Problem 79
Question
Use a graphing utility to obtain the plane curve represented by the given parametric equations. Cycloid: \(x=2(t-\sin t), y=2(1-\cos t)\) \([0,60,5] \times[0,8,1], 0 \leq t<6 \pi\)
Step-by-Step Solution
Verified Answer
Following the above steps, one can generate a set of points by inputting different values of \(t\) from the given interval into the parametric equations and then plot the curve using these points in a graphing calculator. The result would be the graph of a cycloid.
1Step 1: Identify the Formulas for X and Y
We are given the parametric equations for X and Y as functions of t: \(x=2(t-\sin t), y=2(1-\cos t)\).
2Step 2: Identify the Interval for the Parameter t
The interval for \(t\) is specified by \(0 \leq t<6\pi\), and we are working within the viewing window [0,60,5] x [0,8,1]. Here, the first number in each bracket is the lowest value, the second number is the highest value, and the third number is the step increment for plotting points.
3Step 3: Plot the Curve
Plug values of \(t\) from the given interval into the parametric equations. Create a table of these values, then input these points into the graphing calculator. It will plot the curve.
4Step 4: Interpret the Graph
Once the graph is plotted, observe it. A graph created using these parametric equations will give the shape of a cycloid.
Key Concepts
Cycloid CurveGraphing UtilityParametric EquationsTrigonometric Functions
Cycloid Curve
A cycloid curve is the path traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. In mathematics, the cycloid is significant due to its unique properties and historical relevance in calculus and physics. The standard parametric equations for a cycloid generated by a wheel of radius r are given by:
\[ x=r(t-\sin t) \] \[ y=r(1-\cos t) \]
When plotted, it creates a series of arches or loops, characteristic of cycloidal motion. The significance of the cycloid curve is evident in various fields, including mechanical engineering, where it appears in the design of gear teeth, and in physics, where it shows up in discussions of the tautochrone and brachistochrone problems – the paths over which time of descent under gravity is constant and the path of fastest descent, respectively.
\[ x=r(t-\sin t) \] \[ y=r(1-\cos t) \]
When plotted, it creates a series of arches or loops, characteristic of cycloidal motion. The significance of the cycloid curve is evident in various fields, including mechanical engineering, where it appears in the design of gear teeth, and in physics, where it shows up in discussions of the tautochrone and brachistochrone problems – the paths over which time of descent under gravity is constant and the path of fastest descent, respectively.
Graphing Utility
A graphing utility is a powerful tool used to visualize mathematical equations and functions, including the intricate parametric curves like the cycloid. Modern graphing utilities may come as handheld devices, software applications, or online graphing calculators. They allow users to transform complex mathematical expressions into visual graphs. This visual representation aids in understanding the behavior and properties of the equation. For graphing a cycloid, a graphing utility will enable students to input the parametric equations, adjust viewing windows, and observe the unique shape of the cycloid curve as it is generated. This dynamic approach can be instrumental in illustrating concepts that could be difficult to interpret from equations alone.
Parametric Equations
Parametric equations represent a set of related quantities as explicit functions of an independent parameter, usually denoted as t. Unlike standard Cartesian equations that relate x and y variables directly, parametric equations express x and y using t. The beauty of parametric equations lies in their ability to express curves that are difficult or impossible to describe using standard x-y equations.
For example, in the given cycloid problem, the parametric equations \( x=2(t-\sin t) \) and \( y=2(1-\cos t) \) require the independent variable t to calculate the corresponding x and y values at each point along the curve. This framework is particularly beneficial when defining the motion of an object, where the time factor t is essential in plotting the object’s path.
For example, in the given cycloid problem, the parametric equations \( x=2(t-\sin t) \) and \( y=2(1-\cos t) \) require the independent variable t to calculate the corresponding x and y values at each point along the curve. This framework is particularly beneficial when defining the motion of an object, where the time factor t is essential in plotting the object’s path.
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental in the study of periodic phenomena, wave motion, oscillations, and in describing circular or rotational motion. These functions relate an angle in a right-angled triangle to the ratios of two sides of that triangle and have values ranging between -1 and 1.
In parametric equations for the cycloid, \(x=2(t-\sin t)\) and \(y=2(1-\cos t)\), the sine and cosine functions, represented as \(\sin t\) and \(\cos t\), are used to capture the cyclical nature of the rolling motion. The functions account for the circular motion of the wheel’s point and its cyclical return to a baseline, which is essential in constructing the cycloid's unique looping path.
In parametric equations for the cycloid, \(x=2(t-\sin t)\) and \(y=2(1-\cos t)\), the sine and cosine functions, represented as \(\sin t\) and \(\cos t\), are used to capture the cyclical nature of the rolling motion. The functions account for the circular motion of the wheel’s point and its cyclical return to a baseline, which is essential in constructing the cycloid's unique looping path.
Other exercises in this chapter
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