Question

Vertical Motion Two masses hanging side by side from springs have positions \(s_{1}=2 \sin t\) and \(s_{2}=\sin 2 t\) respectively, with \(s_{1}\) and \(s_{2}\) in meters and \(t\) in seconds. (a) At what times in the interval \(t>0\) do the masses pass each other? [Hint: \(\sin 2 t=2 \sin t \cos t ]\) (b) When in the interval \(0 \leq t \leq 2 \pi\) is the vertical distance between the masses the greatest? What is this distance? (Hint: \(\cos 2 t=2 \cos ^{2} t-1 . )\)

Short answer

The masses pass each other at times given by \(t = 2n\pi\). The greatest vertical distance between them, which is 2 meters, is reached when \(t = n\pi/2\) for odd \(n\) in the interval \(0 \leq t \leq 2 \pi\).

Step-by-step solution

Find When The Masses Cross

Given the position equations \(s_{1}=2 \sin t\) and \(s_{2}=\sin 2 t\), set them equal to each other to find the times \(t\) that they cross: \(2 \sin t = \sin 2t\). Use the trigonometric identity \(\sin 2t = 2\sin t \cos t\) to rewrite the equation as \(2 \sin t = 2 \sin t \cos t\). Then, solve it for \(t\) (in the interval \(t>0\)).

Solve for t

Divide both sides of the equation \(2 \sin t = 2 \sin t \cos t\) by \(2 \sin t\) to get \(\cos t = 1\). The values of \(t\) for which \(\cos t = 1\) in the interval \(t > 0\) are \(t = 2n\pi\), where \(n\) is an integer. Thus, the times \(t\) when the masses pass each other are \(t = 2n\pi\).

Find When The Vertical distance is greatest

Determine the absolute difference between \(s_{1}\) and \(s_{2}\), which gives \(|s_{1} - s_{2}| = |2\sin t - \sin 2t| = |2\sin t - 2\sin t \cos t|\). This simplifies to \(|2 \sin t(1 - \cos t)|\). To find when the difference is maximum, solve this equation over the interval \(0 \leq t \leq 2 \pi\).

Solve for maximum t

The maximum value of \(|2 \sin t(1 - \cos t)|\) in the interval \(0 \leq t \leq 2 \pi\) is reached when \(\sin t\) and \(1 - \cos t\) are both maximum. The maximum of \(\sin t\) in the given interval is 1, and the maximum of \(1 - \cos t\) is also 1. Therefore, the greatest vertical distance reached between them in the given interval is \(2 * 1 * 1 = 2\) meters, and it is reached when \(t = n\pi/2\) for odd \(n\).

Key concepts

Trigonometric Identities

Understanding trigonometric identities is essential in solving problems involving periodic and harmonic motions. Identities like \(\sin 2t = 2\sin t \cos t\) and \(\cos 2t = 2\cos^2 t - 1\) are pivotal in manipulating equations to find solutions to problems involving waves, oscillations, and other periodic phenomena.

In our vertical motion exercise, the use of the identity \(\sin 2t = 2\sin t \cos t\) allowed us to set the position functions equal to each other. Simplifying the equation with this identity eventually helped us find at what times the two masses pass each other. Realize that many trigonometric identities stem from the unit circle properties and Euler's formula, making them interrelated and widely useful in various physics and engineering problems.

Harmonic Motion

Harmonic motion is a type of periodic movement where the restoring force is directly proportional to the displacement and acts in the opposite direction. It's characterized by being sinusoidal in time and demonstrates properties such as amplitude, frequency, and phase.

In our exercise, the masses attached to springs exhibit this motion. Their positions as functions of time are sinusoidal (\(s_1=2\sin t\) and \(s_2=\sin 2t\)). Being able to model such systems mathematically provides a powerful framework for predicting the behavior of oscillators and waves, which is fundamental in various fields of science and engineering.

Periodic Functions

Periodic functions repeat their values at regular intervals, which is known as the period. The sine and cosine functions used in our exercise are prime examples of periodic functions, repeating every \(2\pi\) radians. Features like peaks, valleys, and repeating patterns are hallmarks of these functions.

Analyzing the behavior of the two masses over time requires an understanding of these periodic principles. Whether assessing when they intersect or determining the maximum vertical distance, recognizing the inherent repetition in their movements guides the problem-solving process.

Calculus Applications

Calculus plays an integral role in understanding changes and motion. It allows us to find rates of change (derivatives) and total values (integrals) for functions. In the context of vertical motion, we use calculus to analyze velocity, acceleration, and displacement over time.

In our specific scenario, while we did not directly apply derivatives or integrals, the setup for finding maximum values over an interval of time can lead to using calculus concepts like differentiation to find the maximum and minimum values of functions, illustrating the broad utility of calculus in solving real-world problems.