Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by \(y=\frac{1}{4} \cos 16 t, \quad t>0.\) where \(y\) is measured in feet and \(t\) is the time in seconds. (a) Use a graphing utility to graph the function. (b) What is the period of the oscillations? (c) Determine the first time the ball passes the point of equilibrium \((y=0).\)

Step-by-Step Solution

Answer

The period of the oscillation is \( \frac{\pi}{8} \) seconds and the ball first passes the point of equilibrium at \( \frac{\pi}{32} \) seconds.

Step 1 - Graphing the function

Using graphing software or calculator, plug the function \(y=\frac{1}{4}\cos (16t)\) to generate its graph.

Step 2 - Determining the period of oscillation

The period (T) of a cosine function is given by \(T = \frac{2\pi}{|b|}\), where \(b\) is the real number in \(\cos(bt)\). In our case, \(b = 16\) so the period \(T = \frac{2\pi}{16} = \frac{\pi}{8}\) seconds.

Step 3 - Calculate when the ball first passes the point of equilibrium

The ball passes the point of equilibrium when \(y=0\). To determine the time it happens for the first time, we set \(y=\frac{1}{4} \cos 16t\) to zero and solve for \(t\). \nThis gives us the equation \(\frac{1}{4} \cos 16t = 0\), which simplifies to \(\cos 16t = 0\). \nThe cosine function equals zero at \(\frac{1}{4} \text{ of its period and }\frac{3}{4}\text{ of it}.\) Therefore, the first time the ball passes equilibrium is when \(t = \frac{1}{4}T = \frac{1}{4} *\frac{\pi}{8} = \frac{\pi}{32}\) seconds.

Key Concepts

Graphing Trigonometric Functions

Understanding the graphical representation of trigonometric functions is fundamental in grasping the motion of periodic phenomena. In our exercise, the motion of a ball on a spring is described by the function \(y=\frac{1}{4} \cos 16t\), where \(y\) is the displacement in feet and \(t\) is time in seconds. To graph this function, one would typically use a graphing calculator or software.

When graphed, the cosine function creates a wave-like pattern that oscillates above and below the x-axis. This axis often represents the equilibrium position, where the displacement is zero. The maximum and minimum points of the wave correspond to the maximum displacement of the ball from this equilibrium position. Visualizing this pattern is crucial as it helps students understand how the ball moves over time in a clear and tangible way.

Amplitude and Displacement

Here, the amplitude, or the maximum displacement from equilibrium, is \(\frac{1}{4}\) feet or 3 inches. The amplitude is indicative of how far the ball stretches from the rest position at its maximum displacement.

Using Graphs for Comprehension

Graphs are not only useful for visualizing the concept but they also provide a way to verify solutions to problems involving trigonometric functions. They can help to see the practical implications of abstract concepts, making them an invaluable tool in the study of trigonometry.

Period of Oscillation

The period of oscillation is a measure of the time it takes for one complete cycle of repetitive motion. In trigonometry, this is the time it takes for a trigonometric function to repeat its pattern. In our example, the oscillation is described by a cosine function. To determine the period of the ball's oscillations, we examine the multiplicative constant in front of the variable \(t\), which alters the function's frequency.

The formula for calculating the period \(T\) of a cosine function is \(T = \frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(t\) within the function. Here, our \(b\) value is 16, resulting in an oscillation period of \(T = \frac{\pi}{8}\) seconds.

Significance of Period in Motion

The period of oscillation is significant because it reflects how quickly the motion repeats. A smaller period means a higher frequency of oscillation - the ball bounces up and down more rapidly. By calculating the period, students gain a better understanding of the dynamism and speed of the ball's motion in our exercise.

Equilibrium in Harmonic Motion

Equilibrium in harmonic motion refers to the state where the net force acting upon an object is zero, resulting in no acceleration and the object is momentarily at rest. For a ball on a spring or any object in simple harmonic motion, this position is typically at the center of the motion path.

In trigonometric terms, where the function modeling the motion of the ball is \(y=\frac{1}{4} \cos 16t\), equilibrium is reached when the value of \(y\) is zero. To find when the ball first passes through this point of equilibrium, we solve the equation \(\frac{1}{4} \cos 16t = 0\), leading us to determine that the cosine function reaches zero at certain fractions of its period, namely \(\frac{1}{4}T\) and \(\frac{3}{4}T\). Therefore, the first time the ball passes through equilibrium is at \(t = \frac{\pi}{32}\) seconds.

Equilibrium and Real-World Applications

Understanding equilibrium is crucial not only in theoretical exercises but also in real-world applications such as engineering and physics — where the concept is applied to design and analyze systems undergoing oscillatory motion, like suspension bridges, clocks, and even in the study of molecular vibrations in chemistry.