Problem 69 - Precalculus: Functions and Graphs

Question

The formula specifles 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 \(t=0\) ). $$d=4 \cos \frac{3 \pi}{2} t$$

Step-by-Step Solution

Answer

Amplitude: 4 cm; Period: \(\frac{4}{3}\) s; Frequency: \(\frac{3}{4}\) Hz.

Step 1: Identify the Amplitude

The amplitude of a harmonic motion can be found from the coefficient of the cosine function. In the equation \(d = 4 \cos \left(\frac{3\pi}{2} t\right)\), the amplitude is 4 centimeters since it's the coefficient of \(\cos\).

Step 2: Determine the Period

The period of the motion is determined by the argument of the cosine function inside which there is a coefficient multiplying \(t\). The formula for the period \(T\) is given by \(T = \frac{2\pi}{b}\), where \(b\) is the coefficient of \(t\) inside the cosine. Here, \(b = \frac{3\pi}{2}\), so the period is \(T = \frac{2\pi}{\frac{3\pi}{2}} = \frac{4}{3}\) seconds.

Step 3: Calculate the Frequency

Frequency is the reciprocal of the period. The formula is \(f = \frac{1}{T}\). Using the previously calculated period \(T = \frac{4}{3}\) seconds, the frequency \(f = \frac{1}{\left(\frac{4}{3}\right)} = \frac{3}{4}\) Hz.

Step 4: Describe the Motion

At \(t=0\), the point starts at its maximum position \(d=4\) cm. It moves downward through 0 cm to \(-4\) cm (the minimum position) when \(t=\frac{2}{3}\) seconds. Then, it returns upward to 0 cm and back to 4 cm at \(t=\frac{4}{3}\) seconds, completing one full oscillation. This oscillation repeats as harmonic motion.

Key Concepts

Amplitude

In harmonic motion, amplitude refers to the maximum distance the oscillating object travels from its central position, also known as equilibrium. In the provided equation for harmonic motion, the amplitude is derived from the coefficient of the cosine term.

Here, we see the equation is represented as:

• \( d = 4 \cos \left( \frac{3\pi}{2} t \right) \)

The amplitude is the absolute value of the coefficient of the cosine function, which is 4 in this case. This indicates that the point moves from the central position up to 4 centimeters and down to -4 centimeters.

Amplitude plays a crucial role as it dictates how far the object moves from its resting position; a larger amplitude means the object swings farther. Think of a swing set: when you push a swing harder (more amplitude), it travels further away from its resting point.

Period of a Function

The period of a function in harmonic motion tells us how long it takes for the function to repeat its cycle. This is very important for understanding how often certain events, like a pendulum's swing, recur.

In the equation given:

• \( d = 4 \cos \left( \frac{3\pi}{2} t \right) \)

To find the period, we use the formula for the period \( T = \frac{2\pi}{b} \), where \( b \) is the coefficient of \( t \) inside the cosine function. Here, \( b = \frac{3\pi}{2} \), so

• \( T = \frac{2\pi}{\frac{3\pi}{2}} = \frac{4}{3} \) seconds

This means that the object completes one full cycle or oscillation every \( \frac{4}{3} \) seconds. The period tells you how quickly or slowly the object oscillates, which can be crucial when predicting future motion.

Imagine a clock's pendulum: the period determines how fast or slow the clock ticks.

Frequency

Frequency indicates how often the oscillation occurs in one second. It is essentially the number of complete cycles that are repeated per second and is directly related to the period. Frequency is measured in Hertz (Hz).

The relationship between period \( T \) and frequency \( f \) is given by:

• \( f = \frac{1}{T} \)

From the previous calculation, we know the period is \( \frac{4}{3} \) seconds. Therefore, the frequency is:

• \( f = \frac{1}{\left(\frac{4}{3}\right)} = \frac{3}{4} \) Hz

This calculation means that three-quarters of a cycle occurs every second. Frequency is essential as it provides insight into how rapid or spaced out each oscillation is in a given amount of time. For instance, an increased frequency would indicate a faster repetition of swings back and forth.