Trying to find the equation of position in a circular oscillatory motion

[LCSphysicist]

Homework Statement

The problem is how to get a equation of a position of a body that suffer a impulse and now is on a orbit that oscillate. The conditions are:
The body was in a circular orbit at initial.
L remains constant
The circular orbit is stable
The energy increases a little.
The force is such that it has all the necessary conditions above, and, obviously, are central attractive and varies with the distance.

Relevant Equations

E = T + U'
L = mwr^2
Initial radius is ro
f = -kr^n

First of all, i know that the motion will be bounded, is not necessary to know if the motion will be closed or not.
Second, by analyzing the graphic of a effective potential with such conditions, the motion will agree with harmonic motion.

Ok

I don't know how to prove the harmonic oscillation, here i want help.

But, just assuming that it is true, the general equation is
r = M + acos + bsin

Since w = (√(k/m)), and k is the second derivation of the potential energy [here i use the effective potential instead just the potential] in the ro initial.

All of this bring to the final equation:

[Actually the terms between parentheses of the sin is under root , i forget write this.]

I don't know to take off t.
And How to determinate B

 

[kuruman]

I suppose English is not your native language so let me reformulate the statement of the problem as I understand it.

Mass ##m## moves in a circular orbit under the influence of a central force given by ##F=-k r^n.## At time ##t=0## this object receives a small impulse, so that its orbit is perturbed from being circular. Find an expression for the perturbed orbit.

Am I close?

 

[LCSphysicist]

I suppose English is not your native language so let me reformulate the statement of the problem as I understand it.

Mass ##m## moves in a circular orbit under the influence of a central force given by ##F=-k r^n.## At time ##t=0## this object receives a small impulse, so that its orbit is perturbed from being circular. Find an expression for the perturbed orbit.

Am I close?

yes haha sorry

 

[haruspex]

Homework Statement:: The problem is how to get a equation of a position of a body that suffer a impulse and now is on a orbit that oscillate. The conditions are:
The body was in a circular orbit at initial.
L remains constant
The circular orbit is stable
The energy increases a little.
The force is such that it has all the necessary conditions above, and, obviously, are central attractive and varies with the distance.
Relevant Equations:: E = T + U'
L = mwr^2
Initial radius is ro
f = -kr^n

First of all, i know that the motion will be bounded, is not necessary to know if the motion will be closed or not.
Second, by analyzing the graphic of a effective potential with such conditions, the motion will agree with harmonic motion.

Ok

I don't know how to prove the harmonic oscillation, here i want help.

But, just assuming that it is true, the general equation is
r = M + acos + bsin

Since w = (√(k/m)), and k is the second derivation of the potential energy [here i use the effective potential instead just the potential] in the ro initial.

All of this bring to the final equation:

View attachment 262765 [Actually the terms between parentheses of the sin is under root , i forget write this.]

I don't know to take off t.
And How to determinate B

You will be needing to make some approximation for the small perturbation, so I would start from the other end: write the differential equation for the motion based on the force function given.

 

[SammyS]

Relevant Equations:: E = T + U'
L = mwr^2
Initial radius is ro
f = -kr^n
...

Since w = (√(k/m)), and k is the second derivative of the potential energy [here i use the effective potential instead just the potential] in the r₀ initial.

All of this bring to the final equation:

View attachment 262765 [Actually the terms between parentheses of the sin is under root , i forget write this.]

I don't know how to take off t.
And How to determine B .

The expression for angular momentum is incorrect. r should not be squared.
It should be L = mΩr₀, where m is the mass of the orbiting body, Ω is the angular velocity for the orbit and of course, r₀, is the radius of the circular orbit (unperturbed). I use upper case Ω here, because there is no reason to expect that the angular frequency (angular velocity) of the orbit has the same value as the angular frequency of the harmonic oscillation.

I suggest finding the orbital angular frequency, Ω, or equivalently, the orbital period, T, for the circular orbit. Equate centripetal force and the central force, f, where ##\text{f} = -k ~ {r_0}^n ##.

 

[kuruman]

You will be needing to make some approximation for the small perturbation, so I would start from the other end: write the differential equation for the motion based on the force function given.

I would too. The mass is in an effective potential that one writes as the sum of the central potential and the centrifugal potential ##U_{eff}=U_{central}+\dfrac{L^2}{2mr^2}##. This can be used to find (a) the radius of circular orbits; (b) the values of ##n## (positive or negative) for which circular orbits exist; (c) a series expansion which will provide an effective spring constant for small radial oscillations.

 

[LCSphysicist]

Seeing this again, i am trying to solve:
I thought that:
F =

So F = C/(ro + dr)² + l²/m(ro + dr)³
Well, a little of algebra and expansion would lead to:
f = mr'' = C*/ro² - 2*C*L*dr/ro³ + L²/mro³ - L²*3*dr/(m*ro^4)

This is a little ugly.

I assumed n to be minus 2 to facility the problem at first, i think if i understand by this way i can go on.

C is negative

 

[LCSphysicist]

The expression for angular momentum is incorrect. r should not be squared.
It should be L = mΩr₀, where m is the mass of the orbiting body, Ω is the angular velocity for the orbit and of course, r₀, is the radius of the circular orbit (unperturbed). I use upper case Ω here, because there is no reason to expect that the angular frequency (angular velocity) of the orbit has the same value as the angular frequency of the harmonic oscillation.

I suggest finding the orbital angular frequency, Ω, or equivalently, the orbital period, T, for the circular orbit. Equate centripetal force and the central force, f, where ##\text{f} = -k ~ {r_0}^n ##.

But this don't even have the units of L :C kg*m²/s

 

[haruspex]

L = mΩr₀

That RHS has dimension momentum, not angular momentum.

 

[LCSphysicist]

Well, now seems okay my post.

 

[haruspex]

Seeing this again, i am trying to solve:
I thought that:
F = View attachment 264725
So F = C/(ro + dr)² + l²/m(ro + dr)³
Well, a little of algebra and expansion would lead to:
f = mr'' = C*/ro² - 2*C*L*dr/ro³ + L²/mro³ - L²*3*dr/(m*ro^4)

This is a little ugly.

I assumed n to be minus 2 to facility the problem at first, i think if i understand by this way i can go on.

C is negative

The force you need an expression for is the restoring force. This should be zero when dr is zero.

 

[SammyS]

That RHS has dimension momentum, not angular momentum.

Yes, you are correct.

It should be: L = mΩr₀², remembering that Ω is the orbital angular velocity of the initial circular motion, not, ω, the angular frequency of the additional oscillatory which results from the impulse.