(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle of unit mass is projected with a velocity v-(sub 0), at right angle to the radius vector at a distance 'a' from the origin of a center of attractive force given by:

f(r)= -k*(4/(r^3)+ (a^2)/(r^5)).

If (v-(sub 0))^2 = (9*k)/(2*(a^2)) find the polar equation of the resulting orbit.

2. Relevant equations

F=ma

Treat 1/r = u

acceleration in polar coordinates= (r*{double dot} - r*((theta) {dot})^2 * e-sub r) + (r*(theta) {double dotted} + 2*r {dotted} + (theta) {dot})e-sub theta. Where e-sub are units vectors.

theta{dot} = l*u^2, where l is angular momentum per mass.

3. The attempt at a solution

Ok this is kind of long so I may skip few steps (sorry):

m*((r{double dot}) - r*((theta){dot})^2 = f(r), since the angular componet of acceleration is zero for this situtation.

m*(r{double dot} - (theta{dot})^2) = f(u^-1)

m*(r{double dot} - (theta{dot})^2) = -k*(4/(u^3)+ (a^2)/(u^5)).

m* [ -l^2*u^2*d^2*u/d(theta)^2-1/u*(l^2*u^3)]=k*(4/u^3 +a^2/u^5)

d^2*u/d(theta)^2 + u= (k*(4/u^3 + a^2/u^5))/(m*l^2*u^2)

and then I get stuck. I have tried multiple avenues for trying to solve this diff. eq, but none of them seem to cut it.

Anyone have any ideas? If the suggestion for problem goes to using an energy relation, I have tried that too and I get stuck in a similar problem.

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# Determing a polar orbit for a non-inverse square central force

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