A very tall light standard is swaying in an east-west direction in a strong wind. An observer notes that the time difference between the vertical position and the furthest point of sway was 2 seconds. The pole is 40 metres tall. At the furthest point of sway, the top of the pole is 1° out of the vertical position when measured from the bottom of the pole. Create an equation that models the behaviour of the top of the pole.
1 ° (Angstrom) is about the diameter of one atom. That is not much "swaying". The length of the pole does not matter, since that have told you the amplitude. The period of oscillation is in this case
P = 8 seconds. The east-west position of the top will vary according to
x = (Amplitude)*sin (2 pi t/P)
= (Amplitude)* sin (pi t/4)
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To create an equation that models the behavior of the top of the pole, we can use the concept of simple harmonic motion. In this case, the pole is acting as a pendulum, with the wind causing it to sway back and forth.
Let's assume that the position of the top of the pole, when measured from its vertical position, is given by the angle θ at time t.
Step 1: Determine the period of the oscillation.
The period of a pendulum can be found using the formula:
T = 2π√(L/g)
Where:
T = period of the oscillation
L = length of the pendulum (in this case, the height of the pole = 40 meters)
g = acceleration due to gravity (approximately 9.8 m/s²)
Substituting the given values:
T = 2π√(40/9.8)
Step 2: Convert the period to angular frequency.
The angular frequency (ω) can be found using the formula:
ω = 2π/T
Substituting the value of T calculated in Step 1:
ω = 2π / (2π√(40/9.8))
ω = √(9.8/40)
Step 3: Express the position of the top of the pole as a function of time.
The equation that describes the behavior of the top of the pole can be written as:
θ(t) = A * cos(ωt + ϕ)
Where:
A = amplitude of the oscillation (in this case, the maximum angle of 1 degree = π/180 radians)
cos = cosine function
ω = angular frequency (calculated in Step 2)
t = time
ϕ = phase constant (will be determined using the given information)
Step 4: Determine the phase constant (ϕ).
The phase constant can be determined based on the information that the time difference between the vertical position and the furthest point of sway is 2 seconds. At the furthest point of sway, the top of the pole is 1° out of the vertical position when measured from the bottom of the pole.
Since the position is given by the cosine function, the phase constant can be determined by the condition when θ = 0 at t = 2 seconds.
Substituting these values into the equation:
0 = A * cos(ω * 2 + ϕ)
Given that the angle is 1° out of the vertical position (θ = π/180) at the furthest point of sway, we can also write:
π/180 = A * cos(ω * 0 + ϕ)
From these two equations, we can solve for the phase constant ϕ.
Step 5: Substitute the values into the equation.
Now that we have the values for A and ϕ, we can substitute them into the equation:
θ(t) = (π/180) * cos(√(9.8/40) * t + ϕ)
This equation models the behavior of the top of the pole as it sways in the wind.
To create an equation that models the behavior of the top of the pole, we need to consider the characteristics mentioned in the problem: the time difference between the vertical position and the furthest point of sway, the height of the pole, and the angle at the furthest point of sway.
Let's start by considering the time difference. We're told that the time difference between the vertical position and the furthest point of sway is 2 seconds. This means that it takes 2 seconds for the top of the pole to move from the vertical position to the furthest point of sway, and vice versa.
Next, let's consider the height of the pole, which is 40 meters. Since the top of the pole is moving in an east-west direction, we can assume that the displacement from the vertical position when swaying furthest is proportional to the height of the pole.
Finally, we have the angle at the furthest point of sway. We know that the top of the pole is 1° out of the vertical position when measured from the bottom of the pole at the furthest point of sway.
Now, let's create an equation to model the behavior of the top of the pole:
Let x represent the time (in seconds).
Let y represent the displacement (in meters) of the top of the pole from the vertical position.
Since the time difference is 2 seconds, we can write:
x = 0 at the vertical position
x = 2 at the furthest point of sway
The displacement y will be 0 at the vertical position and positive at the furthest point of sway. We can express this using a sinusoidal function. Additionally, the amplitude of the sinusoidal function will be half the height of the pole (since the top of the pole sways back and forth), which is 40/2 = 20 meters.
Therefore, the equation to model the behavior of the top of the pole is:
y = 20 * sin((x/2)π)
This equation captures the vertical displacement of the top of the pole as it sways in an east-west direction in the wind, given the specified conditions.