Hello, I want to know what the amplitude and the phase shift would be for this question to model the equation
Q. A pendulum is connected to a rope 3 m long, which is connected to a ceiling 4 m high. The angle between its widest swing and vertical hanging position is pi/3. If the pendulum swings out to its widest position in 2 seconds, model the height of the pendulum from the ground using a cosine function, considering vertical to be t = 0.
Would it be 4-3/2=1/2 for the amplitude?
lowest height = 4 - 3 = 1
greatest height = 4 - 3/2 = 5/2
amplitude = (5/2 - 1)/2 = 3/4
Ok. I want to know how you got the maxiumum height, I knew the minimum height.
Can you please explain how you got the value of 3/2 for the height when the pendulum was at its widest swing?
Steve, can you please explain how you got 3/2 for the height of the pendulum at its widest position
To model the height of the pendulum from the ground over time, we can use a cosine function of the form:
h(t) = A * cos(B * (t - C))
Where:
- A represents the amplitude of the cosine function,
- B represents the frequency, and
- C represents the phase shift.
To determine the values of A, B, and C, we need to consider the given information.
1. Amplitude (A): The amplitude of the cosine function represents half of the vertical distance between the highest and lowest points of the pendulum's swing. In this case, the pendulum swings out to its widest position, which is 3 meters away from the vertical hanging position. Thus, the amplitude (A) will be half of this distance, which is 3/2 = 1.5 meters.
2. Frequency (B): The frequency of the cosine function represents the number of cycles the pendulum completes in a given time period. Since the pendulum swings out to its widest position in 2 seconds, it completes one full swing in 2 seconds. The frequency (B) can be calculated as 2π divided by the period, where the period is the time taken for one full swing. In this case, the period is 2 seconds, so the frequency (B) will be 2π/2 = π radians per second.
3. Phase Shift (C): The phase shift of the cosine function represents the horizontal shift of the graph. In this case, the vertical hanging position of the pendulum is considered as t = 0. Since the pendulum is already at its widest position when time t = 0, there is no phase shift, and C = 0.
Now that we have determined the values of A, B, and C, the equation to model the height of the pendulum from the ground is:
h(t) = 1.5 * cos(π * t)
This equation will give you the height of the pendulum from the ground at any given time (t) in seconds.