The New England Merchants Bank Building in Boston is 152 high. On windy days it sways with a frequency of 0.12 , and the acceleration of the top of the building can reach 2.5 of the free-fall acceleration, enough to cause discomfort for occupants.
What is the total distance, side to side, that the top of the building moves during such an oscillation?
Could someone pretty please go over this with me step by step? Thanks ahead of time
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1´2
Certainly! Let's break down the problem step by step to calculate the total distance that the top of the building moves during the oscillation.
Step 1: Determine the amplitude of the oscillation.
In this case, the amplitude is not given directly. However, we know that the acceleration of the top of the building can reach 2.5 times the free-fall acceleration. The displacement of an oscillating object is related to the amplitude, so we can use this information to find the amplitude.
The acceleration of the top of the building is given by the equation:
a = -ω^2x
Here, ω is the angular frequency (2πf), x is the displacement, and a is the acceleration. Since the acceleration is the maximum value of 2.5 times the free-fall acceleration, we can rearrange the equation to solve for the displacement:
2.5g = -ω^2x
We know that g (free-fall acceleration) is approximately 9.8 m/s^2. Rearranging the equation gives us:
x = -(2.5g)/(ω^2)
Step 2: Calculate the angular frequency (ω).
The frequency (f) of the oscillation is given as 0.12 Hz. The angular frequency (ω) is related to the frequency by the equation ω = 2πf. Substituting the given frequency:
ω = 2π(0.12)
Step 3: Substitute the values in the displacement equation to find the amplitude.
Using the values we have obtained so far, we can substitute them into the equation for x to find the displacement (amplitude):
x = -(2.5g)/(ω^2)
Step 4: Calculate the total distance moved during oscillation.
The total distance moved during the oscillation is twice the amplitude. This is because the building moves from one extreme point to the other, covering the same distance on either side of its equilibrium position.
Total distance = 2 * amplitude
Step 5: Substitute the amplitude value to find the total distance.
Now, substitute the amplitude value calculated in step 3 into the equation to find the total distance:
Total distance = 2 * |x|
Remember to take the absolute value of x to consider the distance, as displacement can be negative.
Solve this equation to find the total distance moved during the oscillation.
I hope these steps help you solve the problem! Let me know if you need any further assistance.