Problem 47 $$ \text { Prolate cycloid: } ... [FREE SOLUTION]

Chapter 6: Problem 47

$$ \text { Prolate cycloid: } x=\theta-\frac{3}{2} \sin \theta, y=1-\frac{3}{2} \cos \theta $$

Short Answer

Expert verified

The equations $x=\theta - \frac{3}{2}\sin \theta$ and $y=1-\frac{3}{2}\cos \theta$ represent a prolate cycloid. Here, x and y are calculated as functions of the angle $\theta$, creating a cycloid curve due to the periodic behavior of the sine and cosine functions.

Step by step solution

Identify the Parametric Equations

First, you need to identify the parametric equations given in the problem. Here the parametric equations are $x=\theta - \frac{3}{2}\sin \theta$ and $y=1-\frac{3}{2}\cos \theta$. $\theta$ is the parameter and it represents the angle.

Understand the Functions

Second, try to understand how the functions behave for different values of $\theta$. The $\sin$ and $\cos$ functions are periodic, and values will cycle between -1 and 1 as you vary $\theta$. This becomes important when trying to understand or visualize the curve described by these functions in the (x,y) plane.

Interpret the Functions

Finally, interpret the meaning of the equations. The equation for x tells us that the x-coordinate of a point on the cycloid is given by the current angle $\theta$ minus some offset $\frac{3}{2}\sin \theta$. Similarly, the equation for y tells us that the y-coordinate of the point is 1 minus the cosine of the angle times $\frac{3}{2}$. These formulas tell us the exact point on the curve for a given angle $\theta$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric Equations

Parametric equations are a novel way to represent curves in the plane by using an independent parameter, often denoted as $\theta$, rather than directly relating $x$ and $y$ as in traditional Cartesian equations.
This approach allows us to express $x$ and $y$ as separate functions of this parameter, giving more flexibility, especially when dealing with complex curves.

For our prolate cycloid, the parametric equations are $x = \theta - \frac{3}{2}\sin \theta$ and $y = 1 - \frac{3}{2}\cos \theta$.
Here, $\theta$ plays the role of a time-like parameter, controlling how the position changes along the curve.
The key benefit is it allows an easy expression for curves that are difficult to represent using a single function $y = f(x)$.

Each change in $\theta$ uniquely determines a point $(x, y)$ on the curve, with this pair directly summarizing the position on our prolate cycloid.

Trigonometric Functions

Trigonometric functions often play a vital role in parametrized representations of curves, such as cycloids. The sine and cosine functions used here exploit their periodic nature, oscillating between -1 and 1, to determine the specific shape of the path.

Sine function $\sin(\theta)$ shifts the $x$-coordinate by $\frac{3}{2}\sin\theta$, impacting the horizontal stretching of the curve.
Cosine function $\cos(\theta)$ adjusts the $y$-coordinate through $\frac{3}{2}\cos\theta$, influencing the vertical motion.

As $\theta$ moves through its values, these functions create a wave-like effect in the coordinates.
They are crucial in producing the cyclical nature of cycloids, allowing the curve to smoothly extend and loop with each increment of $\theta$. Trigonometric functions embody the inherent periodicity of many natural and engineered systems, making them indispensable in parametric settings.

Curve Visualization

Visualizing parametric curves like the prolate cycloid aids in understanding their formation and movement characteristics. When plotted, these equations create a distinct path where each point relies on $\theta$ to determine the (x, y) coordinates.

By incrementally adjusting $\theta$, each calculated point forms part of the larger picture. This progressive build-up displays the cycloid鈥檚 distinct loop-driven shape.
The symbiotic relationship between sine and cosine in modifying $x$ and $y$ allows the depiction of smooth, continuous movement across the plane.

Visual representation can clarify how these mathematical expressions translate to physical curves.
Such plots make it easier to grasp the cyclical properties, with peaks and troughs influenced by the integrated trigonometric functions. Curve visualization blends the abstract with the tangible, offering a bridge between calculations and their geometrical interpretations.

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$$ \text { Prolate cycloid: } x=\theta-\frac{3}{2} \sin \theta, y=1-\frac{3}{2} \cos \theta $$

Short Answer

Step by step solution

Identify the Parametric Equations

Understand the Functions

Interpret the Functions

Key Concepts

Parametric Equations

Trigonometric Functions

Curve Visualization

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