# In the angular momentum equation $L = r \times p$, which of the remaining variable’s magnitudes is conserved when the magnitude of the radius changes? [closed]

Logic is the cornerstone of physics.

If the premisses are valid and a deduction based solely on those premisses is valid, then the conclusion must also be valid.

Conservation of angular momentum in a variable radii system has not been verified by rigorous scientific experiment. There is no famous experiment. In more than a year of discussions, not one detractor has managed to produce any reference to any experiment which does confirm it.

My efforts here are purely in the interests of exposing a long hidden fault.

If simply reading this presentation is upsetting to you, please consider that the cause might be due to cognitive dissonance.

I would really appreciate it if you could avoid targeting your anger towards me as I am merely the messenger. I have no option, since making this discovery, other than to try and make it known.

I hope you can see that censoring this information is not the most intelligent course of action.

Abstract:

Both angular momentum and momentum are generally accepted by scientists to be conserved values, and both of these variables are contained within the equation L = r x p, where L corresponds to angular momentum, r corresponds to radius, and p corresponds to momentum. Assuming that the implied rotation occurs around a central point, the magnitudes of angular momentum and momentum cannot both be conserved when the magnitude of the radius changes. The generally accepted principle is that the magnitude of the momentum must change in order to conserve angular momentum. However, it is logically proven that it is the magnitude of the component of momentum perpendicular to the radius that must be conserved.

Introduction:

While working on a project that did not achieve the results predicted, I stumbled upon this oversight within the laws of physics.

Proof:

For the equation L = r x p(1) assuming that the implied rotation occurs around a central point.

Premise 1: There is a force at all times directed from the point mass along the radius toward the centre of rotation (centripetal force).

Premise 2: A change in the magnitude of radius is conducted by altering the magnitude of this force.

Premise 3: There can be no component of this force perpendicular to the radius.

Premise 4: In order to affect the magnitude of the component of momentum perpendicular to the radius, one must apply a parallel component of force (Newton’s first law).

Deduction:

A change in the magnitude of the radius cannot affect the magnitude of the component of momentum perpendicular to the radius.

Conclusion:

In the equation L = r x p, assuming that the implied rotation occurs around a central point, it is the magnitude of the component of momentum perpendicular to the radius that must be conserved when the magnitude of the radius changes.

References:

1) D. Halliday & R. Resnick, Fundamentals of Physics, second edition, extended version (John Wiley & Sons, Inc., New York, 1981) p. 181.

## closed as unclear what you're asking by Yashas, ZeroTheHero, David Hammen, John Rennie, honeste_vivereJul 2 '17 at 13:05

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There is no oversight in the laws of physics.

The basic misunderstanding is that conservation laws hold for isolated, closed systems.

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time.

The change of r comes outside of the original closed system, Thus there is no problem. Force = dp/dt ( vectors) and takes account of conservation in the new system "old + force"

• Please see premise 1. There is always a force, so using your explanation, the angular velocity should always be increasing - even if the radius remains constant or is actually increasing. Your explanation does not hold water. – John Jul 2 '17 at 11:03
• You misunderstand my explanation. angular momentum is constant. The centripetal and centrifugal forces cancel each other in an isolated system , otherwise, if there were no string ( or a gravitational force in the case of satellites) there would be no radius to change. The force dp/dt is coming from your hand shortening the string and opening the system. – anna v Jul 2 '17 at 11:09
• Please see premise 2. – John Jul 2 '17 at 11:18
• premise two opens the system. it breaks conservation continuity – anna v Jul 2 '17 at 12:11
• Premise two does not open the system. You are incorrect. Premise two merely states that the only way to alter the radius is to adjust the force which is being applied as centripetal force. Your explanation makes an assumption that there is no force being applied when the radius is constant and that a force is needed to reduce the radius and that force is what accounts for the extra energy required to conserve angular momentum. You are not making sense with that assumption - it is incorrect. – John Jul 2 '17 at 13:14

Newton's second law states that, if $\vec{p}$ is a body's momentum, $\dot{\vec{p}}$ is the force acting on that body, say $\vec{F}$. This is not in general conserved, though the sum of all bodies' momenta will be by Newton's third law. Indeed, if body $i$ exerts force $\vec{F}_{ij}$ on body $j$, $\sum_{i}\dot{\vec{p}}_i=\sum_{ij}\vec{F}_{ij}=\vec{0}$ because $\vec{F}_{ij}=-\vec{F}_{ji}$.

