Angular momentum

In Physical, the angular momentum or kinetic moment is the physical size which in the case of plays a part similar to the Quantité of movement rotations. As the angular momentum depends on the choice of the origin (as of the reference frame of study (R) ) it is always necessary to specify this origin and to never combine angular momenta having different origins.

Case of a material point

One calls Point material or specific Corps a mechanical system whose dimensions are small in front of the distances characteristic of the studied movement (distance covered, ray of an orbit…). The mechanical system is then modelled by a geometrical point M with which its mass m is associated.

Definition

For a material point M of vector position $\ vec {R} = \ vec {OM}$ the kinetic or angular moment $\ vec {L_ {O}}$ compared to the origin O is defined by:

\ vec {L_ {O}} = \ vec {OM} \ wedge \ vec {p} = \ vec {R} \ wedge \ vec {p}

, (1)

where $\ vec {p} =m \ vec {v}$ is the Quantité of movement of the particle. The kinetic moment is thus the Moment of the latter compared to O . $\ wedge$ is the vector operator Product.

A simple example is that of a particle describing a circle of center O and R: $\ vec {L_ {O}}$ is directed according to the axis of the disc and is worth $\ vec {L_ {O}} = \ vec {K} \ cdot mvr$ . The direction $\ vec {K}$ of the kinetic vector moment does not recover physical reality but of a convention, it is a axial Vecteur.

By analogy with the momentum, the kinetic moment makes it possible to define the analog of the mass: the Moment of inertia $I$ . Indeed: $\ vec {L_ {O}} = \ vec {R} \ wedge \ vec {p} =m \ vec {R} \ wedge \ vec {v} = mr^ {2} \ dowry {\ theta} \ vec {K} = I \ dowry {\ theta} \ vec {K}$ , where $\ dowry {\ theta}$ is the angular velocity of the point M, to which one can make correspond the axial Vecteur $\ vec= \ dowry {\ theta} \ vec {K}$ . The kinetic moment is written finally:

\ vec {L_ {O}} = I \ vec {\ dowry {\ theta}}

Theorem of the kinetic moment for a material point

If one derives member with member the définion (1) from the angular momentum, it comes, by supposing O fixes in (R) : $\ frac {\ vec {dL_ {O}}} {dt} = \ frac {\ vec {Dr.}} {dt} \ wedge \ vec {p} + \ vec {R} \ wedge \ frac {\ vec {dp}} {dt} = \ vec {R} \ wedge \ frac {\ vec {dp}} {dt}$ , since $\ frac {\ vec {Dr.}} {dt}$ and $\ vec {p} =m \ vec {v}$ is colinéaires.

In addition for a specific body, one has (fundamental relation of dynamics):

$\ frac {\ vec {dp}} {dt} = \ sum_ {I} \ vec {F_ {I}}$ , (2), the term of right-hand side corresponding to the sum of the forces $\ vec {F_ {I}}$ (real or " of Inertia ") exerted on the body.

Consequently it comes the equation following, known as theorem of the kinetic moment :

\ frac {\ vec {dL_ {O}}} {dt} = \ vec {R} \ wedge \ sum_ {I} \ vec {F_ {I}} = \ sum_ {I} \ vec {\ mathcal {M} _ {O}} \ left (\ vec {F_ {I}} \ right)

, (3)

where $\ vec {\ mathcal {M} _ {O}} \ left (\ vec {F_ {I}} \ right) = \ vec {R} \ wedge \ vec {F_ {I}}$ is the Moment force $\ vec {F_ {I}}$ compared to the point O .

Note:: compared to a mobile point O in (R) , the theorem of the kinetic moment is written: $\ frac {\ vec {dL_ {O}}} {dt} + \ vec {v_ {O}} \ wedge \ vec {p} = \ sum_ {I} \ vec {\ mathcal {M} _ {O}} \ left (\ vec {F_ {I}} \ right)$ .
La only difference comes from the addition 'a complementary term $\ vec {v_ {O}} \ wedge \ vec {p}$ in the member in left from the relation (3).

Examples of application

Movement with central Force: general case

A very important particular case of use of the kinetic moment is that of the movement to central Force , where the material point M is subjected to only one force

\ vec {F}

whose direction passes by a point fixes in (R) , called center of force. Consequently by taking this center of force for origin O , the theorem of the kinetic moment (3) implies that the kinetic moment

\ vec {L_ {O}}

is a Intégrale first movement:

\ frac {\ vec {dL_ {O}}} {dt} = \ vec {0}

, is

\ vec {L_ {O}} = \ vec {R} \ wedge \ vec {p} = \ vec {cte}

, since

\ vec {OM}

and

\ vec {F}

is colinéaires.

