Law of Conservation of Orbital Angular Momentum

"Give me a place to stand on, and I will move the Earth" - Archimedes ( Ἀρχιμήδης, circa 340 AD; Hellenist Greek mathematician of antiquity )

§ The Problem: By what sort of mathematical physics can the following phenomena of nature shown in this video be explained?

source: Jet Propulsion Lab video

§ Definitions:

(i). Radian measurement of angle:

In radian measurement for angle, distance along a circle arc substitutes for an equivalent amount of angle degrees. This then becomes a question of how many "2 π s" are there along a circle's circumference which in any event will complete a circle comprising 360°? That is,

C = 2 π r, traditional circumference equation from Classical Greeks^∗

C = 2π⋅radians, newer radian circumference equation.

^∗note: π or Greek pi comes from the first letter of the classical Greek περιΦερεια 'periphery' and Περιμετρος 'perimeter' or circumference.

In both cases for either 'r ', radius, or radians, the interpretation still is how many "2π - radii" can be laid around a circle's circumference in order to complete 360° degrees.

The definition of radian measure is therefore the unique angle θ which cuts an arc of length S equal to radius r as this following diagram demonstrates:

And, in all cases, a complete circle is divided into 360° degrees so that

The important reason for using the unique radian angle θ instead of the more intuitive ' ° ' degree is because  radian angle θ is much more natural in solving calculus limits such as

(ii). Angular velocity:

(iii). Determining a plane in space:

Given two non - zero, non - parallel vectors

they in turn define a plane as the vector cross product  of two vectors in space as follows:

where is a normal  unit vector perpendicular to the plane and because is a scalar multiple of , hence  is perpendicular to both and and   therefore defines a plane.

note: gives vector direction and gives magnitude of vector .

Determinant formula for :

Given vectors and

,

the distributive vector laws for unit vectors gives us

, magnitude of 'cross' :

is both the magnitude of 'cross' as well as the area of a parallelogram described by vectors and .

(iv). Torque:

Any force applied at a distance away from a pivot axis point is called a torque or torque force.

Example of torque applied about a pivot axis point:

note: if we increase the length of the bar to, say, 30 feet, then the magnitude of torque force becomes 649.50 ft-lbs which is considerably greater given the extension of length by a factor of '6'. Archimedes ( Ἀρχιμήδης, circa 340 AD; one of the most important of the Hellenist Greek mathematicians of antiquity  ) is famously quoted by Pappus of Alexandria as having said "Give me a place to stand on, and I will move the Earth" which is an historical statement of the mathematics of torque and hence the power of leverage!

Example of net - zero torque about a pivot axis point:

note: although we imagine in our mind's eye that a nearby force such as   and   are directed strictly downwards, it is absolutely true that all forces follow gravitational fields and in the case of earth's gravity pull, all nearby earth - forces follow an [ indiscernible ] "orbital curvature" towards earth's center.

(v). System Torque:

We will first give an example or equivalently an "operational definition" in this following diagram:

Each mass exhibits a downward earth gravitational acceleration, , against a pivot axis at point P whose magnitude of torque force is directly determined by the respective distance of away from P.

Therefore, some is given by

which is analogous to linear force

.

For the orbital rotations of planetary bodies or even sub - atomic particles ( waves ),

we therefore define system torque as follows:

note: 'torque' derives from the Latin torquere, to twist.

(vi). Moment:

^∗∗note: in order to accommodate a greater variety of orbital shapes and body densities, we define

(vii). System equilibrium:

For example, in order to balance all of the torque forces so as to reach equilibrium, we have to move the pivot point P as follows:

§ Rotational Kinematics:

Notice that what has already been described up until now are rotational forces about a pivot axis point as opposed to linear, straight - line forces due to acceleration of masses.

Therefore, the angular change of position for a given mass object in rotational motion can be diagrammatically shown as follows:

(i). Derive:

Also,

Notice that  '' and ''  differ only by   ,  a directional radius vector.

Remember also that centripetal acceleration of a body ( acceleration at right angles to the body's motion - i.e., radial or center - directed ) is 

which is radial [ linear ] or centripetal acceleration in terms of angular velocity.

(ii). Another view of rotational dynamics:

So the magnitude of torque about the pivot axis point O is

or simply

,

where torque vector is perpendicular to the the x - y plane given by force vector and radial vector .

 Another way to mathematically write the normal ( Latin: "normalis" or perpendicular ) to the plane given by and   is

,

where

as well as the area of a parallelogram described by   and

(iii). A deeper mathematical understanding of torque:

We also know from the earlier definition of

that for

we get

Notice that

which entirely comports with

^∗∗∗note: torque is also called "moment of force". See definitions above for 'moment' and 'torque'.

§ Rotational Kinetic Energy:

(i). The amount of work performed thru a small change is 

From this we can easily discern the z - axis torque component of force!

Likewise,

for respective torque components of force for work performed thru the same change.

(ii).  Newer definition for system equilibrium:

That is, system equilibrium occurs whenever

The consequences for work performed or, equivalently, energy expended when the system is in equilibrium is expressed as

and therefore no work or expended energy.

