Mechanics - Orbits under Central Force

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Homework Help Overview

The discussion revolves around a mechanics problem involving a particle attracted to a fixed point by a force proportional to zr^-5. The original poster attempts to show that the orbit of the particle can be expressed as r=acos(theta) using a specific orbit equation.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster outlines their approach to solving the problem and seeks confirmation on their progress. Another participant questions the formulation of the force term and suggests an alternative method of substitution to verify the proposed solution.

Discussion Status

The discussion is active, with participants engaging in clarifying the mathematical expressions involved. There is a suggestion to explore a substitution method, indicating a potential direction for further exploration without reaching a consensus on the correctness of the original poster's approach.

Contextual Notes

Participants are examining the implications of the force's formulation and its effect on the orbit equation. There is an underlying assumption regarding the nature of the force and its representation in the equations being discussed.

Master J
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Homework Statement


A particle moving in a plane is attracted towards a point (fixed) by a force zr^-5. The particle is projected from an apse at distance a with speed SQRT(z/(2a^4)).

Show that the orbit is r=acos(theta)

Using the Orbit Eq: d^2 u/dC^2 + u = za^-5/(hu)^2

h = angular momentum, or r(speed) in this case, u = 1/r



I get down to (du/dC)^2 + u^2 = -4/[(a^3)u] + A

where C = theta, A = integration constant.

Am I correct so far?
 
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za^-5/(hu)^2 should be zu^5/(hu)^2 shouldn't it? Rather than solving the equation, why don't you just substitute u=1/(a*cos(theta)) and see if the solution works?
 
Why is that?
 
Because if the radial force per unit mass is z/r^5 and r=1/u that turns into zu^5.
 

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