Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by \(y=\frac{1}{3} \sin 2 t+\frac{1}{4} \cos 2 t\) where \(y\) is the distance from equilibrium (in feet) and \(t\) is the time (in seconds). (a) Use a graphing utility to graph the model. (b) Use the identity \(a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\) where \(C=\arctan (b / a), a>0,\) to write the model in the form \(y=\sqrt{a^{2}+b^{2}} \sin (B t+C) .\) Use the graphing utility to verify your result. (c) Find the amplitude of the oscillations of the weight. (d) Find the frequency of the oscillations of the weight.

Step-by-Step Solution

Answer

The amplitude of the oscillations of the weight is 'a'. And the frequency of the oscillations of the weight is 'B'

Step 1: Transforming the model

First off, transform the equation of the weight's motion, \(y=\frac{1}{3} \sin 2 t+\frac{1}{4} \cos 2 t\), to the desired form \(y=\sqrt{a^{2}+b^{2}} \sin (B t+C)\). We use the given trigonometric identity, \(a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\), where \(C=\arctan (\frac{b}{a})\), \(a>0\). For now, we can say \(a = \frac{1}{3}\), \(b = \frac{1}{4}\), and \(B = 2\).

Step 2: Calculation of 'a' and 'C'

Now we can calculate \(a\) and \(C\). We have to find \(a = \sqrt{(\frac{1}{3})^2 + (\frac{1}{4})^2}\) and \( C = \arctan(\frac{b}{a})\).

Step 3: Rewriting the model

Substitute the calculated values of 'a' and 'C' in the equation. So, the model becomes \(y = a \sin(Bt + C)\)

Step 4: Finding the amplitude of oscillations

The amplitude of oscillations is the maximum distance the weight moves from its equilibrium position. This is represented by our calculated 'a' in the transformed equation.

Step 5: Finding the frequency of oscillations

The frequency of the oscillations is the number of waves that pass a fixed point in unit time. In this case, 'B' in the transformed equation is representing the frequency of oscillations.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in mathematical modeling, especially when dealing with periodic phenomena like oscillations of a spring. In our exercise, the equation describing the motion of the weight involves a combination of sine and cosine terms. These can be transformed into a single sine function using identities. The key identity we use here is: \[ a \sin B\theta + b \cos B\theta = \sqrt{a^2 + b^2} \sin(B\theta + C) \] where \(C = \arctan\left(\frac{b}{a}\right)\), assuming \(a > 0\). This transformation is beneficial because it simplifies the expression and makes it easier to interpret the amplitude and phase of the oscillation. It also helps verify the accuracy of the model graphically. The identity highlights the elegance of trigonometric transformations and their capacity to unify expressions into a single, coherent wave function.

Amplitude

Amplitude in trigonometric modeling refers to the peak value of a wave's displacement from its equilibrium point. In the context of our spring-and-weight system, it signifies the maximum distance the weight moves vertically from the equilibrium position. After converting the given equation using the identity, the amplitude is found as the coefficient of the transformed sine function: \[ \text{Amplitude} = \sqrt{\left(\frac{1}{3}\right)^2 + \left(\frac{1}{4}\right)^2} \] Solving this gives the value of the amplitude, which is crucial for understanding how far the spring stretches during oscillation. The amplitude represents:

• The height of each peak from the center line (equilibrium).
• The strength or severity of the periodic motion.

Understanding amplitude allows us to grasp the energy and force exerted during each oscillation.

Frequency

Frequency in trigonometric models is a measure of how many complete oscillations occur in a unit time, usually seconds. It tells us how fast the system is oscillating. In formula terms, frequency is tied to the coefficient 'B' in the equation \(\sin(Bt + C)\). For this scenario, where the model of motion is: \[ y = \sqrt{a^2 + b^2} \sin(2t + C) \] 'B' is 2, meaning the oscillation frequency is dictated by this value. This tells us:

• The system completes 2 cycles per second.
• The rapidity of the back-and-forth motion.

Frequency is vital for understanding repetitive processes, helping us predict future states of the system based on current observations.

Trigonometric Transformation

Trigonometric transformation is the process we use to change the form of trigonometric expressions to simplify solving and understanding them. By applying transformations, we can convert complex equations into more manageable forms that reveal key attributes like amplitude and frequency.In our spring-and-weight system, the given equation: \[ y = \frac{1}{3} \sin 2t + \frac{1}{4} \cos 2t \] is transformed to: \[ y = \sqrt{\left(\frac{1}{3}\right)^2 + \left(\frac{1}{4}\right)^2} \sin(2t + C) \] using identities and transformations.Benefits of transformations include:

• Simplifying complex expressions.
• Identifying crucial features, such as phase shifts and amplitudes, quickly.
• Making it easier to visualize and predict behavior using graphs.

Trigonometric transformations are foundational in fields like signal processing, physics, and engineering, offering clarity in the theoretical and applied aspects of wave behaviors.