Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 67-70: The formula specifies the position of a point \(P\) that is moving harmonically on a vertical axis, where \(t\) is in seconds and \(d\) is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at \(t=0\) ). $$ d=6 \sin \frac{2 \pi}{3} t $$

Step-by-Step Solution

Answer

Amplitude: 6 cm, Period: 3 s, Frequency: 1/3 Hz; the point oscillates vertically between +6 cm and -6 cm.

Step 1: Identifying the Equation and Components

The given equation for the motion of the point is \( d = 6 \sin \left( \frac{2\pi}{3} t \right) \). Here, \(6\) is the amplitude, \(\frac{2\pi}{3}\) is the angular frequency \(\omega\), and \(t\) is the time.

Step 2: Calculating the Amplitude

The amplitude of the motion is the coefficient of the sine function in the equation. Here, the amplitude is \(6\) centimeters.

Step 3: Determining the Period

The period \(T\) of the motion is determined by the formula \(T = \frac{2\pi}{\omega}\). Substitute \(\omega = \frac{2\pi}{3}\) to find \(T = \frac{2\pi}{\frac{2\pi}{3}} = 3\) seconds.

Step 4: Finding the Frequency

The frequency \(f\) is the reciprocal of the period, calculated by \(f = \frac{1}{T}\). With \(T = 3\) seconds, the frequency is \(f = \frac{1}{3}\) Hz.

Step 5: Describing the Motion for One Complete Oscillation

In one complete oscillation starting at \(t=0\), the point moves from the equilibrium position \(d=0\) up to \(d=6\), back through \(d=0\), down to \(d=-6\), and returns to \(d=0\). This motion completes within 3 seconds, moving harmonically.

Key Concepts

Amplitude

When discussing harmonic motion, amplitude is a key component. In simple terms, amplitude refers to how far a point moves from its resting or equilibrium position. Imagine it as the height of a wave from its center line. In the given equation \( d = 6 \sin \frac{2\pi}{3} t \), the amplitude is represented by the number before the sine function, which is 6 centimeters. This means that the point \( P \) will move 6 centimeters above and below the equilibrium position during its motion.
Amplitude is vital because it tells us the maximum extent of the motion. Unlike other aspects such as frequency or period, amplitude is solely about the distance covered during this motion.

• Maximum displacement: 6 cm
• Measured from the equilibrium position
• Determines the energy of the oscillation

Period

The period of harmonic motion denotes the time it takes for a complete cycle or oscillation to occur. In essence, it's how long it takes for the point to return to its starting position and start over. For the equation \( d = 6 \sin \frac{2\pi}{3} t \), the period \( T \) can be found using the formula \( T = \frac{2\pi}{\omega} \), where \( \omega \) represents the angular frequency. Substituting \( \omega = \frac{2\pi}{3} \), we find \( T = 3 \) seconds.
This means it takes 3 seconds for the point to complete one full oscillation. Understanding the period is crucial, as it gives insight into the speed of the harmonic motion.

• Period \( T = 3 \) seconds
• Time for one complete cycle
• Influences the frequency

Frequency

Frequency is a measure of how often oscillations occur in a specific time frame, typically measured in hertz (Hz), which denotes cycles per second. It is the reciprocal of the period and can be calculated using the formula \( f = \frac{1}{T} \). For the given motion equation, since the period \( T \) is 3 seconds, the frequency \( f \) becomes \( \frac{1}{3} \) Hz.
Frequency provides a sense of urgency or repetition for the motion. In this context, it tells us that the point completes \( \frac{1}{3} \) of a cycle each second, which frames how quickly the harmonic motion repeats.

• Frequency \( f = \frac{1}{3} \) Hz
• Number of cycles per second
• Determines how frequent the motion is

Sine Function

The sine function in the context of this harmonic motion controls the shape and pattern of the oscillation. It's a smooth periodic function that is critical to describing the motion under consideration. In the equation \( d = 6 \sin \frac{2\pi}{3} t \), the sine function means the motion will follow a sinusoidal path.
The properties of the sine function include:

• Periodic behavior: Repeats over intervals of \( 2\pi \)
• Shape: Creates a smooth, wave-like oscillation
• Range: Values between -1 and 1, scaled by amplitude

In harmonic motion, this function ensures that the movement is both smooth and predictable. It starts at zero, reaches the maximum, returns to equilibrium, then goes to the minimum and back again, repeating this cycle as dictated by the period and frequency.