(iii).  Total kinetic energy of system:

Suppose some point mass is moving about some axis with a radial distance

where each  possesses a radial distance

to the given axis of rotation.

Now the total mass, , simply consists of all the little pieces of mass , or

and the total kinetic energy of the entire rotating mass is simply the sum of all the little kinetic energies of each of the little pieces of mass; hence,

,

where each point - mass has a tangential linear velocity perpendicular to    with angular velocity    and therefore  .

Because the entire rotating mass is now being observed in the aggregate,   angular velocity is the same for all point - masses and is therefore a constant as follows:

If we let

and then call "moment of inertia" analogous to

thus, we arrive at a deeper understanding of rotating bodies in motion.

That is, whereas total mass, , itself is constant for , "relativistically speaking", becomes a "rotational mass" analogous to the traditional concept of mass during linear translational motion. Thus when speaking of rotating bodies, velocity is replaced by angular velocity and traditional mass in linear motion is replaced by moment  of inertia analogous to mass.

Moreover, moment of inertia depends not only on traditional mass but equally upon how far away that mass is from a given axis of rotation !

And, hence, total rotational kinetic energy can now be understood as

analogous to

in linear translational motion.

(iv).  Components of :

The axial components of

can easily be obtained about each x - , y - , and z - axis as follows: Suppose there is a rotating planar figure in the x - y plane perpendicular to the z - axis, therefore

That is, the moment of inertia about the z - axis is equal to the sums of the separate moments of inertia about the individual x - and y - axes! The Perpendicular - Axis Theorem is thus

,

where each   is the 2 - dimensional distance squared, even for a 3 - dimensional object.

[ note: click here to see examples of moments of inertia for several types of bodies ]

§ Angular Momentum:

The analogue of torque

is angular momentum

where

What the analogue of torque shows is that

Also,

Here also is a vector derivation for time change of angular momentum in terms of torque force:

(i). Notice several things are occurring here:

 Angular momentum again depends upon distance away from an axis of rotation;

Angular momentum is given in terms of linear momentum which is solely a function of tangential velocity, therefore only the tangential component of the angular momentum vector to the radial axis is taken into consideration. This is precisely why

since sin 90° = 1.

  Any given component of angular momentum can also be understood as

By Newton's 3rd Law of Motion, all internal torque forces are equal and opposite and so cancel each other out; therefore what remains to be observed for the overall system is that the system responds only to external [ unbalancing ] torque forces which in turn can change the system's angular momentum.

The entire physical mathematics for torque, moment of inertia and angular momentum is entirely predicated upon the ideas of homogeneous and isotropic space - time which in turn are mathematically based upon Emmy Noether's Theorem, namely that the physical laws for energy conservation are fundamentally derived from  the proposition that they are invariant with respect to time and therefore can be derived for any space - time continuum.

Finally, a simple proof for the law of the conservation of angular momentum in a closed system is now easily derived:

implies that without any additional external torque forces acting upon a closed system of particles or orbiting bodies about a central axis,

That is, if any system experiences no external unbalancing torque forces, then also the time rate of change of angular momentum is also zero which immediately means that the

angular momentum itself is unchanging or constant!

(ii).  Several examples of the law of conservation of angular momentum:

That is, angular velocity    must counterbalance moment of inertia in order to conserve angular momentum whenever there is no additional torque applied to the rotating system.

Nevertheless, Britney does speed up as arms are drawn inward and so what occurs regarding kinetic energy,

,

is something still to be examined. In other words, what does balance, or rather is conserved, is , not  !

So, here's how it's analyzed:

where

but obviously

produces

Question: from where, therefore, does the extra energy come? The answer is that extra work and hence an expenditure of extra energy occurs by drawing in the arms and thereby countervailing the energy contained in the outward centrifugal forces of rotating masses - i.e., the outstretched arms.

As a planet goes around the sun, the only meaningful force upon the planet is that of the inward force of the sun's gravitational pull. Torque therefore becomes

In other words, the angular momentum of a planet going around the sun is constant since there is no torque about an axis at the sun.

Now according to Kepler's 1st Law ( Planetary Law of Ellipses ), all planetary orbits are ellipses with the Sun at one of the two foci with nearer positions ( the perihelion ) and farther positions ( the aphelion ) from the center of mass, the sun.

Hence,

We also know, for example, that earth possesses these following values:

Some gas - dust - rock, spherical planet defines its "day" ( 1.0 rev ) as a full 360° rotation about its axis.^∗∗∗∗

Now, as the gas - dust - rock sphere contracts inward to form a more consolidated planet by, say 1%, therefore by how much does its "day" lengthen? Shorten? Stays the same?

But,

which means that a 1% contraction of the initial gas - dust - rock spherical planet will result in a 2% decrease in the "time length" ( or time duration ) of a planet's "day"!!

This again completely comports with Britney's faster angular velocity when this marvelous young ice skater pulls in her arms. In fact, if Britney were a rotating clock approaching the speed of light, c, her "time" would slow down ( or time dilate ) compared to a relatively stationary clock!

^∗∗∗∗

note: this example is the one used in the future upcoming Relativity Science Calculator Mac application with the exception of substituting an imploding gaseous hydrogen star for the hypothetically gas - dust - rock planet posited here.

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