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Angular frequency, often symbolized as \( \omega \), is a fundamental concept in simple harmonic motion (SHM). This term measures how rapidly an object oscillates or cycles through its motion. It is typically expressed in radians per second (rad/s).
The formula to calculate angular frequency \( \omega \) is:

• \( \omega = \frac{2\pi}{T} \)

where \( T \) is the period of oscillation, which is the time it takes for one complete cycle of motion.
In our example of the Sears Tower, the period \( T \) is given as 7 seconds. So, \( \omega = \frac{2\pi}{7} \approx 0.897 \text{ rad/s} \).
This means that the top of the building completes about 0.897 radians of its oscillation per second. Understanding angular frequency helps us describe the rhythm of motion and is essential for finding other important motion parameters like maximum velocity and acceleration.

The amplitude of oscillation is a term used to explain the maximum extent of displacement from the equilibrium position during oscillation. It is a crucial aspect of simple harmonic motion that defines the peak range or distance that the object travels from its center or resting state.
In simpler terms, amplitude indicates how far, at maximum, the object swings away from its median position during each cycle of motion. This is typically conveyed in meters or centimeters depending on the context.
In this scenario, the amplitude for the Sears Tower is 15 cm or 0.15 meters. This figure reflects how much the top of the tower sways to one side at its furthest reach. Why is understanding amplitude important? It helps us identify how dramatic or mild the sway is, which is pivotal in assessing the impact on the structure and the comfort of the people inside.

Maximum acceleration in simple harmonic motion refers to the highest rate of velocity change that occurs during oscillation. It gives us insight into the forces exerted on the object at the extremes of its motion.
For any object undergoing SHM, the formula to determine this maximum acceleration, \( a_{\text{max}} \), is:

• \( a_{\text{max}} = \omega^2 A \)

where \( \omega \) is the angular frequency and \( A \) is the amplitude of oscillation.
In our building sway example, using \( \omega \approx 0.897 \text{ rad/s} \) and \( A = 0.15 \text{ m} \), we get \( a_{\text{max}} \approx 0.1205 \text{ m/s}^2 \).
Understanding maximum acceleration is crucial since it tells us how intense the forces can be during the oscillation. In structural engineering, it helps in comparing how those forces measure against gravitational acceleration (\( g = 9.81 \text{ m/s}^2 \)), allowing for a safety assessment as seen by the calculation of it as a percentage of gravity, here about 1.23%.