Question

An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations $$x(\theta)=a \cos \theta+\cos n \theta, y(\theta)=a \sin \theta+\sin n \theta.$$ The distance from the moon to the planet is taken to be \(1,\) the distance from the planet to the Sun is \(a,\) and \(n\) is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of \(n\) produce loops for a fixed value of \(a\) a. \(a=4, n=3\) b. \(a=4, n=4 \) c. \(a=4, n=5\)

Short answer

Provide a brief explanation for your answer. Answer: For \(a=4\), values of \(n\) that produce loops in the moon's path are 3, 4, and 5. This observation is based on the plotted graphs of the parametric equations with these values of \(n\). In each case, the moon's path crossed itself at certain points, indicating loops. However, these conclusions might not be universally true for all possible values of 'a' and 'n'.

Step-by-step solution

Understanding the equations

The parametric equations given are: \(x(\theta) = a \cos\theta + \cos(n\theta)\) \(y(\theta) = a \sin\theta + \sin(n\theta)\) Here, \(a\) is the distance of the planet from the Sun, and \(n\) is the number of moon orbits during a single planet revolution around the Sun. The variable \(\theta\) represents the angle. Step 2: Plot the graph of the path of the moon for given constants

Plotting the graph

For this step, you will use any graphing tool or software you prefer. You should input the given parametric equations and use the given values of \(a\) and \(n\). For each part of the problem, here are the parametric equation variations: a. \(x(\theta) = 4\cos\theta + \cos(3\theta), y(\theta) = 4\sin\theta + \sin(3\theta)\) b. \(x(\theta) = 4\cos\theta + \cos(4\theta), y(\theta) = 4\sin\theta + \sin(4\theta)\) c. \(x(\theta) = 4\cos\theta + \cos(5\theta), y(\theta) = 4\sin\theta + \sin(5\theta)\) Step 3: Observe the loops in the path

Observing loops

After plotting the graphs for each part, analyze them and look for loops in the moon's path. Loops are observed when the moon's path crosses itself at some points. Step 4: Conjecture which values of 'n' produce loops

Conjecturing values of 'n' that produce loops

Based on the observations made in the previous step, conjecture which values of 'n' produce loops for a fixed value of \(a\) among the given options. Remember that this conjecture is based on your observation of the plotted graphs, so it might not be universally true for all possible values of 'a$' and 'n'.

Key concepts

Circular Motion

Circular motion is an important concept when studying the paths of celestial bodies, such as moons and planets. It refers to the movement of an object along a circular path. In the problem above, both the moon and the planet exhibit circular motion as they orbit around each other and the Sun, respectively.
The moon follows a circular orbit around the planet, and the planet revolves around the Sun in a circular path as well. Each of these circular motions can be represented using parametric equations, as shown in the exercise. Such equations capture the position of an object in terms of the angle \(\theta\) that describes its location along the circular path.
Understanding the dynamics of circular motion is crucial for plotting accurate paths of celestial bodies. It is also foundational to concepts in physics, such as angular velocity and centripetal force. These forces explain how objects stay in their circular paths by continuously changing direction to maintain a constant radius from the center of the circle.

Graphs of Parametric Equations

Graphs of parametric equations offer a unique way of visualizing motion in a plane, particularly when it comes to depicting circular paths and other complex trajectories. In the context of the given exercise, the parametric equations:

• \(x(\theta) = a \cos\theta + \cos(n\theta)\)
• \(y(\theta) = a \sin\theta + \sin(n\theta)\)

help represent the complex path of a moon relative to its planet and the Sun. Here, the parameter \(\theta\) is the angle that increases as the moon orbits, capturing both the planet's orbit and the moon's orbit corresponding to the parameter \(n\).
The beauty of parametric equations lies in their ability to express two-dimensional motion without constraints seen in traditional \(y = f(x)\) functions. This feature makes them especially suitable for capturing more intricate paths, like looping motions observed in different values of \(n\). By plotting these equations, students can visually analyze and compare how different parameters affect the motion.

Orbital Mechanics

Orbital mechanics is a fascinating branch of physics and mathematics that studies how objects move under the influence of gravity. Through it, we understand how moons orbit planets and how planets, in turn, orbit stars like our Sun.
The parametric equations described simulate orbital paths in two dimensions. They balance gravitational forces to create stable orbits. Key details include the number of orbits the moon makes relative to the planet's orbit, represented by \(n\), and the varying loops that may arise from different \(n\) values. This contributes to the creation of interesting graphs and insights into orbital dynamics.
Understanding these dynamics allows scientists to predict planetary positions, allocate satellite orbits, and craft trajectories for spacecraft. It provides explanations for why moons don't simply fly off into space and underpins the essentials of space travel. The insights from orbital mechanics are foundational to planning missions, understanding tidal patterns, and exploring the universe.