- #1

azabak

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Can one show that the work done by the force is equal to the variation of kinetic energy solely due the conservation of angular momentum?

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- Thread starter azabak
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In summary, if a mass is in an elliptical orbit around a central force that is inversely proportional to the distance squared, the mass will accelerate when moving closer to the focus. The angular momentum of the orbit remains constant. It is possible to show that the work done by the force is equal to the change in kinetic energy solely due to the conservation of angular momentum. There must be an additional work done in order to conserve angular momentum. The radius, velocity, and distance from the focus are all interdependent and must be chosen carefully to maintain conservation of energy and angular momentum.

- #1

azabak

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Can one show that the work done by the force is equal to the variation of kinetic energy solely due the conservation of angular momentum?

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- #2

tiny-tim

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azabak said:Can one show that the work done by the force is equal to the variation of kinetic energy solely due the conservation of angular momentum?

work done is

- #3

azabak

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Correct me if I'm wrong.

- #4

Doc Al

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Why is that?azabak said:If you calculate just the work done due the variation of potential energy angular momentum is not conserved.

What do you mean?For it to happen the force must do an "extra" work.

- #5

azabak

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The central force is F = -k*m/R² where k is a constant.

The radius varies from R to r.

The work done W = -ΔU and the potential energy U = -k*m*/R.

The final kinetic energy due variation of potential energy is K = (m*v²/2)+(k*m*((1/R)-(1/r))).

To conserve the angular momentum the final kinetic energy should be K = (m*v²*R²)/(2*r²), which will be larger than the calculated just from variation of potential energy.

Therefore there must be an additional work in order to conserve angular momentum.

- #6

Doc Al

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Careful. Angular momentum is a vector quantity:azabak said:The angular momentum of a mass m is L = m*v*R.

[tex]\vec{L} = \vec{r}\times m\vec{v}[/tex]

The angle between the vectors v and r is important.

Angular momentum, properly defined as the vector product above, is conserved, but the quantity m*v*R is not.

- #7

azabak

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Oh, I see. Thank you very much.

- #8

willem2

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azabak said:

The central force is F = -k*m/R² where k is a constant.

The radius varies from R to r.

The work done W = -ΔU and the potential energy U = -k*m*/R.

The final kinetic energy due variation of potential energy is K = (m*v²/2)+(k*m*((1/R)-(1/r))).

To conserve the angular momentum the final kinetic energy should be K = (m*v²*R²)/(2*r²), which will be larger than the calculated just from variation of potential energy.

Therefore there must be an additional work in order to conserve angular momentum.

If you use the apihelion and the perihelion distance for r and R, so as to escape Doc Al's objection, you will find that you can't freely choose v, R and r. If you give the apihelion distance, and the initial speed, the perihelion distance is fixed so that both energy and angular momentum are conserved, and the 2 energies you computed are equal.

(At the perihelion and apihelion the distance vector r is perpendicular to v, so the magnitude of the angular momentum is equal to m*r*v)

- #9

azabak

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- #10

willem2

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azabak said:

There's always an unique wordline if you specify an initial position and velocity. If you have a central force and a conservative force field, angular momentum and total energy will be conserved.

- #11

tiny-tim

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hi azabak!

i don't understand this

potential energy is*defined* as minus the work done by a conservative force

angular momentum is*always* conserved if the force is central

and*any* central force which is a sensible function of r will be the gradient of a scalar, and so will be conservative

azabak said:If you calculate just the work done due the variation of potential energy angular momentum is not conserved. For it to happen the force must do an "extra" work. In any situation of rotation conservation of angular momentum is the important factor. Therefore accepting that angular momentum is always conserved implies accepting that any central force does more work than the variation of potential energy.

i don't understand this

potential energy is

angular momentum is

(proof: d/dt(**r** x m**r**') = **r** x m**r**'' = **r** x **F**

which by definition is 0 if**F** is central)

which by definition is 0 if

and

as **willem2** says …

willem2 said:If you have a central force and a conservative force field, angular momentum and total energy will be conserved.

Last edited:

- #12

azabak

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I previously considered that the energy and angular momentum were conserved for

- #13

tiny-tim

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azabak said:… Then willem2 cleared me that it can only happen for a certain defined values of the radius.

no he

once you choose two of them, the third is automatically decided

A central force is a type of force that acts on an object towards or away from a fixed point, known as the center of force. This force is always directed along the line connecting the object and the center of force, and its magnitude depends only on the distance between them.

A central force causes an object to move in a curved path, either towards or away from the center of force, depending on the direction of the force. This results in a change in the object's velocity, leading to a change in its angular momentum.

Angular momentum is a measure of an object's rotational motion around a fixed axis. It is defined as the product of an object's moment of inertia and its angular velocity. In simpler terms, it is the tendency of an object to continue rotating at a constant rate.

In a central force system, the magnitude and direction of angular momentum remain constant as long as there is no external torque acting on the object. This is known as the law of conservation of angular momentum.

Kepler's laws of planetary motion describe the motion of planets around the sun. The first law states that the planets move in elliptical orbits with the sun at one focus. This can be explained by the central force of gravity between the sun and the planets. The second law states that a line joining a planet and the sun sweeps out equal areas in equal time intervals, which is a consequence of the conservation of angular momentum. The third law states that the square of a planet's orbital period is proportional to the cube of its average distance from the sun, which can be derived using the law of universal gravitation and the concept of central forces.

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