# Period of circular orbit with in terms of varying speeds and radius.

1. Apr 5, 2013

### DocHoliday

1. The problem statement, all variables and given/known data
A particle of unknown mass moves in a circular orbit of radius R under the influence of a central force centred at some point inside the orbit. The minimum and maximum speeds of the particle are vmin and vmax respectively. Find the orbital period T in terms of these speeds and the radius of the orbit.
Hint : Use the second Kepler's law.
We are going over central forces in classical mechanics.
The picture has the centred force offset from the middle of the circular orbit.
No equations were given, just the ones I think are relevant.

2. Relevant equations
Conservation of Energy:
E = T + U = 0
T = (1/2)mvmin2 + (1/2)mvmax2
U = l2/(2u(R+x)2) + l2/(2u(R-x)2)
= (1/2)u(R+x)2(dθ/dt)2 + (1/2)u(R-x)2(dθ/dt)2
Conservation of angular momentum:
l = uR2(dθ/dt) = constant
*u is reduced mass

Centrifugal Energy:
U = l2/(2uR2)

Area swept or aerial velocity:
dA = (1/2)R2

Divide by time to get:
(dA/dt)=(1/2)R2(dθ/dt)

3. The attempt at a solution

Attached but I will also go through some steps.

I assumed that I could solve this by using conservation of energy and conservation of angular momentum. I drew a line between the random point in the orbit and the center calling it "x".
I assumed two points on either side of the circle where (1/2)mv2 for the kinetic energy while the potential centrifugal energy would be l2/(2u(R+x)2). m would represent the particles mass. I picked (R+x) and (R-x) to represent the distances at the points where the kinetic energy would be measured. Setting the Energy = 0 and solving gave

(dθ/dt)2 = m(V2min + V2max)/(u((R+x)2 + (R-x)2))

I used this to replace the (dθ/dt) in the aerial velocity equation. Next I moved dt to the right hand side of the equation and integrated from 0 to A on the left and 0 to T on the right.
Then replacing A by ∏R2 (area of a circle), and solving for the period I assumed would give me an answer. I don't know if it's right or wrong, my prof. emphasizes analytical solutions. If there's an easier way I'm all ears.

My final solution was T = sqrt((8pi2(R2+x2))/(um(v2min+v2max)))

Thanks

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• ###### Phy321.9.3.jpg
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Last edited: Apr 5, 2013