertical position, which is nonsense.
There is doubled error in it, since vertical position of the pendulum is not a sinusoidal function,
at all, and the conception of an amplitude is not applicable to it.
In plain words, this solution is a mess and is good only to throw it to a garbage bin.
==========================
Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.
The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.
It has no feeling of shame - it is shameless.
This time, again, it made an error.
Although the @CPhill' solution are copy-paste Google AI solutions, there is one essential difference.
Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.
All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.
Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.
And the last my comment.
When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.
Without it, their reliability is ZERO and their creadability is ZERO, too.
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