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

[DocHoliday]

Homework Statement

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 v_min and v_max 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.

Homework Equations

Conservation of Energy:
E = T + U = 0
T = (1/2)mv_min² + (1/2)mv_max²
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)R²dθ

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

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(V²_min + V²_max)/(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(v²_min+v²_max)))

Thanks

 

[mark726]

for sharing your approach to this problem! It seems like you have a good understanding of the conservation laws and the relevant equations. Your approach of using the conservation of energy and angular momentum is definitely a valid way to solve this problem.

I would suggest double checking your final solution, as it seems like there may be a small error in your calculations. I would also recommend checking the units of your solution to make sure they are consistent.

Another approach to solving this problem could be to use the second Kepler's law, which states that the area swept out by the radius vector of an orbiting body is equal to half the orbital period times the angular momentum. This could be another way to arrive at the same solution.

Overall, it's great to see you using your understanding of the conservation laws to solve this problem. Keep up the good work!