Theorem of the kinetic moment

In Mechanical traditional, the theorem of the kinetic moment is a fundamental result, useful Corollaire of the Lois of the movement of Newton. It appears very practical in the study of the problems to two bodies and in Mécanique of the solid.

The theorem of the kinetic moment is in particular used in the study of the problems to central forces, because those have a null contribution, which simplifies the analysis sometimes largely: it can in this case being included/understood like a Principe of conservation. It can in addition be used to show the Lois of Kepler.

It connects two physical quantities: the kinetic Moment and the moment of a force.

Mechanics of the point

Statement

One places oneself in a Référentiel galiléen. One considers a point fixes O , origin of the selected Repère. That is to say a point M of Mass m . The kinetic Moment of the point M in this Référentiel is defined by:

$\backslash \; overrightarrow\; \{L\_O\}\; =\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{p\}\; =\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; m\; \backslash \; overrightarrow\; \{v\}$

with p the Impulse of the specific mass, v the Speed of the point M and where $\backslash \; wedge$ indicates the vector Product usual.

That is to say a force F applying to the point M . Then one defines the moment F by:

$\backslash \; overrightarrow\; \{\backslash \; mathcal\; M\_O\}\; =\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{F\}$

If the point is subjected to N forces, of moment $\backslash \; overrightarrow\; \{\backslash \; mathcal\; M\_\; \{I,\; O\}\}$ each one, then the theorem of the kinetic moment states that:

Demonstration

One shows here for reasons of clearness the theorem of the kinetic moment for only one force. The demonstration is done in the same way for several forces, by noting F the resultant of all the forces.

By carrying out the operation of derivation at the kinetic time, one a:

$\backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \{L\_O\}\}\; \{\backslash \; mathrm\; dt\}\; =\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; left\; (\backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{p\}\; \backslash \; right)\}\{\backslash \; mathrm\; dt\}\; =\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \{p\}\}\; \{\backslash \; mathrm\; dt\}\; +\; \backslash \; overrightarrow\; \{p\}\; \backslash \; wedge\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \{OM\}\}\; \{\backslash \; mathrm\; dt\}$

However, according to the basic principle of dynamics, the Dérivée from the Impulsion is equal to the sum of the force S. In addition, the Impulsion is colinéaire with the Speed, therefore the vector second Product is null. One has as follows:

$\backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \{L\_O\}\}\; \{\backslash \; mathrm\; dt\}\; =\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{F\}\; =\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; M\_\; \{O\}\}$

Use: equation of the simple pendulum

See also: simple Pendulum

One considers a Pendule simple, made up of a mass m , located by his position in polar Coordonnées ( R , θ ) and hung by a perfect inextensible wire length L at a point fixes O . The pendulum is supposed to be only subjected to the field of gravity, considered uniform and equal to G . The wire forms a Angle θ with the vertical.

One works in the Frame of reference polar ( e_r , e_θ , e_z ).

The moment of the force S other than the weight (reaction on the level of the Connection pivot and forces tension of the wire) is null. The moment exerted by the Poids at the point O is:

$\backslash \; overrightarrow\; \{\backslash \; mathcal\; M\_\; \{O\}\}\; =\; -\; mgl\; \backslash \; sin\; \backslash \; left\; (\backslash \; theta\; \backslash \; right)\; \backslash \; overrightarrow\; \{e\_z\}$

The kinetic Moment of the mass is:

$\backslash \; overrightarrow\; \{L\_O\}\; =\; L\; \backslash \; overrightarrow\; \{e\_r\}\; \backslash \; wedge\; ml\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; theta\}\; \{\backslash \; mathrm\; dt\}\; \backslash \; overrightarrow\; \{e\_\; \{\backslash \; Theta\}\}\; =\; ml^2\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; theta\}\; \{\backslash \; mathrm\; dt\}\; \backslash \; overrightarrow\; \{e\_z\}$

By deriving this relation, one has, according to the theorem of the kinetic moment:

$ml^2\; \backslash \; frac\; \{\backslash \; mathrm\; d^2\; \backslash \; theta\}\; \{\backslash \; mathrm\; dt^2\}\; \backslash \; overrightarrow\; \{e\_z\}\; =\; -\; mgl\; \backslash \; sin\; \backslash \; left\; (\backslash \; theta\; \backslash \; right)\; \backslash \; overrightarrow\; \{e\_z\}$

What one can simplify to write, by noting the Dérivée compared to the Temps by a point:

$\backslash \; ddot\; \{\backslash \; theta\}\; +\; \backslash \; frac\; \{G\}\; \{L\}\; \backslash \; sin\; \backslash \; left\; (\backslash \; theta\; \backslash \; right)\; =\; 0$

One finds well the general formula of the equation of the simple plane pendulum.

Mechanics of the solid

In Mechanical of the solid, one does not treat any more in any rigor of the material points. Nevertheless, the concept of Center of gravity locally makes it possible to compare part of the solid to a point, in order to apply the theorem of the kinetic moment to him. One can thus summon the contributions on a Volume Infinitésimal, where the theorem of the kinetic moment remains valid.

