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law of conservation of angular momentum


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 = 2πradians, 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:

law of conservation of angular momentum

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

law of conservation of angular momentum

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

law of conservation of angular momentum

(ii). Angular velocity:

law of conservation of angular momentum

(iii). Determining a plane in space:

law of conservation of angular momentum Given two non - zero, non - parallel vectors

law of conservation of angular momentum

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

law of conservation of angular momentum

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

law of conservation of angular momentum

note: law of conservation of angular momentum gives vector direction and law of conservation of angular momentum gives magnitude of vector law of conservation of angular momentum.

law of conservation of angular momentum Determinant formula for law of conservation of angular momentum:

Given vectors law of conservation of angular momentum and law of conservation of angular momentum

law of conservation of angular momentum,

the distributive vector laws for unit vectors law of conservation of angular momentum gives us

law of conservation of angular momentum

law of conservation of angular momentum law of conservation of angular momentum, magnitude of law of conservation of angular momentum 'cross' law of conservation of angular momentum:

law of conservation of angular momentum

law of conservation of angular momentum

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

(iv). Torque:

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

law of conservation of angular momentum Example of torque applied about a pivot axis point:

law of conservation of angular momentum

law of conservation of angular momentum

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!

law of conservation of angular momentum Example of net - zero torque about a pivot axis point:

law of conservation of angular momentum

law of conservation of angular momentum

note: although we imagine in our mind's eye that a nearby force such as law of conservation of angular momentum  and law of conservation of angular momentum  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:

law of conservation of angular momentum

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

Therefore, some law of conservation of angular momentum is given by

law of conservation of angular momentum

which is analogous to linear force

law of conservation of angular momentum.

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

law of conservation of angular momentum

we therefore define system torque as follows:

law of conservation of angular momentum

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

(vi). Moment:

law of conservation of angular momentum

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

law of conservation of angular momentum

(vii). System equilibrium:

law of conservation of angular momentum

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:

law of conservation of angular momentum

law of conservation of angular momentum

§ 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:

law of conservation of angular momentum

(i). Derive:

law of conservation of angular momentum

law of conservation of angular momentum

law of conservation of angular momentum Also,

law of conservation of angular momentum

Notice that  'law of conservation of angular momentum' and 'law of conservation of angular momentum'  differ only by  law of conservation of angular momentum ,  a directional radius vector.

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

law of conservation of angular momentum

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

(ii). Another view of rotational dynamics:

law of conservation of angular momentum

law of conservation of angular momentum

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

law of conservation of angular momentum

or simply

law of conservation of angular momentum,

where torque vector law of conservation of angular momentum is perpendicular to the the x - y plane given by force vector law of conservation of angular momentum and radial vector law of conservation of angular momentum.

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

law of conservation of angular momentum,

where

law of conservation of angular momentum

as well as the area of a parallelogram described by  law of conservation of angular momentum and law of conservation of angular momentum

(iii). A deeper mathematical understanding of torque:

We also know from the earlier definition of

law of conservation of angular momentum

that for

law of conservation of angular momentum

we get

law of conservation of angular momentum

Notice that

law of conservation of angular momentum

which entirely comports with

law of conservation of angular momentum

∗∗∗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 delta_theta.png change is 

law of conservation of angular momentum

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

Likewise,

law of conservation of angular momentum

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

(ii).  Newer definition for system equilibrium:

law of conservation of angular momentum

That is, system equilibrium occurs whenever

law of conservation of angular momentum

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

law of conservation of angular momentum

and therefore no work or expended energy.

(iii).  Total kinetic energy of system:

Suppose some point mass law of conservation of angular momentum is moving about some axis with a radial distance

law of conservation of angular momentum

where each  law of conservation of angular momentum possesses a radial distance

law of conservation of angular momentum

to the given axis of rotation.

Now the total mass, law of conservation of angular momentum, simply consists of all the little pieces of mass law of conservation of angular momentum, or

law of conservation of angular momentum

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,

law of conservation of angular momentum,

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

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

law of conservation of angular momentum

If we let

law of conservation of angular momentum

and then call law of conservation of angular momentum "moment of inertia" analogous to

law of conservation of angular momentum

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

That is, whereas total mass, law of conservation of angular momentum, itself is constant for law of conservation of angular momentum, "relativistically speaking", law of conservation of angular momentum 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 law of conservation of angular momentum depends not only on traditional mass law of conservation of angular momentum 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

law of conservation of angular momentum

analogous to

law of conservation of angular momentum

in linear translational motion.

(iv).  Components of law of conservation of angular momentum:

The axial components of

law of conservation of angular momentum

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

law of conservation of angular momentum

That is, the moment of inertia law of conservation of angular momentum 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

law of conservation of angular momentum,

where each  law of conservation of angular momentum  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

law of conservation of angular momentum

is angular momentum

law of conservation of angular momentum

where

law of conservation of angular momentum

What the analogue of torque law of conservation of angular momentumshows is that

law of conservation of angular momentum

Also,

law of conservation of angular momentum

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

law of conservation of angular momentum

(i). Notice several things are occurring here:

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


law of conservation of angular momentum 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

law of conservation of angular momentum

since sin 90° = 1.

law of conservation of angular momentum  Any given component law of conservation of angular momentum of angular momentum can also be understood as

law of conservation of angular momentum

law of conservation of angular momentum 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.

law of conservation of angular momentumThe 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.

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

law of conservation of angular momentum

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

law of conservation of angular momentum

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 law of conservation of angular momentum itself is unchanging or constant!

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

law of conservation of angular momentum

law of conservation of angular momentum

That is, angular velocity  law of conservation of angular momentum  must counterbalance moment of inertia law of conservation of angular momentum in order to conserve angular momentum law of conservation of 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,

law of conservation of angular momentum ,

is something still to be examined. In other words, what does balance, or rather is conserved, is law of conservation of angular momentum, not  law of conservation of angular momentum!

So, here's how it's analyzed:

law of conservation of angular momentum

where

law of conservation of angular momentum

but obviously

law of conservation of angular momentum

produces

law of conservation of angular momentum

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.

law of conservation of angular momentum 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

law of conservation of angular momentum

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,

law of conservation of angular momentum

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

law of conservation of angular momentum

law of conservation of angular momentum

law of conservation of angular momentum 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?

law of conservation of angular momentummoment of inertial for a spherical planet

But,

law of conservation of angular momentum

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.


law of conservation of angular momentum

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