Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

In many places in the solar system, a moon orbits a planet, which in turn orbits the sun. In some cases the orbits are very close to circular. We will assume that these orbits are circular with the sun at the center of the planet's orbit and the planet at the center of the moon's orbit. We will further assume that all motion is in a single \(x y\) -plane. Suppose that in the time the planet orbits the sun once the moon orbits the planet ten times. (a) If the radius of the moon's orbit is \(R_{m}\) and the radius of the planet's orbit about the sun is \(R_{p}\), show that the motion of the moon with respect to the sun at the origin could be given by $$ x=R_{p} \cos t+R_{m} \cos 10 t, \quad y=R_{p} \sin t+R_{m} \sin 10 t $$ CAS (b) For \(R_{p}=1\) and \(R_{m}=0.1\), plot the path traced by the moon as the planet makes one revolution around the sun. (c) Find one set of values for \(R_{p}, R_{m}\) and \(t\) so that at time \(t\) the moon is motionless with respect to the sun.

Step-by-Step Solution

Answer

(a) Parametric form of moon's orbit given. (b) Plot shows a spirograph-like pattern. (c) Moon is motionless w.r.t. the sun at \\(t=0\\).

Step 1: Understanding the System

We have a planet orbiting the sun and a moon orbiting the planet. Both orbits are circular and happen in the xy-plane. The planet completes one revolution around the sun while the moon completes ten revolutions around the planet.

Step 2: Derive Moon's Coordinates w.r.t. Sun

The moon’s position can be expressed using the parametric equations for circular motion. For the planet around the sun, it is \(x = R_p \cos t\) and \(y = R_p \sin t\). For the moon around the planet, we have \(x = R_m \cos 10t \) and \(y = R_m \sin 10t\). Adding these gives the moon’s path relative to the sun as \(x = R_p \cos t + R_m \cos 10t\) and \(y = R_p \sin t + R_m \sin 10t\).

Step 3: Set Constants for Plotting

We are given \(R_p = 1\) and \(R_m = 0.1\) for plotting. The parametric equations become \(x = \cos t + 0.1 \cos 10t\) and \(y = \sin t + 0.1 \sin 10t\).

Step 4: Plot the Path

Using the values from Step 3, plot the equations \(x = \cos t + 0.1 \cos 10t\) and \(y = \sin t + 0.1 \sin 10t\) over a range of \(t\) from 0 to \(2\pi\). This can be done using graphing software or a graphing calculator.

Step 5: Condition for Moon Being Motionless

For the moon to be motionless relative to the sun, its velocity, which is the derivative of its position \(x(t), y(t)\), should be zero. Thus, we need computed derivatives: \(\frac{dx}{dt} = -R_p \sin t - 10 R_m \sin 10t\) and \(\frac{dy}{dt} = R_p \cos t + 10 R_m \cos 10t\). Set both derivatives to zero and solve for \(t\).

Step 6: Solve for t When Velocity is Zero

From Step 5, solve the system: \(-R_p \sin t = 10 R_m \sin 10t\) and \(R_p \cos t = -10 R_m \cos 10t\) given \(R_p = 1\) and \(R_m = 0.1\). A simple solution can be found: \(t = 0\), since it could satisfy both equations in special cases (for instance symmetry points).

Key Concepts

Parametric Equations

Parametric equations are a powerful mathematical tool used to represent curves and paths in a plane. Unlike standard equations, which express a direct relationship between two variables, parametric equations introduce a third variable, typically time (denoted as \( t \)), to describe the evolution of each coordinate separately.
This is particularly useful in situations where the relationship between variables is more complex or involves motion, such as in planetary orbits.

With parametric equations, we can express the horizontal ( \( x \) ) and vertical ( \( y \) ) positions of an object over time. For circular motion, these equations are typically of the form:

• \( x = R \cos(t) \)
• \( y = R \sin(t) \)

Here, \( R \) represents the radius of the circle, and \( t \) parameterizes time over which the motion occurs. These equations determine the location of a point moving in a circular path, providing an effective solution for handling problems involving circular motion like that of our planet and moon example.

Planetary Motion

Planetary motion refers to the movement of planets and other celestial bodies around a star, most notably around our sun. This motion is governed by gravitational forces and results in orbits that are typically elliptical or circular.

In the context of the given problem, we're dealing with simplified circular orbits. This means that the moon orbits the planet and the planet orbits the sun, both in perfect circles. The beauty of this simplification is that it allows for the use of parametric equations to describe the motion efficiently.

The equations used to define the moon's position, relative to an origin at the sun, incorporate both the revolution of the planet and the moon. By observing:

• \( x = R_{p} \cos(t) + R_{m} \cos(10t) \)
• \( y = R_{p} \sin(t) + R_{m} \sin(10t) \)

we see how the motion intricately depends on both the orbit of the planet around the sun (first terms) and the orbit of the moon around the planet (second terms). Understanding these interactions through parametric equations unveils the complex dance of celestial bodies in our solar system.

Circular Orbits

Circular orbits are a special case of orbital motion where an object follows a path with a constant radius around another object, typically under the influence of gravity. In astronomy, many celestial bodies, like moons and planets, exhibit circular or near-circular orbits due to these gravitational interactions.

In a perfect circular orbit, the speed of the orbiting body is constant, and it traces a path defined by simple geometric parameters. This assumption of circular paths makes calculations simpler and helps us understand the broader concepts of motion and attraction in space.

In our problem, we are dealing with the idealized scenario where both the planet and the moon have perfectly circular orbits. This allows us to seamlessly apply the principles of parametric equations to describe the moon's path as influenced both by its orbit around the planet and the planet's orbit around the sun.

• The concept relies on constant radii: \( R_{p} \) for the planet's orbit and \( R_{m} \) for the moon's orbit.
• The expressions \( \cos \) and \( \sin \) embody the cyclical nature of the motion, capturing the continuous rotation and position of the bodies.

By leveraging these assumptions, we can model the intricate pathways traveled by celestial bodies within our solar system.