One thus poses a new definition of the kinetic Moment. That is to say a Solid S , whose each point M is animated a speed v (M) and is affected of a Density ρ (M) . Then the kinetic moment in O is:

$\backslash \; overrightarrow\; \{L\_O\}\; =\; \backslash \; int\_\; \{S\}\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; rho\; \backslash \; left\; (M\; \backslash \; right)\; \backslash \; overrightarrow\; \{v\}\; \backslash \; left\; (M\; \backslash \; right)\; \backslash \; mathrm\; D\; \backslash \; tau$

One can check that for an object with spherical and homogeneous symmetry — i.e. comparable to a specific mass — this formula is strictly equivalent to that given in Mécanique of the point.

If the solid is homogeneous and in rotation according to the fixed vector $\backslash \; overrightarrow\; \backslash \; omega$ of direction, then the expression of the kinetic moment can be simplified by the introduction of the Moment of inertia J :

$\backslash \; overrightarrow\; \{L\_O\}\; =\; J\; \backslash \; cdot\; \backslash \; overrightarrow\; \backslash \; omega$.

It then appears a remarkable formal analogy between the expression of the theorem of the kinetic moment and that of the theorem of the center of inertia which are the two principal tools of the study Dynamique in Mécanique of the solid. Indeed, the theorem of the kinetic moment is written:

$J\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \backslash \; Omega\}\; \{\backslash \; mathrm\; dt\}\; =\; \backslash \; sum\_i\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; M\_\; \{I,\; O\}\}$

Whereas the theorem of the center of inertia is written:

$m\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \{v\}\}\; \{\backslash \; mathrm\; dt\}\; =\; \backslash \; sum\_i\; \backslash \; overrightarrow\; \{F\_\; \{I,\; G\}\}$

with v the Speed, m the Mass (inertial) and F_{i, G} forces applied in its Center of gravity G — analogues respectively of the angular Velocity ω , of the Moment of inertia J and of the moments $\backslash \; mathcal\; M\_\; \{I,\; O\}$ at the point fixes O .

Mechanics of the fluids

The theorem of the kinetic moment can keep a direction in Mécanique fluids, but the phenomena of convection deprive to us of a statement similar to that of the Mécanique of the point or of the solid. It is not possible, indeed, to define in a clear way the kinetic Moment of a fluid Particule independently of its neighbors.

Nevertheless, one can consider the system studied as a whole, to carry out a Bilan total kinetic moment. One has then:

$\backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \{L\_\; \{early\}\}\}\; \{\backslash \; mathrm\; dt\}\; =\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; M\_\; \{early\}\}$

One often prefers to express such an assessment like a difference on an infinitesimal time interval D T :

$\backslash \; mathrm\; D\; \backslash \; overrightarrow\; \{L\_\; \{early\}\}\; =\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; M\_\; \{early\}\}\; \backslash \; cdot\; \backslash \; mathrm\; dt$

The principal disadvantage of this formulation is that it gives access only to the total properties of the system, which amounts regarding the unit as a problem of mechanics of the solid. One does not reach, indeed, at the times of each particle. Moreover, certain properties, like the Density, can depend on the Temps. The theorem is valid only if these variations are negligible, uniform or sufficiently slow. One gives here expressions independent of time.

The total kinetic moment is given by a formula similar to that of Mécanique of the solid:

$\backslash \; overrightarrow\; \{L\_\; \{early\}\}\; =\; \backslash \; int\_\; \{S\}\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; rho\; \backslash \; left\; (M\; \backslash \; right)\; \backslash \; overrightarrow\; \{v\}\; \backslash \; left\; (M\; \backslash \; right)\; \backslash \; mathrm\; D\; \backslash \; tau$

With the difference of the solid, the speed of a point is not related to the speed of another point: it is not possible to define the equivalent of the Moment of inertia, therefore to simplify this expression. It is also current to consider the moments voluminal forces. For a voluminal force F , one defines the moment of F on all the system by:

$\backslash \; overrightarrow\; \{\backslash \; mathcal\; M\}\; =\; \backslash \; int\_\; \{S\}\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; overrightarrow\; F\; \backslash \; left\; (M\; \backslash \; right)\; \backslash \; mathrm\; D\; \backslash \; tau$

According to the Theorem of Noether, a fluid system whose kinetic Moment total taken in its Center of gravity is preserved presents a spherical Symétrie: it is at first approximation the case of the star S or the drop S of Eau in the Vide.

The complete expression of the theorem in this form is:

$\backslash \; int\_\; \{S\}\; \backslash \; frac\; \{\backslash \; mathrm\; D\; \backslash \; left\; (\backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; rho\; \backslash \; overrightarrow\; \{v\}\; \backslash \; right)\}\{\backslash \; mathrm\; dt\}\; \backslash ,\; \backslash \; mathrm\; D\; \backslash \; tau\; =\; \backslash \; int\_\; \{S\}\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; sum\_i\; \backslash \; overrightarrow\; f\_i\; \backslash ,\; \backslash \; mathrm\; D\; \backslash \; tau$

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