Problem 63 - Precalculus: A Concise Course

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 \(y=\frac{1}{4} \cos 16 t(t>0),\) where \(y\) is measured in feet and \(t\) is the time in seconds. (a) Graph the function. (b) What is the period of the oscillations? (c) Determine the first time the weight passes the point of equilibrium \((y=0)\).

Step-by-Step Solution

Answer

(a) Graph resembles a wave with amplitude of 1/4 centered around the x-axis. (b) The period of the oscillations is \(P=\frac{\pi}{8}\) seconds. (c) The weight first passes the point of equilibrium at \(t = \frac{\pi}{32}\) seconds.

Step 1: Graph the function

Plot the function \(y=\frac{1}{4} \cos 16t\) over the interval [0,2]. This function will resemble a wave starting at \(y=\frac{1}{4}\) at \(t=0\), descending to a minimum of \(y=-\frac{1}{4}\) in the middle of the interval, and then rising back to \(y=\frac{1}{4}\) at the end.

Step 2: Find the period of the oscillations

The period of a cosine funtion is given by \(P=\frac{2\pi}{|B|}\), where in this case \(B=16\). Substitute \(B\) into the equation to find the period \(P\): \(P = \frac{2\pi}{16} = \frac{\pi}{8}\) seconds.

Step 3: Determine the first time the weight passes the point of equilibrium

The equation of cosine equals zero at \(\frac{\pi}{2}, \frac{3\pi}{2},\) etc. But in our case, we wish to find when \(16t=\frac{\pi}{2} \Rightarrow t=\frac{\pi}{32}\).

Key Concepts

Graphing Trigonometric Functions

Understanding how to graph trigonometric functions is crucial in examining harmonic motion like that of a spring or pendulum. To begin with, let's consider the problem where we graph the function \(y=\frac{1}{4} \cos 16t\) for a ball bobbing on a spring. This particular function oscillates between a maximum of \(\frac{1}{4}\) foot and a minimum of \(-\frac{1}{4}\) foot, showing the displacement of the ball over time.

When graphing this cosine function, you'll notice a wave-like pattern that reflects the ball's movement. At \(t=0\), the graph starts at its maximum because cos(0) equals 1. The function then decreases to its minimum displacement and rises again, completing one full cycle of motion. To aid in understanding, we can visualize this function by plotting key points at intervals like \(\frac{\pi}{32}\), \(\frac{\pi}{16}\), and so on to see the ups and downs of the ball as time passes. Remember that for each full cycle of a cosine or sine function, the graph returns to its starting point, resembling a continuous wave.

Period of Oscillation

The period of oscillation is a term describing the time it takes to complete one full cycle of motion, such as one up-and-down movement of a ball on a spring. In trigonometric terms, this is closely tied to the concept of the period of a function. In our exercise, the period is determined by the function \(P=\frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(t\) in the cosine function.

For the given function \(y=\frac{1}{4} \cos 16t\), the value of \(B\) is 16, so the period of oscillation is \(P = \frac{2\pi}{16} = \frac{\pi}{8}\) seconds. This tells us that the ball completes one full bobbing cycle every \(\frac{\pi}{8}\) seconds. Understanding the period is essential for analysing the regularity and speed of oscillatory systems, and it applies to various physical contexts beyond just springs and pendulums, including electrical circuits and waves.

Equilibrium in Harmonic Motion

In the context of harmonic motion, equilibrium refers to the position where the forces in the system are balanced, and there is no net motion from these forces. It is, in essence, the 'resting' position of the system. For a ball in oscillatory motion on a spring, this is when the ball is at the midpoint of its up-and-down travel, not influenced by gravity or the spring's restoring force.

In trigonometric functions that model such motion, equilibrium occurs when the function's value is zero. To find when our bobbing ball first passes through the equilibrium point, we set the cosine function equal to zero and solve for \(t\). For the function \(y=\frac{1}{4} \cos 16t\), we find equilibrium by solving \(16t = \frac{\pi}{2}\), yielding \(t = \frac{\pi}{32}\) seconds. This calculation is a foundational one in physics and engineering, where determining the points of equilibrium is essential for analysing the stability and behavior of systems under oscillatory motion.