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 π/3. If the pendulum swings out to its widest position in 2 seconds, model the horizontal displacement of the pendulum using a sinusoidal function, considering vertical to be x = 0.
so the period is 2*4 = 8 seconds, making
y = 3sin(π/3) sin(π/4 x)
To model the horizontal displacement of the pendulum using a sinusoidal function, we need to understand the relationship between the angle of swing and the horizontal position. Let's break down the given information step by step:
1. The length of the rope is 3 meters.
This means that the total length of the pendulum's path is 3 meters. When the pendulum swings to its widest position, the rope forms a hypotenuse of a right triangle with the vertical (ceiling) and horizontal distances as its other two sides.
2. The height of the ceiling is 4 meters.
This information is needed because it helps determine the vertical component of the pendulum's position, which is not relevant to the horizontal displacement we want to model.
3. The angle between the pendulum's widest swing and the vertical hanging position is π/3 (60 degrees).
This is the maximum angle of swing for the pendulum. At this angle, the pendulum is at its widest position. We will use this information to identify the maximum horizontal displacement.
4. The pendulum swings out to its widest position in 2 seconds.
This information represents the time it takes for the pendulum to complete one full back-and-forth swing.
Now, let's find the maximum horizontal displacement of the pendulum:
Using the length of the rope (3 meters) and the maximum swing angle (π/3), we can calculate the maximum horizontal displacement using trigonometry. Since the vertical component (ceiling height of 4 meters) does not affect the horizontal displacement, we only need to consider the horizontal component.
The horizontal component can be found using the formula:
horizontal displacement = length of the rope * sin(angle of swing)
horizontal displacement = 3 * sin(π/3) = 3 * (√3/2) = √3 * 3/2 = (3√3)/2
Therefore, the maximum horizontal displacement of the pendulum is (3√3)/2 meters.
To model the horizontal displacement of the pendulum as a sinusoidal function, we know that the period is 2 seconds (one complete back-and-forth swing).
The general equation for a sinusoidal function is:
y = A * sin(B * x + C) + D
To fit this equation to the given scenario:
- A represents the amplitude, which is the maximum distance of the horizontal displacement, so A = (3√3)/2.
- B is related to the period of the function, and since the period is 2 seconds, we can use the formula B = 2π/period.
B = 2π/2 = π
- C represents the phase shift, which determines the starting position of the pendulum. Since the horizontal displacement is measured from the vertical position, C = 0.
- D represents the vertical displacement, but in our case, we are only interested in the horizontal displacement, so we don't need to consider D.
The equation for modeling the horizontal displacement of the pendulum becomes:
x = (3√3)/2 * sin(π * t)
Where x represents the horizontal displacement at time t.