Problem 23 - Precalculus Essentials

Question

An object moves in simple harmonic motion described by the given equation, where \(t\) is measured in seconds and \(d\) in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. $$d=-6 \cos 2 \pi t$$

Step-by-Step Solution

Answer

a. The maximum displacement is 6 inches. b. The frequency is 1 cycle per second. c. The time required for one cycle is 1 second.

Step 1: Find the Maximum Displacement

The maximum displacement is given by the absolute value of the amplitude of the cosine function. The amplitude here is -6, so the maximum displacement is \(|-6| = 6\) inches.

Step 2: Find the Frequency

The frequency is given by the coefficient of \(t\). Here the coefficient is \(2\pi\). However, in the standard formula, the frequency \(f\) is measured in cycles/seconds and the coefficient of \(t\) equals \(2\pi f\). So, we equate \(2\pi f\) to the coefficient \(2\pi\) it is observed that \(f=1\). Hence, the frequency is 1 cycle per second.

Step 3: Find the Time Required for One Cycle

The time required for one complete cycle, otherwise known as the period, is the reciprocal of the frequency. For this problem, the frequency \(f\) is 1 (from step 2), hence the period of the function is \(1/f = 1/1 = 1\) second. Therefore, the time required for one cycle is 1 second.

Key Concepts

Maximum Displacement

In simple harmonic motion, maximum displacement, also known as "amplitude," refers to how far an object moves from its equilibrium position, which is the central point of motion. Imagine it as the peak height of a wave. In the given equation, the maximum displacement is influenced by the amplitude's absolute value.

In the function \(d = -6 \cos 2\pi t\), the amplitude is -6. However, when considering displacement value, we only care about the absolute size of this movement, not the direction. Thus, the maximum displacement is \( |-6| = 6 \) inches.

This tells us that, at its peak, the object moving with this motion travels 6 inches from its central position in either direction.

Frequency of Oscillation

The frequency of oscillation in simple harmonic motion indicates how often the motion repeats every second. It's measured in Hertz (Hz), which means cycles per second. In simpler terms, frequency tells us how many times the wave pattern completes in a second.

With the equation \(d = -6 \cos 2\pi t\), the frequency value is derived from the coefficient of \(t\). The standard form of a cosine function for simple harmonic motion is \(d = A \cos(2\pi f t)\), where \(f\) is the frequency.

In this problem, the term \(2\pi\) is exactly the cycle rate for \(t\), suggesting it completes one full cycle each second. Therefore, the frequency \(f = 1\, \text{Hz}\), indicating one cycle occurs per second.

Remember, a higher frequency means a faster repeating motion.

Period of a Cycle

The period of a cycle is the duration it takes to complete one full cycle of motion. It's a critical measure in periodic functions, indicating the time for one complete oscillation.

To find the period, take the reciprocal of the frequency. In our problem, since the frequency \(f\) is 1 cycle per second, the period \(T = \frac{1}{f} = \frac{1}{1} = 1\, \text{second}\).

This means each wave pattern in the motion described by \(d = -6 \cos 2\pi t\) starts and ends within one second before repeating.

Understanding the period is crucial because it helps in anticipating when the motion will repeat, a fundamental aspect when dealing with periodic events in physics or engineering.