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?

Short answer

The farthest distance is 2 units.

Step-by-step solution

Identify the Equation Format

The equation for the position of the weight is given in the form of a trigonometric function: \( x = \sin 2t + \sqrt{3} \cos 2t \). This resembles a linear combination of sine and cosine functions.

Convert to Single Trigonometric Function

Our goal is to express the given trigonometric function in a form like \( R \sin(2t + \alpha) \) to easily determine its maximum value. For this, identify \( R \) using the formula \( R = \sqrt{A^2 + B^2} \), where \( A = 1 \) and \( B = \sqrt{3} \).

Calculate Magnitude R

Calculate \( R \) using the formula from the previous step: \( R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \).

Determine Phase Shift \( \alpha \)

\( \alpha \) can be found using the equations: \( \cos \alpha = \frac{A}{R} \) and \( \sin \alpha = \frac{B}{R} \). Apply these: \( \cos \alpha = \frac{1}{2} \) and \( \sin \alpha = \frac{\sqrt{3}}{2} \).

Identify Maximum Value of the Function

The expression can now be written as \( x = 2 \sin(2t + \frac{\pi}{3}) \). The sine function reaches its maximum value of 1, hence the maximum value of the expression is \( 2 \cdot 1 = 2 \).

Conclusion

Thus, the farthest distance the weight gets from the origin along the x-axis is 2 units.

Key concepts

Trigonometric Functions

In the world of mathematics, especially when dealing with phenomena like harmonic motion, trigonometric functions are of paramount importance. These functions, sine and cosine, allow us to describe oscillatory behaviors that are fundamental in physics and engineering. For example, the trigonometric function given in the original exercise, \( x = \sin(2t) + \sqrt{3}\cos(2t) \), is a combination that represents how a weight moves back and forth along the x-axis over time. Such expressions involve the standard trigonometric functions sine \( \sin \) and cosine \( \cos \), which oscillate between -1 and 1, describing the repeating nature of waves.When these functions are combined, like in our exercise, they create a new composite waveform that can be analyzed via techniques like converting them into a single sine or cosine function with a new amplitude and a phase shift, which simplifies the understanding of their behavior.

Phase Shift

To better understand composite trigonometric functions, converting them involves finding a phase shift, which essentially tells us how much the graph of the function is shifted horizontally. In the exercise given, our function \( x = \sin(2t) + \sqrt{3}\cos(2t) \) is converted into the form \( R \sin(2t + \alpha) \) where \( \alpha \) represents the phase shift. ### Calculating Phase Shift- The formula for the phase shift \( \alpha \) is derived from the relationships \( \cos \alpha = \frac{A}{R} \) and \( \sin \alpha = \frac{B}{R} \).- Here, \( A = 1 \) and \( B = \sqrt{3} \), with the calculated magnitude \( R = 2 \).- Thus, \( \cos \alpha = \frac{1}{2} \) and \( \sin \alpha = \frac{\sqrt{3}}{2} \), which corresponds to well-known angles in trigonometry, leading to \( \alpha = \frac{\pi}{3} \).Understanding this shift is crucial as it dictates the position of maxima and minima in the wave pattern, indicating the points where the weight reaches its farthest and nearest distances from the origin.

Amplitude Calculation

Amplitude, in the context of trigonometric functions used for harmonic motion, represents the maximum extent of oscillation, which in this case is the farthest distance that a weight moves from its point of equilibrium. From the equation \( x = \sin(2t) + \sqrt{3}\cos(2t) \), we wish to find the amplitude or maximum distance from the origin, which simplifies our calculation of the "reach" of oscillation.### Calculating Amplitude- The transformation into a single sine function form, \( R \sin(2t + \alpha) \), provides a clear path.- We use \( R = \sqrt{A^2 + B^2} \).- Substituting \( A = 1 \) and \( B = \sqrt{3} \) results in \( R = \sqrt{1 + 3} = 2 \).Thus, the calculated amplitude \( R = 2 \) means the maximum distance the weight achieves from the origin is 2 units. This tells us the strength of oscillation in the system, which can be observed as the peak movement during a cycle.