Problem 10 - 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 \(t=1 ?\) \(s(0)=-4\) inches; \(P=1.2\) seconds

Step-by-Step Solution

Answer

s(t) = -4 \cos\left(\frac{5\pi}{3} t\right); s(1) = -2; Moving downward.

Step 1: Understanding the Problem

We are given the initial position of a weight on a spring, which is a harmonic motion. We need to find the function that models this displacement and evaluate the displacement at a specific time to determine the direction of motion.

Step 2: Determine the Amplitude

The amplitude of the displacement, denoted by \(a\), is the absolute value of the initial displacement. Since \(s(0) = -4\) inches, the amplitude is \(|a| = 4\). Thus, \(a = -4\).

Step 3: Determine the Angular Frequency

The angular frequency \(\omega\) is related to the period \(P\) by the formula \(\omega = \frac{2\pi}{P}\). Given that \(P = 1.2\) seconds, \(\omega = \frac{2\pi}{1.2} = \frac{5\pi}{3}\).

Step 4: Write the Function for Displacement

The function for the displacement of the weight is \[s(t) = a \cos(\omega t)\]. Substituting the values for \(a\) and \(\omega\), we get \[s(t) = -4 \cos\left(\frac{5\pi}{3} t\right)\].

Step 5: Evaluate the Function at t=1

Substitute \(t = 1\) into the function to find \[s(1) = -4 \cos\left(\frac{5\pi}{3} \times 1\right) = -4 \cos\left(\frac{5\pi}{3}\right)\]. Because \(\cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}\), we get \[s(1) = -4 \times \frac{1}{2} = -2\].

Step 6: Determine the Direction of Motion

The direction of motion depends on the derivative of \(s(t)\) at \(t=1\). The derivative is \[s'(t) = 4 \cdot \frac{5\pi}{3} \sin\left(\frac{5\pi}{3} t\right)\]. Substituting \(t = 1\), we find \[s'(1) = 4 \times \frac{5\pi}{3} \times \sin\left(\frac{5\pi}{3}\right)\]. Since \(\sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}\), \(s'(1) < 0\), indicating that the weight is moving downward.

Key Concepts

Amplitude

In harmonic motion, amplitude is a key concept which represents the maximum extent of the displacement from the equilibrium position. Think of it as how far the weight on the spring can move from its resting position.
The amplitude is always a positive number because it measures size, not direction. In our problem, the initial position is given as (-4) inches, but the amplitude's magnitude will be taken as 4 inches since amplitude is the absolute value of the initial displacement.

• The amplitude tells us how intense the movement of the system is.
• You can think of it as the 'height' of the oscillation.

In the formula, the amplitude is represented by the letter 'a', which in our case equals 4 inches, indicating that the spring can extend or compress by up to 4 inches from its midpoint.

Angular Frequency

Angular frequency (\(\omega\)) is an important parameter when dealing with periodic functions like harmonic motion. It describes how quickly the system cycles through its motion. More technically, it's how many radians are swept per second.
For our weight on a spring, we can calculate \(\omega\) using the period (\(P\)), which is the time taken to complete one full oscillation. The relationship is given by:\[\omega = \frac{2\pi}{P}\]
In our exercise, the period (\(P\)) is 1.2 seconds. Therefore, the angular frequency is:\(\omega = \frac{2\pi}{1.2} = \frac{5\pi}{3}\)
This means that the weight cycles through \(\frac{5\pi}{3}\) radians in one second.

• It tells us how "fast" the oscillation occurs.
• High angular frequency means rapid motion; low means slower.

Keep in mind that angular frequency is a constant for a given periodic motion unless the period changes.

Cosine Function

The cosine function is part of the family of trigonometric functions, and it is especially key to describing periodic phenomena like harmonic oscillation. In our displacement function, cosine captures the oscillating nature of the weight as it moves up and down.
The basic cosine function oscillates between 1 and -1. When applied in our context with the function \(s(t) = a\cos(\omega t)\), the cosine term is scaled by the amplitude (a) and shifted by the function argument (\(\omega t\)).

• Cosine starts at its maximum value; at \(t = 0\), its value is 1.
• It then decreases to -1, representing a full oscillation of one cycle.

The nature of the cosine curve makes it ideal for modeling the smooth, continuous motion typical of weights on springs.

Displacement Function

The displacement function \(s(t) = a \cos(\omega t)\) provides a comprehensive way to model the position of the weight on its spring at any given time (t).
This function is generated using the amplitude (a) and angular frequency (\(\omega\)), making it an all-in-one tool to predict the position of the weight. The cosine function is integral to this, capturing the regular, repeated path of the oscillating system.

• We use this function to calculate \(s(1)\), the displacement at \(t = 1\).
• This provides a practical way to understand how the weight really behaves over time.

This particular function not only tells us how high or low the weight is at any time but also includes the math to calculate velocity and direction of motion, which are key for understanding the system's dynamics. In our example, after evaluating (s(1)), we find the weight is moving downward, demonstrating the displacement's application in real-world prediction.