Problem 46 - Thomas Calculus in SI Units

Question

Two masses hanging side by side from springs have positions \(s_{1}=2 \sin t\) and \(s_{2}=\sin 2 t,\) respectively. a. At what times in the interval \(0

Step-by-Step Solution

Answer

a. No crossing times were found in 0 < t. b. The greatest vertical distance is 2, at t = \( \frac{\pi}{2} \).

Step 1: Analyze the condition for masses passing each other

For two masses to pass each other, their positions should be equal, i.e., \( s_1 = s_2 \). This implies \( 2 \sin t = \sin 2t \).

Step 2: Use identity for sin 2t

Using the identity \( \sin 2t = 2 \sin t \cos t \), substitute in the equation. It becomes \( 2 \sin t = 2 \sin t \cos t \).

Step 3: Simplify the equation

Cancel out \( 2 \sin t \) from both sides, assuming \( \sin t eq 0 \), which gives \( 1 = \cos t \).

Step 4: Solve for t when 0 < t

\( \cos t = 1 \) only when \( t = 0 \); within the interval \( 0 < t \), \( t = 0 \) doesn't satisfy, so check other solutions by considering when \( \sin t = 0 \). This leads to \( t = \pi, 2\pi \), but these are outside the strict inequality, or are considered separately within the next task.

Step 5: Finding maximum distance

The vertical distance \( d \) is given by \( d = |s_1 - s_2| = |2 \sin t - \sin 2t| \). Substitute \( \sin 2t = 2 \sin t \cos t \) into \( d \): \( d = |2 \sin t - 2 \sin t \cos t| = |2 \sin t (1 - \cos t)| \).

Step 6: Simplify and maximize

Since \( 0 \leq \cos t \leq 1 \), maximizing \( d = 2 \sin t (1 - \cos t) \) requires considering when \( \sin t = 1 \) at \( t = \frac{\pi}{2} \) and with \( \cos t = 0 \), the distance simplifies to 2.

Step 7: Calculate maximum value

Evaluate at points where potential extrema can occur, like critical points or endpoints in \(0 \leq t \leq 2\pi\). Evaluation gives \( d = 2 \) as maximum when \(t = \frac{\pi}{2}\).

Key Concepts

Harmonic Motion

Harmonic motion refers to the repetitive back and forth movement of an object. It's a classic concept often seen in the physics of objects attached to springs. In this exercise, we have two masses hanging from springs, each oscillating in a vertical plane. Their motion can be described using trigonometric functions, specifically sine functions. The positions of these masses at any time, represented as functions of time, depict harmonic motion. This predictable pattern allows us to explore points of intersection and measure distances easily. Key examples include calculating when these masses pass each other, by matching their position functions, and understanding their motion dynamics over time.

• Dependent on time: The motion is described as a function of time.
• Repetitive nature: Masses move back and forth in a periodic manner.

Understanding harmonic motion allows us to delve deeper into more complex calculations, like finding the relative positions of oscillating masses.

Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, are crucial for modeling oscillatory behaviors in physics. When we look at the positions of masses, such as in our exercise, we use sinusoidal functions like \(s_1 = 2\sin(t)\) and \(s_2 = \sin(2t)\). These functions describe how the position of each mass changes over time. The sinusoidal nature means these functions repeat their values in a predictable cycle known as a period.
By employing trigonometric identities, it becomes easier to manipulate these functions to find times when certain conditions, like a particular position or distance, are met.

• Amplitudes: Govern how high or low the oscillations go.
• Periods: The interval required to complete one full cycle.

Mastering sinusoidal functions enhances our ability to anticipate and calculate specific motion characteristics.

Maxima and Minima

Finding the maxima and minima in trigonometric functions can help determine critical points, like peak positions or distances. In the exercise, determining when the masses are the farthest apart calls for finding the maximum value of the distance function. This is implemented by evaluating the expression \(d = |2\sin t - 2\sin t \cos t|\). To maximize \(d\), identify values of \(t\) that cause \(\sin t\) to reach its highest value within the given bounds of \(0 \leq t \leq 2\pi\). These critical points occur at specific angles or time intervals within the motion's cycle.

• Critical Points: Times when the function reaches its peaks or troughs.
• Intervals: Key ranges where these maxima or minima occur.

Gaining insight into maxima and minima not only simplifies finding maximum distances but is also essential in many scientific applications.

Trigonometric Identities

Trigonometric identities are the tools that simplify complex expressions into manageable forms. They reveal relationships between trigonometric functions that allow us to solve equations and assess conditions within the context of harmonic motions and sinusoidal functions. In our exercise, we use identities such as \(\sin 2t = 2\sin t \cos t \) and \(\cos 2t = 2\cos^2 t - 1 \), making it possible to compare or equate different functions.
These identities remove complexities by transforming a product of functions into sums or vice versa, making it far easier to solve.

• Simplification: Turning complex expressions into simpler forms.
• Problem-solving: Enable the determination of specific conditions in equations.

Understanding these relationships is vital across math and physics for predicting outcomes and solving intricate problems.