Problem 14 - Algebra and Trigonometry

Question

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 35 cm, period 8 s

Step-by-Step Solution

Answer

The function modeling this simple harmonic motion is \( y(t) = 35 \cos\left(\frac{\pi}{4} t\right) \).

Step 1: Understand Simple Harmonic Motion

Simple harmonic motion is typically modeled by a trigonometric function such as a sine or cosine function. Since the maximum displacement occurs at time \( t = 0 \), we will use a cosine function, because \( \cos(0) = 1 \).

Step 2: Write the General Cosine Function for SHM

The general equation for simple harmonic motion using a cosine function is \( y(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase shift. In this problem, the phase shift \( \phi = 0 \) because the maximum displacement is at \( t = 0 \).

Step 3: Determine Amplitude

We are given the amplitude as 35 cm. Therefore, in our function, \( A = 35 \).

Step 4: Calculate the Angular Frequency

The angular frequency \( \omega \) is related to the period \( T \) by the equation \( \omega = \frac{2\pi}{T} \). Here, the period \( T = 8 \) s, thus \( \omega = \frac{2\pi}{8} = \frac{\pi}{4} \).

Step 5: Write the Complete Function

Substituting the values calculated: amplitude \( A = 35 \), angular frequency \( \omega = \frac{\pi}{4} \), and no phase shift, the function is given by \( y(t) = 35 \cos\left(\frac{\pi}{4} t\right) \).

Key Concepts

Amplitude: The Height of Vibration

In simple harmonic motion, amplitude refers to the maximum displacement of an object from its equilibrium position. Imagine it as the distance from the center to the farthest point of a swinging pendulum. Amplitude is important because it tells us how far the object travels during its motion.

• In our exercise, the amplitude is given as 35 cm. This value determines the peak displacements the function will achieve, above and below the equilibrium position.
• The larger the amplitude, the more energy the system has since it is moving a longer distance.

Amplitude is always a positive value, as it represents a distance. Think of it as the reach of the motion's stretch or swing, showing the limits of its vibration.

Trigonometric Function: The Motion Describers

Trigonometric functions like sine and cosine are tools used to model oscillatory or wave-like movements, making them perfect for simple harmonic motion. These functions repeat their values in a cyclical pattern because of their periodic nature.

• The cosine function is often used when the motion starts at the maximum displacement or peak because \(\cos(0) = 1\).
• In the context of the problem, the cosine function is ideal since the displacement is maximal at time \(t = 0\).

The general form of a trigonometric function for simple harmonic motion is \(y(t) = A \cos(\omega t + \phi)\). Here, \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase shift. By choosing a cosine function, the modeling aligns with how the motion behaves at the beginning of its cycle. This cyclic nature allows the prediction of future positions of the moving object using these functions.

Angular Frequency: The Speed of the Cycle

Angular frequency is a measure of how quickly the motion cycles through its pattern. It tells us the rate at which the object goes through its full cycle, equivalent to \(2\pi\) radians per cycle. Mathematically, it's calculated using the relationship \(\omega = \frac{2\pi}{T}\), where \(T\) is the period of the motion in seconds.

• In the given problem, the period \(T\) is 8 seconds, leading to an angular frequency of \(\omega = \frac{\pi}{4}\).
• This means every 8 seconds, the system completes a full oscillation, moving swiftly or slowly depending on the period.

Angular frequency is crucial as it determines how fast the object swings back and forth. A higher \(\omega\) means the system oscillates more quickly. It helps predict the timing of the peaks and troughs in the simple harmonic motion model.