Problem 43 - Algebra and Trigonometry

Question

A strong gust of wind strikes a tall building, causing it to sway back and forth in damped harmonic motion. The frequency of the oscillation is 0.5 cycle per second and the damping constant is \(c=0.9 .\) Find an equation that describes the motion of the building. (Assume \(k=1\) and take \(t=0\) to be the instant when the gust of wind strikes the building.)

Step-by-Step Solution

Answer

The equation is \( x(t) = e^{-0.9t} \cos(\pi t) \).

Step 1: Identify the Damped Harmonic Motion Equation

The equation for damped harmonic motion is given by \[ x(t) = A e^{-ct} \cos(\omega t + \phi) \]where \( A \) is the amplitude, \( c \) is the damping constant, \( \omega \) is the angular frequency, and \( \phi \) is the phase angle.

Step 2: Calculate Angular Frequency

Given the frequency of oscillation \( f = 0.5 \) cycles per second, convert it to angular frequency using the formula: \[ \omega = 2\pi f \]Substituting the values, we get:\[ \omega = 2\pi \times 0.5 = \pi \]

Step 3: Substitute Known Values

We know \( c = 0.9 \), \( \omega = \pi \), and from the problem statement, assume initial amplitude \( A = 1 \) and phase angle \( \phi = 0 \). Substitute these into the equation:\[ x(t) = e^{-0.9t} \cos(\pi t) \]

Step 4: Write the Final Equation

The equation describing the motion of the building swaying due to the wind gust is:\[ x(t) = e^{-0.9t} \cos(\pi t) \]

Key Concepts

Frequency of Oscillation

The frequency of oscillation plays a crucial role in describing how often an oscillating system repeats its cycle. In the context of damped harmonic motion, frequency refers to the number of times the system completes its motion in a second. For example, if a building sways back and forth 0.5 times per second due to a gust of wind, the frequency is 0.5 cycles per second (Hz).

Understanding the frequency is important because it gives insight into the behavior of the oscillating system:

• A higher frequency indicates a faster oscillation, meaning the system completes more cycles in a given time.
• A lower frequency signifies slower motion, with fewer cycles in the same interval.

In damped harmonic motion, the frequency is influenced by external factors such as the system's natural properties and external forces like wind gusts. Knowing the frequency helps us calculate other important parameters like the angular frequency.

Angular Frequency

Angular frequency, which is denoted by the symbol \( \omega \), is closely related to the frequency of oscillation. It offers a different perspective, focusing on the rotational aspect of oscillatory motion. Angular frequency is calculated by transforming the frequency into a radian measure, using the formula \( \omega = 2\pi f \). Here, \( f \) represents the frequency in cycles per second.

To visualize this, think of angular frequency as how rapidly the system moves through its cycle in terms of angles:

• If \( f = 0.5 \), then the angular frequency \( \omega = 2\pi \times 0.5 = \pi \). This tells us that for each complete cycle, the system moves through \( \pi \) radians per second.
• Angular frequency is essential in determining the system's dynamic response to forces, including the effects of damping.

By understanding angular frequency, we can better predict the motion of systems under various conditions, such as the swaying building, which resonates at a particular angular frequency when disturbed.

Damping Constant

The damping constant, designated as \( c \), is a parameter that describes how quickly the oscillations of a system diminish due to factors like friction or air resistance. In damped harmonic motion, the damping constant is crucial for understanding how energy loss affects the system's behavior over time.

Think of the damping constant as a measure of "motion resistance":

• A larger damping constant means the system loses energy more rapidly, hence oscillations will die out faster.
• With a smaller damping constant, the system retains its energy longer, resulting in prolonged oscillations before they come to a halt.

In the case of the building swaying due to the wind, a damping constant of 0.9 signifies moderate resistance against motion. This ensures that while the building sways visibly, it slowly comes to rest over time without continuous motion, ensuring structural stability and integrity.