Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

A weight connected to a spring moves along the \(x\)-axis so that its \(x\)-coordinate at time \(t\) is $$ x=\sin 2 t+\sqrt{3} \cos 2 t $$ What is the farthest that the weight gets from the origin?

Step-by-Step Solution

Answer

The farthest the weight gets from the origin is 2 units.

Step 1: Understand the problem

We need to find the farthest distance of the weight from the origin. The position of the weight is given by the function \(x = \sin 2t + \sqrt{3} \cos 2t\). We need to find the maximum value of \(|x|\).

Step 2: Express the function as a single trigonometric function

We can express the function in the form \(x = R \cos(2t - \phi)\). Here, \(R\) is the amplitude, which represents the maximum distance, and \(\phi\) is the phase shift. This transformation will help us find the maximum distance.

Step 3: Calculate the amplitude \(R\)

Using the identity \(R = \sqrt{a^2 + b^2}\), where \(a = 1\) and \(b = \sqrt{3}\), we calculate \[R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2.\]Thus, the amplitude is 2.

Step 4: Conclude the maximum distance

The maximum distance from the origin is equal to the maximum absolute value of the function \(|x|\). Since we have determined the amplitude to be 2, the maximum distance the weight gets from the origin is 2.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding harmonic motion, as they describe oscillations and waves. In this exercise, we are particularly leveraging sine and cosine functions. The given equation \(x = \sin 2t + \sqrt{3} \cos 2t\) is a sum of two trigonometric functions, where the argument \(2t\) indicates their frequency component, doubling the variability per unit time.

These functions are periodic, meaning they repeat values in regular intervals, a property crucial for studying oscillatory systems. Here, both sine and cosine have a period of \(\pi\) when the argument is \(2t\), implying that the cycle completes twice as fast as their standard period of \(2\pi\).

Understanding how these two functions combine can help us convert them into a single compact form, making it easier to find characteristics like amplitude and phase shift.

Amplitude Calculation

Amplitude refers to the maximum extent of oscillation in harmonic motion. This is essentially the "height" of the wave from its mean position. In our exercise, we need to express the combined function as a single trigonometric function to find the amplitude. The formula to express the function as \(x = R \cos(2t - \phi)\) helps us convert the sum into a simpler model, where \(R\) represents the amplitude.

Calculating the amplitude \(R\) involves the expression \(R = \sqrt{a^2 + b^2}\). Here, \(a = 1\) and \(b = \sqrt{3}\), derived from the coefficients of \(\sin 2t\) and \(\cos 2t\) respectively. Plugging these values in, we find:

• \(R = \sqrt{1^2 + (\sqrt{3})^2}\)


• \(R = \sqrt{1 + 3}\)


• \(R = 2\)

This shows that the amplitude is 2, thus the maximum distance the weight reaches from the origin.

Phase Shift

Phase shift is an important concept in wave mechanics, indicating how far the wave is shifted from its standard position. In the trigonometric form \(x = R \cos(2t - \phi)\), the term \(-\phi\) represents the phase shift.

To find the phase shift in our equation, we need to look into the transformation required to convert the original equation into its compact form. The formula \(\tan \phi = \frac{a}{b}\), where \(a\) and \(b\) are coefficients of the sine and cosine terms, helps identify \(\phi\).

Here, the equation \(\tan \phi = \frac{1}{\sqrt{3}}\) is solved to find \(\phi\):

• \(\phi = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\)

This process gives us the phase shift angle, adjusting the wave's starting position. Understanding phase shift is key to predicting the motion of oscillatory systems, affecting when and how the motion will occur at specific points in time.