Problem 62 - Precalculus : Building Concepts and Connections

Question

The position \(d\) of a block that is attached to a spring is given by the formula \(d=5 \sin \frac{\pi}{4} t\) where \(t\) is in seconds. What is the maximum distance of the block from its equilibrium position (the position at which \(d=0\) )? Find the period of the motion.

Step-by-Step Solution

Answer

The maximum distance of the block from its equilibrium position is 5 units. The period of the motion is 8 seconds.

Step 1: Identify the Amplitude

The amplitude of the sinusoidal wave is indicated by the coefficient of the sine function. In our equation, this is 5. Therefore, the maximum distance of the block from its equilibrium position is 5 units.

Step 2: Identify the Angular Frequency

The angular frequency of the wave is the coefficient of the variable \(t\) inside the sine function. In our equation, this is \(\pi / 4\). This represents how many total cycles the wave completes in a unit of time.

Step 3: Calculate the Period

The period of a sinusoidal wave is given by the reciprocal of the angular frequency, in this case, \(T = 2\pi / \omega\). Substituting our known value of \(\omega = \pi / 4\), we get that \(T = 2\pi / (\pi / 4) = 8\) seconds. This is how long one complete cycle of the motion takes.

Key Concepts

Amplitude of Sine Function

Understanding the amplitude of a sine function is crucial when analyzing sinusoidal motion in precalculus. In simple terms, the amplitude is the height of the wave from its mean, or equilibrium, position to its peak. For the function d=5 \sin \frac{\pi}{4} t, the number in front of the sine function, in this case 5, represents the amplitude of the wave.

This value tells us the maximum distance the block attached to a spring will move from its equilibrium position, which is when \(d=0\). Therefore, the amplitude of the sine function in this example is 5 units, indicating the block can reach 5 units away from the center in both positive and negative directions.

Angular Frequency

The concept of angular frequency is a bit more abstract than amplitude but just as important. It is denoted by \(\omega\) and describes how many radians a sinusoidal wave completes per unit of time. In the context of the provided function d=5 \sin \frac{\pi}{4} t, the coefficient \(\frac{\pi}{4}\) represents the angular frequency. Angular frequency tells us the rate at which the block oscillates back and forth about its equilibrium position.

When you identify the coefficient \(\frac{\pi}{4}\) inside the sine function, you discover the angular frequency of the block's motion, which equates to \frac{\raise.17ex\text{{ormalfont\textasciitilde}}}4 radians per second. The term 'radians' is crucial here; unlike cycles or revolutions, it is a standard unit of angular measurement used in mathematics.

Period of a Sinusoidal Wave

Last but not least, the period of a sinusoidal wave is a measure of time it takes for one complete oscillation or cycle of the wave to occur. It's the time interval after which the sinusoidal function starts repeating its values again. For the given function d=5 \sin \frac{\pi}{4} t, we can calculate the period by taking the reciprocal of the angular frequency \(\omega\), which gives us T = 2\pi / \omega.

By plugging the angular frequency value into this formula, \(T = 2\pi / (\pi / 4)\), we determine the period of the block's motion to be 8 seconds. This means that the block will take 8 seconds to complete one full back-and-forth motion from its starting position through its maximum displacement and back to the start.