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

2A=

2A stands for the total amplitude of oscillation. In this case, the amplitude is the maximum distance from the equilibrium position (the centerline) to either side of the building. So, if we want to find 2A, we can use the given height of the building.

2A = 152 feet

Since one complete oscillation covers the distance from one extreme to the other and back again, the total distance side to side that the top of the building moves will be twice the amplitude:

Total distance = 2 * 2A

Now we just need to calculate the value of 2A with the given information.

To find the total distance, side to side, that the top of the building moves during an oscillation, we can use the formula for the amplitude of oscillation.

The formula for the amplitude (A) of oscillation is:

A = (acceleration of the top of the building) / (angular frequency)^2

Where:
Acceleration of the top of the building = 2.5 times the free-fall acceleration
Angular frequency (ω) = 2π times the frequency

First, let's calculate the angular frequency (ω):

Angular frequency (ω) = 2π times the frequency
ω = 2π * 0.12
ω = 0.24π

Next, let's calculate the acceleration of the top of the building:

Acceleration of the top of the building = 2.5 times the free-fall acceleration
Acceleration of the top of the building = 2.5 * 9.8 (considering the acceleration due to gravity as 9.8 m/s^2)
Acceleration of the top of the building = 24.5 m/s^2

Now, let's calculate the amplitude (A) using the formula:

A = (acceleration of the top of the building) / (angular frequency)^2

A = 24.5 / (0.24π)^2

To calculate the total distance, side to side, we need to double the amplitude:

Total distance = 2 * A

Total distance = 2 * (24.5 / (0.24π)^2)

Simplifying the equation gives us the total distance, side to side, that the top of the building moves during an oscillation.

To find the total distance, side to side, that the top of the building moves during its oscillation, we need to use the amplitude of the oscillation, which is represented by the letter "A." In this case, the amplitude is not given directly, but we can find it using the information given.

First, let's identify the relevant information given:

- Height of the building: 152 meters
- Frequency of oscillation: 0.12 Hz
- Maximum acceleration of the top of the building: 2.5 times the free-fall acceleration

Now, let's proceed with the step-by-step explanation:

Step 1: Calculate the period (T) of the oscillation.
The period (T) is the time taken for one complete cycle of oscillation and is calculated as the reciprocal of the frequency (f), represented as T = 1/f.

In this case, the frequency (f) is given as 0.12 Hz, so we can calculate the period as T = 1/0.12 = 8.33 seconds (rounded to two decimal places).

Step 2: Determine the angular frequency (ω).
The angular frequency (ω) is calculated as the reciprocal of the period (T), represented as ω = 2π/T.

In this case, the period (T) is 8.33 seconds, so we can calculate the angular frequency as ω = 2π/8.33 = 0.753 rad/s (rounded to three decimal places).

Step 3: Calculate the amplitude (A).
The amplitude (A) is the maximum displacement from the equilibrium position reached by the top of the building.

In this case, it is mentioned that the maximum acceleration of the top of the building is 2.5 times the free-fall acceleration. The free-fall acceleration is approximately 9.8 m/s². So, the maximum acceleration during the oscillation is 2.5 * 9.8 = 24.5 m/s².

The amplitude (A) can be calculated using the formula A = (maximum acceleration) / (angular frequency)².

Substituting the values, A = 24.5 / (0.753)² = 43.86 meters (rounded to two decimal places).

Step 4: Calculate the total distance (d) moved side-to-side by the top of the building.
The total distance moved side-to-side is equal to twice the amplitude, represented as d = 2A.

In this case, the amplitude (A) is 43.86 meters, so the total distance moved is d = 2 * 43.86 = 87.72 meters (rounded to two decimal places).

Therefore, the total distance, side to side, that the top of the building moves during its oscillation is approximately 87.72 meters.

period=1/.12 sec

a=2.5*9.8 m/s^2

acceleration=-w^2* displacement

w=2PI*f=2PI*.12

solve for displacement