Consequently the vector position $\ vec {R}$ and the Quantité of movement $\ vec {p}$ of the body are at any moment perpendicular to a vector of constant direction: the trajectory is thus plane , entirely contained in the plan perpendicular to $\ vec {L_ {O}} = \ vec {r_ {0}} \ wedge \ vec {p_ {0}}$ (the index " 0" indicate the initial values of the sizes).

The movement comprising only two Degrees of freedom one places in polar Coordonnées (R, θ) in the plan of the Trajectoire. It comes as follows:

$\ vec {L_ {O}} =L \ vec {e_ {Z}}$ , with $L \ equiv mr^ {2} \ constant dowry {\ theta}$ .

Taking into account $v^ {2} = \ dowry {R} ^ {2} +r^ {2} \ dowry {\ theta} ^ {2}$ in polar coordinates, the kinetic energy of the material point can separate in a radial part and an angular part. She is written then $E_ {K} = \ frac {1} {2} m \ dowry {R} ^ {2} + \ frac {L^ {2}} {2mr^ {2}}$ .

Movement with central Force: case where the force derives from a potential energy

If the central force

\ vec {F}

drift of a potential energy

V (R)

, the mechanical energy of the body is put in the form:

E_ {m} = \ frac {1} {2} m \ dowry {R} ^ {2} +U_ {EFF} (R)

with

U_ {EFF} (R) \ equiv V (R) + \ frac {L^ {2}} {2mr^ {2}}

, potential energy effective .

One brings back to a unidimensional movement of a fictitious particle in a Potentiel $U_ {EFF} (R)$ . the term $\ frac {L^ {2}} {2mr^ {2}}$ being positive and crescent with short of distance, it plays the part of " barrier of Potentiel Centrifuges ".

Some additional remarks and references

Of many authors supposes that a central Force drift always of a potential energy: this is false in general. For example, for the simple Pendulum, the force of tension of the wire is a central force because it always passes by the point of fixing O of the Pendule, BUT it does not derive from a potential energy.
an important application of the preceding developments is in the study of the Keplerian Mouvement of planets and the satellites. The Trajectoire S are then closed Courbe S: ellipse S.
It should be stressed that in general the trajectories obtained for a potential energy V (R) unspecified are not not closed curves: only the Coulomb potential gravitational $V (R) = \ frac {K} {R}$ ( K constant) and the harmonic potential $V (R) = \ alpha r^ {2}$ will give some. That comes from the existence, for these potentials, of an integral first additional (for the Coulomb potential, it is about the Invariant of Runge Lenz), associated with a additional Symétrie (by transformation of the group O (4) ).

Case of a material system

Definition in the general case

If a system consists of several particles (discrete model), the total angular momentum is obtained by adding or integral the angular momentum with each one of its components. It is also possible to place within the limit of the continuous mediums to describe certain mechanical systems (solid S, in particular).

According to whether one adopts a discrete or continuous model, the kinetic moment of the system (S) compared to a point O is written:

$L_ {O} = \ sum_ {I} \ vec {OM_ {I}} \ wedge \ vec {p_ {I}}$ or $L_ {O} = \ int_ {(S)} \ vec {OM} \ wedge \ rho (M) \ vec {v_ {M}} D \ tau$

These general expressions are hardly usable directly. The theorem of Koenig relating to the kinetic moment makes it possible to give a more comprehensible form of it physically.

In the case of a system with $\ constant dowry {\ theta}$ (case of the solids in particular), one can also write:

$L_ {O} = I \ dowry {\ theta}$ where $I= \ sum_ {I} m_ {I} r_ {I} ^2$ or $I= \ int r^2 \ rho (M) D \ tau$ .

Theorem of Koenig for the moment kinetic

Case of a solid : tensor of inertia

forces motion . In exchange forces motion, two bodies form year isolated system not influenced by outside forces, and the origin is placed somewhere one the line between the two bodies. Since any force the bodies exert one each other must Be directed along this line, there edge Be No torque Net, with respect to the afore-mentioned origin, one either body. Thus, angular momentum is conserved. -->

See too

Moment (mechanical)
quantum kinetic Moment
orbital kinetic Moment and Spin in quantum mechanics
Moment of inertia
Pseudovecteur

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