If the body's position is $\vec{r}$, its angular momentum $\vec{L}=\vec{r}\times\vec{p}$ has time derivative $\dot{\vec{r}}\times\vec{p}+\vec{r}\times\dot{\vec{p}}$. Since the cross product of parallel vectors is zero, the first term vanishes because $\vec{p}=m\dot{r}$, with $m$ the body's mass. By Newton's second law, $\dot{\vec{L}}=\vec{r}\times\vec{F}$. Thus $\vec{L}$ is conserved if $\vec{F}$ is parallel to $\vec{r}$. If the only force acting on the body is along the line joining it to a point it's orbiting, and if that point is taken as the coordinate system's origin, we say $\vec{F}$ is radial. Then $\vec{L}$ is conserved by the above argument. If there are at least three bodies in a system, and each force between two bodies is along the line between them, you can prove the sum of all bodies' angular momenta will be conserved (though individual angular momenta in general are not).

• Presenting an alternative theory says nothing about my work. I am well aware that there are alternative theories and derivations, my view is that there must be errors in those theories because they do not agree with my findings. Please address my argument? – John Jul 2 '17 at 11:07
• @John I think you could benefit from learning more about how polar coordinates describe planar orbits, viz. ucl.ac.uk/~ucahad0/1302week3Polar08.pdf You can then understand the implications for your premises (which weren't there when I wrote my answer), especially premise 4. The momentum perpendicular to $\vec{r}$ is $\vec{p}-(\vec{p}\cdot\hat{\vec{r}})\hat{\vec{r}}$, but the time-dependence of the unit vector $\hat{\vec{r}}$ complicates matters. – J.G. Jul 2 '17 at 11:39
• Presenting an alternative theory and making a reference to my argument and stating that something might complicate matters is not addressing my argument. Premise 4 is merely Newtons first law. Are you suggesting that there is a problem with it ? – John Jul 2 '17 at 13:19
• @John I'm saying that the vector against which you're resolving parallel and perpendicular components is itself time-dependent, and you need to work through the calculus properly instead of doing everything with words. – J.G. Jul 2 '17 at 13:34
• @John The problem is that $\vec{r}\cdot\vec{p}_\perp=0$ doesn't imply $\vec{r}\cdot\frac{d}{dt}\vec{p}_\perp=0$ for general time-dependent $\vec{r}$. So resolving momenta into components doesn't properly map to how the force is resolved. – J.G. Jul 2 '17 at 15:44

We have to be careful when discussing the conservation of linear and angular momentum. Both Linear and angular momentum are always conserved.Changing a variable only implies changing a system, work must be done to do, so you must incorporate this work into the system to maintain the conservation as per computation.Conservation of both linear and angular momentum encompasses all forces acting on the system, that it is true for any system.So if, you change a variable in a system of-course, you will have to assume the conservation for your new system in the light of the changed variable.By changing a variable,in this case radius, you no longer have a new isolated system or perhaps the kinematics of your isolated system has changed.Conservation has nothing to do with computational values, you can have different values for a conserved quantity in different systems.You know researchers usually say that, conservation of angular momentum is the reason that the angular velocity of individuals involved in figure skating increases, as they fold up(changing moment of inertial). Infact the value of angular momentum should change,when you consider that you do work on the system by changing your moment of inertia.

• Comments are not for extended discussion; this conversation has been moved to chat. – ACuriousMind Jul 4 '17 at 20:56