# CH3D - 3.7 The Kepler Problem: Inverse-Square Law Force The...

This preview shows pages 1–3. Sign up to view the full content.

By Prof. Juyang Huang, Texas Tech University. All rights reserved. 3.7 The Kepler Problem: Inverse-Square Law Force Inverse square law force : F(r) = -k/r 2 ; V(r) = -k/r Newton’s Universal Law of Gravitation F = Gm 1 m 2 /r 2 ; Coulomb’s Law: F = k e q 1 q 2 /r 2 ; It is the most important case of Central Force motion. It describes the motions of comets and planets about Sun, the motions of Moon and satellites about Earth. The general equation of orbit for any central force (section 3.5) " = ' # du 2 mE l 2 # 2 mV l 2 # u 2 \$ , where u = 1/r and ! is the angle at which r = r min . Substitute V(r) = -k/r = ' # du 2 mE l 2 + 2 mku l 2 # u 2 \$ Use the indefinite integral dx + # x + \$ x 2 % = 1 & arccos & + 2 x 2 & 4 "\$ ' ( ) ) * + , , By Prof. Juyang Huang, Texas Tech University. All rights reserved. The equation of the orbit becomes 1 r = mk l 2 1 + + 2 El 2 mk 2 cos( # ') \$ % & & ' ( ) ) The general equation of a conic with one focus at the origin is 1/ r = C [1 + e cos( ! - ! ´ )] where e is the eccentricity of the conic section: e ! + 2 El 2 mk 2 , and C ! mk/l 2 . All orbits for 1/ r 2 forces (attractive or repulsive) are conic sections. Conic Sections (Analytic Geometry Review)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
By Prof. Juyang Huang, Texas Tech University. All rights reserved. Conic sections: Curves formed by the intersection of a plane and a cone. A conic section: A curve formed by the loci of points (in a plane) where the ratio of the distance from a fixed point ( the focus ) to a fixed line ( the directorix ) is a constant.
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/22/2011 for the course PHYSICS 116 taught by Professor Scopel during the Spring '11 term at Seoul National.

### Page1 / 6

CH3D - 3.7 The Kepler Problem: Inverse-Square Law Force The...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online