Skyscrapers sway in high-wind conditions. in one case, at t=2s, the top floor swayed 30cm to the left (-30cm), and at t=12, the top floor swayed 30 cm to the right (+30cm) of its starting position.
a) what is the equation of a sinusoidal function that describes the motion of the building in terms of time?
b) if a damper is added to this building, it will reduce the sway (not the period) by 70%. what is the equation of the new function that describes the motion of the building in terms of time?
a) |a|= 30 cm
1/2 period = 10
period = 20
k= 360/ period
= 360/20
= 18
p = 2
a = -30 (as a reflected cosine)
therefore, d(t)= -30 cos{18(t-2)}
b) Reduce sway = reduce of amplitude
|a|= 100% - 70%
= 30%
|a| = 30 (.3)
= 9 cos{ 18(t-2)}
a) Well, it seems like this skyscraper is quite the dancer! To describe its swaying motion, we can use a sinusoidal function. Let's assume the function relates the displacement (d) of the top floor to time (t):
d(t) = A * sin(B * t + C) + D
In this equation, A is the amplitude, B determines the period, C represents the phase shift, and D is any vertical shift. To find the values of these parameters, let's analyze the given information:
At t = 2s, the top floor swayed 30cm to the left (-30cm). This means d(2) = -30.
At t = 12s, the top floor swayed 30cm to the right (+30cm). This means d(12) = 30.
Plugging these into the equation, we have two equations:
-30 = A * sin(2B + C) + D
30 = A * sin(12B + C) + D
With these two equations, we can solve for A, B, C, and D. I'll let you do the honors of solving this system of equations!
b) Ah, the building is getting a damper treatment! Now, if the sway is reduced by 70%, we need to adjust the amplitude (A). The new amplitude will be 70% of the original amplitude. So, the new amplitude (A') would be:
A' = 0.7 * A
But the period (B), phase shift (C), and vertical shift (D) will remain the same. Thus, the new equation for the building's motion with the damper can be represented as:
d'(t) = A' * sin(B * t + C) + D
Substituting the value of A' back in, we have:
d'(t) = 0.7 * A * sin(B * t + C) + D
There you have it! The new equation describes the building's swaying motion with the added damper.
a) To describe the motion of the building in terms of time using a sinusoidal function, we can use the following equation:
y(t) = A * sin(ωt + φ) + C
Where:
- A is the amplitude (half the distance between the maximum and minimum positions).
- ω is the angular frequency (2π divided by the period).
- φ is the phase shift.
- C is the vertical shift.
In this case, since the top floor swayed 30 cm to the left and 30 cm to the right, the amplitude is 30 cm. The period can be calculated by finding the time difference between the two instances of maximum displacement. In this case, the time difference is 12 - 2 = 10 seconds. Therefore, the period is 10 s.
The amplitude is half the distance between the maximum and minimum positions, so it is (30 cm)/2 = 15 cm.
The angular frequency, ω, can be calculated as ω = 2π / T, where T is the period. Substituting the given period value, we get ω = 2π / 10.
The phase shift, φ, can be determined by observing that the building starts swaying to the left (negative displacement) at t = 2 s. This means there is a phase shift of -2 seconds.
The vertical shift, C, is the equilibrium position of the building, which is considered to be the starting position. Here, the starting position is 0 cm, so C = 0.
Therefore, the equation of the sinusoidal function that describes the motion of the building in terms of time is:
y(t) = 15 * sin((2π / 10) * (t - 2))
b) To account for the reduction in sway by 70% when a damper is added, we need to multiply the amplitude by 0.3 (since 70% reduction is equivalent to 30% remaining).
The new amplitude, A_new, is 0.3 * 15 = 4.5 cm.
Therefore, the equation of the new function that describes the motion of the building in terms of time is:
y(t) = 4.5 * sin((2π / 10) * (t - 2))
To find the equation of a sinusoidal function that describes the motion of the building in terms of time, we can start by analyzing the given information.
a) The general form of a sinusoidal function that describes motion is given by:
y = A*sin(B(t - C)) + D
Where:
- A is the amplitude (half the vertical distance between the maximum and minimum values).
- B determines the frequency or period of the function.
- C represents a horizontal shift or time delay of the function.
- D is the vertical shift or offset of the function.
Given that the top floor swayed 30 cm to the left at t=2s and then 30 cm to the right at t=12s, we can determine the values of A, B, C, and D.
1. Amplitude (A):
The amplitude is half the vertical distance between the maximum and minimum values. In this case, the maximum displacement is +30cm and the minimum displacement is -30cm. Therefore, the amplitude is (30 + 30)/2 = 30 cm.
2. Frequency or Period (B):
The period is the time it takes for one complete cycle of the motion. Since the building swayed to the left and then back to its starting position, the time interval for one complete cycle is 12s - 2s = 10s. Therefore, the period (T) is 10s, and the frequency (f) is 1/T = 1/10 = 0.1 Hz.
3. Horizontal Shift or Time Delay (C):
Given that the building swayed to the left at t=2s, we can use this as the reference point. Therefore, the horizontal shift or time delay (C) is 2s.
4. Vertical Shift or Offset (D):
Since the building swayed to the left at t=2s and returned to its starting position, the vertical shift or offset (D) is 0 cm.
Applying these values to the sinusoidal function, the equation that describes the motion of the building is:
y = 30*sin(2π*0.1(t - 2))
b) If a damper is added to the building, it will reduce the sway by 70%. This means the new amplitude (A') will be 30 cm * 0.7 = 21 cm.
The other parameters, frequency (B), horizontal shift (C), and vertical shift (D) remain unchanged. Therefore, the equation for the new motion of the building with the damper can be written as follows:
y = 21*sin(2π*0.1(t - 2))
I assume that the building goes directly from one extreme to the other in 12-2=10 seconds. Therefore the half-period is 10 seconds, or the period T=20 s.
The neutral position is at (2+10)/2=6 seconds, when the displacement is zero.
Thus we have
Xo=30 cm = 0.3m
X(t)=Xo sin((t-6)*2π/T), or
X(t)=0.3sin((t-6)π/10)