The relation between angular and linear momentum

In summary, the conversation discusses the concept of angular momentum and its relationship with linear momentum. The laws of motion are presented as theorems in a modernized formulation, in which the third law only pertains to dynamics. The use of angular momentum in physics computations is discussed, but it is acknowledged that it can always be reduced to instantaneous linear momentum. The comparison is made to calculus and computer language levels in terms of efficiency and understanding.
  • #1
Cleonis
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In the image above the line ABCDE represents the trajectory of a point mass, in inertial motion. Point S is in inertial motion also. As we know, in inertial motion equal distances are covered in equal intervals of time.

All of the consecutive triangles (SAB, SBC, etc) have the same area, as they have the same base and same height. This implies that equal areas are swept out in equal intervals of time.


The image below reproduces Newton's demonstration that Kepler's law of areas follows from the laws of motion.

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A point mass is traveling along the curvilinear trajectory BCDE
Point S represents the common center of mass of two point masses that attract each other. It follows from the third law that point S is in inertial motion. The force upon the traveling point mass is at all times towards point S.

Without any force the point mass would proceed to point c. At point B the point mass receives an impulse towards S. The resulting motion is the vector sum of the original velocity and the added velocity component. This algorithm is repeated over the length of the curve.

The consecutive triangles all have the same area. In the limit of infinitisimal time increments the segments produce the curve.


I think these geometric demonstrations illustrate that angular momentum is a compound entity, reducible to elements. The elements of angular momentum are Newton's laws of motion.
 

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  • #2
Cleonis, I replied in the other thread because the OP in this post seems a little bit different from what we were talking about there. You may be correct that our disagreement is just about definitions. Anyways --

I agree with you about the interpretation of the angular momentum laws as theorems. That is a nice way of putting it. However, force laws being an axiom and angular momentum laws being theorems does not imply they are dependent, mathematically speaking. For example, you wouldn't say calculus is really dependent on the axioms of set theory. In most cases in calculus, you don't care at all about set theory, even though strictly speaking they are obviously dependent.

What I really meant was that angular momentum and linear momentum are linearly independent, in the sense that [tex] ( r \times (mv) ) \cdot (mv) = 0 [/tex].
 
  • #3
mordechai9 said:
What I really meant was that angular momentum and linear momentum are linearly independent, in the sense that [tex] ( r \times (mv) ) \cdot (mv) = 0 [/tex].

Isn't it the definitions that are linearly independent? The cross product gives us the norm to an imaginary surface where v is the actual path of a massive particle.
 
  • #4
mordechai9 said:
I agree with you about the interpretation of the angular momentum laws as theorems. That is a nice way of putting it. However, force laws being an axiom and angular momentum laws being theorems does not imply they are dependent, mathematically speaking. For example, you wouldn't say calculus is really dependent on the axioms of set theory. In most cases in calculus, you don't care at all about set theory, even though strictly speaking they are obviously dependent.

What I really meant was that angular momentum and linear momentum are linearly independent, in the sense that [tex] ( r \times (mv) ) \cdot (mv) = 0 [/tex].


OK, we agree as to the status of angular momentum as a theorem of the laws of motion.

For completeness: when I refer to laws of motion I think of a modernized formulation of them. This modernized formulation is designed to be logically equivalent with the traditional formulation, but more suited to the purpose.

1. When an object is in inertial motion it covers equal distances in equal intervals of time (This law asserts uniformity of space and time.)
2. F=m*a (to define the concept of force)
3. When two objects interact (attracting or repelling each other) then their common center of mass remains in inertial motion. (Once again asserting uniformity of space and time.)

(This version of the 3rd law is exclusively about dynamics. Newton's formulation of the 3rd law attempts so cover statics also, which is a superfluous attempt.)



(Pressed for time now, I'll post some more comment tomorrow.)
 
  • #5
mordechai9 said:
I agree with you about the interpretation of the angular momentum laws as theorems. That is a nice way of putting it. However, force laws being an axiom and angular momentum laws being theorems does not imply they are dependent, mathematically speaking. For example, you wouldn't say calculus is really dependent on the axioms of set theory. In most cases in calculus, you don't care at all about set theory, even though strictly speaking they are obviously dependent.

What I really meant was that angular momentum and linear momentum are linearly independent, in the sense that [tex] ( r \times (mv) ) \cdot (mv) = 0 [/tex].


I'd like to take a closer look at the comparison with calculus. The first post of this thread presents a diagram that shows Newton's derivation of Kepler's second law. Newton uses a precursor of differential calculus. His derivation rests on taking the limit of infinitisimal time increments. Whereas Newton had to take the limit explicitly in all his proofs, today we can expect everyone to be familiar with differential calculus, allowing more complicated problems to be addressed.

(Yet another comparison: levels of computer language. The very first computers were programmed in machine language, then assembler languages were developed, then third generation languages. Each language level is dependent on services rendered by the level below it. Using the higher level environment is much more efficient use of programmer man-hours.)

In physics computations angular momentum is used in situation where it gives an efficiency advantage. Still, if a phenomenon can be accounted for in terms of angular momentum it can also be accounted for in terms of instantaneous linear momentum of two or more interacting objects. This is always valid, since angular momentum is reducible, with the laws of motion as elements.


In physics I distinguish between what matters for efficiency advantage and what matters for understanding at gut level. What is best for computational efficiency is not necessarily suitable for gut level understanding.

In the case of gyroscopic precession the computation that uses the angular momentum of the spinning top and the vector cross product you readily get the correct prediction of the motion, but it doesn't illuminate why the spinning top tends to precess.

Interestingly, understanding why the vector cross product arises in the first place is in fact accessible. That is https://www.physicsforums.com/showpost.php?p=2697666&postcount=3" in the thread about how a rolling hoops remain upright.
 
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What is angular momentum and how does it relate to linear momentum?

Angular momentum is a measure of an object's rotational motion. It is the product of an object's moment of inertia and its angular velocity. Linear momentum, on the other hand, is a measure of an object's linear motion. The two are related by the formula L = Iω, where L is angular momentum, I is moment of inertia, and ω is angular velocity.

How is angular momentum conserved in a closed system?

In a closed system, the total angular momentum remains constant. This means that if no external torques act on the system, the initial angular momentum will be equal to the final angular momentum. This is known as the law of conservation of angular momentum.

Can an object have angular momentum without having linear momentum?

Yes, an object can have angular momentum without having linear momentum. This can occur when an object is rotating around its center of mass, but its center of mass remains stationary. In this case, the object has angular momentum due to its rotation, but no linear momentum.

How does the moment of inertia affect the angular momentum of an object?

The moment of inertia is a measure of an object's resistance to rotational motion. The larger the moment of inertia, the more difficult it is to change the object's rotational speed. As a result, a larger moment of inertia will result in a larger angular momentum for a given angular velocity.

Can angular momentum be transferred between objects in a collision?

Yes, angular momentum can be transferred between objects in a collision. This is similar to how linear momentum is transferred between objects in a collision. The total angular momentum before the collision will be equal to the total angular momentum after the collision, even if the distribution of angular momentum among the objects changes.

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