Question

A trochoid is the path followed by a point $b$ units from the center of a wheel of radius $a$ as the wheel rolls along the $x$ -axis. Its parametric description is $x=at-b\sin t,y=a-b\cos t.$ Choose specific values of $a$ and $b,$ and use a graphing utility to plot different trochoids. In particular, explore the difference between the cases $a>b$ and \(a

Short answer

Answer: When a > b (radius of the wheel is larger than the distance of the point b from the center), the trochoid will have wider loops with smaller cusps occurring in the same period. In contrast, when a < b (radius of the wheel is smaller than the distance of the point b from the center), the trochoid will have more elongated loops with more prominent cusps in the same period. The differences are due to the proportion of the radius of the wheel to the distance of the point from the center, affecting the shape of the curve and the size of the loops.

Step-by-step solution

Choose Values for $a$ and $b$

We will need to choose specific values for $a$ and $b$ and use those to plot a graph of the trochoid. To do that, let's first choose a pair of values for $a>b$. Let $a=5$, and $b=3$. Now, let's choose a pair of values for $a<b$. Let $a=3$ and $b=5$.

Calculate Parametric Equations for each Case

For each case, we need to plug in our chosen values of $a$ and $b$ into the parametric equations for $x$ and $y$: - Case $a>b$, $a=5$, $b=3$ $$x=5t-3\sin t$$ $$y=5-3\cos t$$ - Case $a<b$, $a=3$, $b=5$ $$x=3t-5\sin t$$ $$y=3-5\cos t$$

Plot Trochoid for each Case on a Graphing Utility

With a graphing utility, plot each set of parametric equations as a curve: - Case $a>b$, $a=5$, $b=3$ - Parametric Equations: $$x=5t-3\sin t$$ $$y=5-3\cos t$$ - Graph the curve for $t$ in the interval $[0,2\pi ]$ or other interval depending on the visualization. - Case $a<b$, $a=3$, $b=5$ - Parametric Equations: $$x=3t-5\sin t$$ $$y=3-5\cos t$$ - Graph the curve for $t$ in the interval $[0,2\pi ]$ or other interval depending on the visualization.

Analyze the Differences in Trochoid Shapes

Compare the trochoid shapes for each case to analyze the differences: - Case $a>b$: When $a>b$ (radius of the wheel is larger than the distance of the point $b$ from the center), the trochoid will have wider loops with smaller cusps occurring in the same period. - Case $a<b$: When $a<b$ (radius of the wheel is smaller than the distance of the point $b$ from the center), the trochoid will have more elongated loops with more prominent cusps in the same period. These differences are due to the proportion of the radius of the wheel to the distance of the point from the center, affecting the shape of the curve and the size of the loops.

Key concepts

Parametric Equations

When studying geometry or calculus, you may encounter curves that can't be described as a function in the traditional form of $y=f(x)$. This is where parametric equations come in handy. Parametric equations break the single equation into a system of equations that define both $x$ and $y$ as separate functions of a third variable, usually denoted as $t$, which is known as the parameter.

For example, in the case of a trochoid, where a point is tracing a path as a wheel rolls, parametric equations accurately capture the motion along the curve. The trochoid's equations, $x=at-b\sin t$ and $y=a-b\cos t$, represent the horizontal and vertical positions of a point at time $t$.

• $t$ can correspond to time, making it incredibly practical for real-world applications such as physics and engineering.
• The variable $a$ relates to the wheel's radius while $b$ represents the distance from the wheel's center.
• By varying $a$ and $b$, trochoids demonstrate different kinds of periodic motion, which is beautifully illustrated when graphed.

Graphing Parametric Curves

The process of graphing parametric curves is distinct because you're plotting points $(x(t),y(t))$ for various values of the parameter $t$. When you graph these points, you can visualize the curve that a moving point follows.

To graph a trochoid, you need a graphing utility that allows you to input these parametric equations. As you change the parameter $t$, you map out the motion of the point and thereby 'draw' the shape of the trochoid. Often, the range of $t$ is chosen to complete one full cycle, such as $[0,2\pi ]$, corresponding to the complete turn of the wheel.

Graphing parametric curves is critical in areas like computer graphics, where they're used to animate motion, or in design, where they help in constructing intricate patterns and figures. The difference in the curves for different values of $a$ and $b$ in trochoids offers a perfect demonstration of how these parameters define the nature and shape of the resulting graph.

Trigonometric Functions in Calculus

Trigonometric functions such as $\sin$ and $\cos$ are fundamental to calculus. They have unique properties that make them perfectly suitable for describing periodic phenomena like waves, oscillations, and circular motion. In the context of trochoids, which are curves based on circular motion, trigonometric functions in calculus play a crucial role.

Periodicity and Amplitude

• The trigonometric terms in the parametric equations represent the periodic nature of the movement of the point on the rim or outside/inside of the wheel.
• The coefficient $b$ in front of the $\sin t$ and $\cos t$ functions determines the amplitude. For instance, in a trochoid, the value of $b$ controls the vertical and horizontal deviations from the center, affecting the 'loopiness' of the curve.
• In analyzing trochoids, the interplay between the trigonometric functions and the parameters $a$ and $b$ influence the overall shape and complexities of the curves.

The application of trigonometric functions extends far beyond simple right-angle triangle problems, becoming a powerful tool in creating and understanding complex mathematical models in calculus.