Question

The piston engine is the most commonly used engine in the world. The height of the piston over time can be modelled by a sine curve. Given the equation for a sine curve, \(y=a \sin b(x-c)+d,\) which parameter(s) would be affected as the piston moves faster?

Short answer

The parameter \(b\) (frequency) is affected as the piston moves faster.

Step-by-step solution

Identify the Sine Curve Components

The equation for the sine curve is given as \(y = a \sin b(x-c) + d.\) Identify each component: \(a\) is the amplitude, \(b\) is the frequency, \(c\) is the horizontal shift (phase shift), and \(d\) is the vertical shift.

Understand the Impact of Piston Speed

When the piston moves faster, it completes its cycles more quickly. This change affects the frequency of the sine wave.

Relate Frequency to the Parameter

Frequency in the sine curve equation is represented by the \(b\) parameter. As the speed of the piston increases, the value of \(b\) increases, indicating more cycles per unit time.

Summarize the Affected Parameter

In the context of the piston speed, only the \(b\) parameter, which represents frequency, is affected.

Key concepts

Sine Wave Frequency

In trigonometry, the frequency of a sine wave tells us how many cycles or oscillations occur in a given time period. This is vital for understanding the motion of any periodic system, such as a piston in an engine. The sine function, represented as \(y = a \sin b(x-c) + d\), uses the parameter \(b\) to denote frequency. When \(b\) increases, the wave completes more cycles in a shorter time, indicating a higher frequency.
For example, if a piston moves faster, the frequency of its motion increases. This directly correlates to an increase in the \(b\) parameter in the sine function equation.

Modeling Periodic Motion

Modeling periodic motion is essential in understanding many real-world systems like engines, musical instruments, and even the human heart. The sine function conveniently models such behaviors because it is inherently periodic, repeating its pattern over regular intervals.
To visualize this for a piston, imagine the up and down motion the piston undergoes as it operates. This repetitive action can be depicted using the sine curve, where each cycle corresponds to a complete upward and downward motion. By adjusting the parameters in \(y = a \sin b(x-c) + d\), we can accurately represent the behavior of the piston:

• \(a\): amplitude, or the maximum height reached
• \(b\): frequency, or how quickly the piston cycles
• \(c\): phase shift, or the starting point of the cycle
• \(d\): vertical shift, or the baseline level of the motion

Amplitude and Frequency in Trigonometry

Amplitude and frequency are two crucial components in trigonometric functions used to model waves. The amplitude \(a\) measures how far the wave peaks from its central value, reflecting the maximum displacement. In other words, it shows how 'tall' or 'short' the wave is.
In the context of our piston model, a higher amplitude indicates a greater range of motion of the piston. Conversely, frequency \(b\) measures how many wave cycles occur within a certain time frame. This is especially relevant when analyzing how fast the piston moves. Increased frequency means the piston completes its up-and-down cycles more quickly, suggesting a faster engine speed.

Phase Shift and Vertical Shift in Sine Functions

Phase shift \(c\) in sine functions determines the horizontal displacement, or where the wave starts its cycle. This means that if you change \(c\), you effectively move the entire wave left or right along the x-axis without altering its shape.
Vertical shift \(d\) alters the wave's mean position along the y-axis. By adjusting \(d\), you move the entire wave up or down without changing its frequency or amplitude.
In the piston model, a phase shift might represent the timing adjustment, while a vertical shift adjusts for any constant offset in the piston's position. Understanding these shifts helps in fine-tuning models to match real-world data precisely.