- #1

- 32

- 0

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?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter azabak
- Start date

- #1

- 32

- 0

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?

- #2

tiny-tim

Science Advisor

Homework Helper

- 25,836

- 252

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

- 32

- 0

Correct me if I'm wrong.

- #4

Doc Al

Mentor

- 45,261

- 1,616

Why is that?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

- 32

- 0

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

Mentor

- 45,261

- 1,616

Careful. Angular momentum is a vector quantity: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

- 32

- 0

Oh, I see. Thank you very much.

- #8

- 2,056

- 319

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

- 32

- 0

- #10

- 2,056

- 319

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

Science Advisor

Homework Helper

- 25,836

- 252

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

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 …

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

Last edited:

- #12

- 32

- 0

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

- #13

tiny-tim

Science Advisor

Homework Helper

- 25,836

- 252

… 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

Share: