Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Suppose that a weight on a spring has initial position s(0) and period \(P .\) Do not use a calculator. (a) Find a finction \(s\) given by \(s(t)=a\) cos \(\omega t\) that models the displacement of the weight. (b) Evaluate \(s(1)\). Is the weight moving upward, downward, or neither when \(t=1 ?\) \(s(0)=5\) inches; \(P=1.5\) seconds

Step-by-Step Solution

Answer

The displacement function is \( s(t) = 5 \cos\left(\frac{4\pi}{3}t\right) \). At \( t=1 \), \( s(1) = -2.5 \) and the weight is moving upward.

Step 1: Understand the problem

We are given a spring problem where a weight oscillates back and forth. We need to model this oscillation using a cosine function. The spring's initial position and period are provided, and we will use these to determine the function.

Step 2: Find the angular frequency

The period of the function, denoted by \( P \), is related to angular frequency \( \omega \) by the formula \( P = \frac{2\pi}{\omega} \). Since \( P = 1.5 \) seconds, we can solve for \( \omega \) by rearranging this equation to \( \omega = \frac{2\pi}{P} \), resulting in \( \omega = \frac{2\pi}{1.5} \). Calculating this gives \( \omega = \frac{4\pi}{3} \).

Step 3: Form the displacement function

The displacement function is given by \( s(t) = a \cos(\omega t) \). We know \( s(0) = a \cos(0) = 5 \), meaning \( a = 5 \). Thus, the function becomes \( s(t) = 5 \cos\left(\frac{4\pi}{3}t\right) \).

Step 4: Evaluate the function at t=1

Substitute \( t = 1 \) into the displacement function: \( s(1) = 5 \cos\left(\frac{4\pi}{3} \times 1\right) = 5 \cos\left(\frac{4\pi}{3}\right) \). Since \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \), we find \( s(1) = 5 \times -\frac{1}{2} = -2.5 \).

Step 5: Assess motion direction at t=1

To determine the motion at \( t = 1 \), check the derivative's sign. The derivative \( s'(t) = -5 \cdot \frac{4\pi}{3} \sin\left(\frac{4\pi}{3}t\right) \). This is evaluated at \( t = 1 \): \( s'(1) = -5 \cdot \frac{4\pi}{3} \sin\left(\frac{4\pi}{3}\right) \). Since \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \), \( s'(1) > 0 \), indicating the weight is moving upward at \( t = 1 \).

Key Concepts

Oscillations

In physics, oscillations refer to any repetitive back-and-forth motion about an equilibrium position. This type of motion is often seen in systems like a pendulum swinging, or in this case, a weight attached to a spring. The nature of oscillations is that they are cyclical and can often be described using trigonometric functions such as sine and cosine. These functions are ideal for modeling oscillations because they themselves repeat after a certain interval known as the period.

In this context, we express the displacement of the weight from its equilibrium position using the cosine function. The displacement function that models the oscillation is given by:

• \( s(t) = a \cos(\omega t) \)

Here, \( a \) represents the amplitude, or the maximum displacement from equilibrium, and \( \omega \) is the angular frequency. This formula results from the simplicity and periodic properties of cosine, making it straightforward to capture the constant rhythm of the weight's oscillations. Oscillations are an essential concept, since they appear in various natural phenomena and engineering applications.

Angular Frequency

Angular frequency is a crucial component in the study of periodic motion, including oscillations. It indicates how rapidly something is oscillating in terms of radians per unit of time. Calculated using the formula:

• \( \omega = \frac{2\pi}{P} \)

The angular frequency \( \omega \) relates to the period \( P \) of the motion. The period is the time taken for one complete cycle of oscillation. In the context of a cosine function modeling a spring motion, \( \omega \) determines how fast the weight on the spring moves back and forth.

For the exercise at hand, we calculated \( \omega \) to be \( \frac{4\pi}{3} \), reflecting how many radians the weight travels in one second. The angular frequency is fundamental to understanding the timing and speed of oscillations. It provides insight into the system's dynamics and ensures precision in predicting future positions of the weight attached to the spring.

Periodic Motion

Periodic motion is any motion that repeats at regular time intervals, known as periods. Common examples include the movement of swinging pendulums, vibrating strings, and in our exercise, a weight moving up and down on a spring. The periodic nature of such motion allows it to be expressed neatly in terms of trigonometric functions, which repeat over consistent intervals.

Periodic motion is characterized by:

• Period (P): The time taken to complete one complete cycle of motion. It's crucial as it determines the length of each repetitive sequence.
• Amplitude (a): The maximum extent of the vibration or oscillation, measured from the equilibrium position.
• Phase: A particular stage in the cycle of motion at a specific point in time.

In the given exercise, the period of 1.5 seconds controls the rhythm of the spring's oscillation. Since periodic functions are predictable, they are fundamental to accurately modeling cycles in mechanical systems, waves, and even in economic models.