Problem 50 - Thomas Calculus

Question

Two masses hanging side by side from springs have positions \(s_{1}=2 \sin t\) and \(s_{2}=\sin 2 t,\) respectively. \begin{equation}\begin{array}{l}{\text { a. At what times in the interval } 0 < t \text { do the masses pass each }} \\ \quad {\text { other? (Hint: sin } 2 t=2 \sin t \cos t )} \\ {\text { b. When in the interval } 0 \leq t \leq 2 \pi \text { is the vertical distance between }} \\ \quad {\text { the masses the greatest? What is this distance? (Hint: }} \\ {\quad \cos 2 t=2 \cos ^{2} t-1 . )}\end{array}\end{equation}

Step-by-Step Solution

Answer

The masses pass each other at no non-trivial points between 0 to 2π. Greatest distance occurs at t = π with distance 2.

Step 1: Setting the Equation for Passing Each Other

To determine when the masses pass each other, set the positions equal: \(s_1 = s_2\). This gives the equation: \(2 \sin t = \sin 2t\). Using the identity \(\sin 2t = 2 \sin t \cos t\), we substitute and simplify to get: \[2 \sin t = 2 \sin t \cos t\] This simplifies to \[1 = \cos t\].

Step 2: Solving for Time Instances

Solve \( \cos t = 1 \) to find the value of \(t\). \(\cos t = 1\) at \(t = 0, 2\pi, 4\pi, \ldots\), but we only need \(t\) in the open interval \((0, 2\pi)\). Thus, no solution strictly within the open interval exists.

Step 3: Establishing the Equation for Maximum Distance

The vertical distance between the masses is given by \(|s_1 - s_2| = |2 \sin t - \sin 2t|\). Using \(\sin 2t = 2 \sin t \cos t\), this becomes \(|2 \sin t - 2 \sin t \cos t| = 2|\sin t (1 - \cos t)|\).

Step 4: Maximizing the Distance

The expression \(2 \sin t (1 - \cos t)\) needs to be maximized. Recognizing that the maximum magnitude of \(\sin\) and \(\cos\) is 1, we find that \(1 - \cos t\) is maximized when \(\cos t = -1\), leading to \(\sin t = 0\). This occurs at \(t = \pi/2\), where \(\sin(\pi/2) = 1\). Thus, maximal distance is \(2 \times 1 \times 2 = 2\).

Step 5: Answering and Verifying the Greatest Distance

Verify which of the values within \(0 \leq t \leq 2\pi\) provide greatest distance \(2|\sin t (1 - \cos t)|\), found to be \(2\) at \(t = \pi\).

Key Concepts

Harmonic Motion

Harmonic motion is a type of periodic motion where an object, such as a mass on a spring, moves back and forth through an equilibrium position. This motion is characterized by a restoring force that is proportional to the displacement from the equilibrium position. The springs in the given exercise cause the masses to oscillate in a manner described by harmonic motion.

In our problem, the positions of the masses are given by trigonometric functions, which are commonly used to model harmonic motion because they naturally repeat over time. This is crucial since it allows us to predict the positions and interactions among the masses at any given moment. The trigonometric sine and cosine functions are particularly valuable as they describe the vertical and horizontal components of this oscillation. Understanding these concepts helps in analyzing the behavior and interaction of the two masses over time.

Sine Function

The sine function, often represented as \(\sin \theta\\), is a fundamental trigonometric function significant in describing periodic phenomena like waves, circular motion, and harmonic oscillations. In this exercise, the sine function plays a key role in expressing the positions of the suspended masses. The position of one mass is given by \(\ s_1 = 2 \sin t \), where \(t\\) represents time.

The sine function oscillates between -1 and 1, defining the matter moves in one dimension up and down in reference to a central point. Its amplitude, multiplied by the frequency, controls how far up or down the object moves in each cycle. The coefficient in front of the sine, in our case 2, determines the amplitude of the motion. Higher amplitudes result in larger oscillations, meaning the mass moves further from its original position with every cycle.

Cosine Function

The cosine function is another pivotal trigonometric function represented as \(\ \cos \theta \ \). Like the sine function, its values range between -1 and 1, making it vital for harmonic motion analysis.

In the given exercise, the cosine function appears indirectly through the identity \(\ \sin 2t = 2 \sin t \cos t\ \). This relation helps simplify complex trigonometric expressions and solve equations involving harmonic oscillations. Often associated with the horizontal component of circular motion, the cosine function helps us determine the phase shifts and calculate the interactions between two oscillating objects. By setting \(\ \cos t = 1 \ \) in our problem, we are looking for when these phase shifts lead to specific positional matches for the masses.

Vertical Distance

The vertical distance between two oscillating masses is crucial to understanding their relative movement. In our exercise, the vertical distance is defined as \(\ |s_1 - s_2|\ \), which translates to \(\ |2 \sin t - \sin 2t|\ \). This expression tells us how far apart the two masses are from each other at any given moment in time.

An essential aspect of trigonometry in harmonic motion is the ability to compute this distance efficiently, since it provides insight into potential collisions, synchronization, or maximum separation events. Solving for when this vertical distance reaches its maximum is key to understanding how the two masses interact dynamics and helps shed light on their independent yet linked motions.

Maximization Problem

A maximization problem involves finding the point in time or space where a particular function reaches its greatest value. Here, we aim to find when the vertical distance between the two masses – given by \(\ 2 \sin t (1 - \cos t) \ \) – is maximized.

To solve this, we identify that \(\ \sin t = 0 \ \) and \(\ \cos t = -1 \ \) are critical points, where the function could potentially reach its peak. Using trigonometric identities, we can zero in on an optimal configuration at \(\ t = \pi\ \) within the prescribed interval, yielding a maximal distance. This solution technique leverages calculus and trigonometric properties for optimization, providing critical insights into the behavior of the masses over time.