The Earth and the Moon form a two-body system interacting through their mutual gravita- tional attraction. In addition, each body is attracted by the gravitational field of the Sun, which here we will treat as an external force to the two-body Earth-Moon system. Take the Sun as the origin and write down the equations of motion for the center of mass X and the relative position x of the Earth-Moon system. Expand the resulting expressions in powers of r/X, the ratio of their magnitudes. Show that to lowest order in x/X the center of mass and relative position are uncoupled, but that in higher orders they are coupled because the Sun's gravitational force is not constant.

The Earth and the Moon form a two-body system interacting through their mutual gravita- tional attraction. In addition, each body is attracted by the gravitational field of the Sun, which here we will treat as an external force to the two-body Earth-Moon system. Take the Sun as the origin and write down the equations of motion for the center of mass X and the relative position x of the Earth-Moon system. Expand the resulting expressions in powers of r/X, the ratio of their magnitudes. Show that to lowest order in x/X the center of mass and relative position are uncoupled, but that in higher orders they are coupled because the Sun's gravitational force is not constant.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Concept explainers

Rigid Body

A rigid body is an object which does not change its shape or undergo any significant deformation due to an external force or movement. Mathematically speaking, the distance between any two points inside the body doesn't change in any situation.

Rigid Body Dynamics

Rigid bodies are defined as inelastic shapes with negligible deformation, giving them an unchanging center of mass. It is also generally assumed that the mass of a rigid body is uniformly distributed. This property of rigid bodies comes in handy when we deal with concepts like momentum, angular momentum, force and torque. The study of these properties – viz., force, torque, momentum, and angular momentum – of a rigid body, is collectively known as rigid body dynamics (RBD).

Question

The Earth and the Moon form a two-body system interacting through their mutual gravita-
tional attraction. In addition, each body is attracted by the gravitational field of the Sun,
which here we will treat as an external force to the two-body Earth-Moon system. Take the
Sun as the origin and write down the equations of motion for the center of mass X and the
relative position x of the Earth-Moon system. Expand the resulting expressions in powers
of x/X, the ratio of their magnitudes. Show that to lowest order in x/X the center of mass
and relative position are uncoupled, but that in higher orders they are coupled because the
Sun's gravitational force is not constant.

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