König's Theorem (kinetics)

In kinetics, König's theorem or König's decomposition is a mathematical relation derived by Johann Samuel König that assists with the calculations of angular momentum and kinetic energy of bodies and systems of particles.

For a system of particles

The theorem is divided in two parts.

First part of König's theorem

The first part expresses the angular momentum of a system as the sum of the angular momentum of the centre of mass and the angular momentum applied to the particles relative to the center of mass.

$\displaystyle \vec = \vec_ \times \sum\limits_ m_ \vec_ + \vec'= \vec_ + \vec'$

Proof

Considering an inertial reference frame with origin O, the angular momentum of the system can be defined as:

$\vec = \sum\limits_ (\vec_ \times m_ \vec_ )$

The position of a single particle can be expressed as:

$\vec_ = \vec_ + \vec'_$

And so we can define the velocity of a single particle:

$\vec_ = \vec_ + \vec'_$

The first equation becomes:

:$\vec = \sum\limits_ (\vec_ + \vec'_) \times m_ (\vec_ + \vec'_)$

:$\vec = \sum\limits_ \vec'_ \times m_ \vec'_ + \left( \sum\limits_ m_\vec'_\right) \times \vec_ + \vec_ \times \sum\limits_ m_ \vec'_ + \sum\limits_ \vec_ \times m_ \vec_$

But the following terms are equal to zero:

$\sum\limits_ m_ \vec'_ = 0$

$\sum\limits_ m_ \vec'_ = 0$

So we prove that:

$\vec = \sum\limits_ \vec'_ \times m_ \vec'_+M \vec_ \times \vec_$

where M is the total mass of the system.

Second part of König's theorem

The second part expresses the kinetic energy of a system of particles in terms of the velocities of the individual particles and the centre of mass.

Specifically, it states that the kinetic energy of a system of particles is the sum of the kinetic energy associated to the movement of the center of mass and the kinetic energy associated to the movement of the particles relative to the center of mass.

$K = K' + K_$

Proof

The total kinetic energy of the system is:

$K = \sum_i \frac m_i v_i^2$

Like we did in the first part, we substitute the velocity:

:$K = \sum_i \frac m_i , \bar v'_i + \bar v_\text, ^2$

:$K = \sum_i \frac m_i (\bar v'_i + \bar v_\text)\cdot(\bar v'_i + \bar v_\text) = \sum_i \frac m_i ^2 + \bar v_\text \cdot \sum_i m_i \bar v'_i + \sum_i \frac m_i v_\text^2$

We know that $\bar v_\cdot\sum_im_i \bar v'_i = 0,$ so if we define:

$K' = \sum_i \frac m_i ^2$

$K_\text = \sum_i \frac m_i v_\text^2 = \frac 12 M v_\text^2$

we're left with:

$K = K' + K_\text$

For a rigid body

The theorem can also be applied to rigid bodies, stating that the kinetic energy K of a rigid body, as viewed by an observer fixed in some inertial reference frame N, can be written as:

$^K = \frac m \cdot \cdot + \frac \cdot ^^R$

where $$ is the mass of the rigid body; $$ is the velocity of the center of mass of the rigid body, as viewed by an observer fixed in an inertial frame N; $$ is the angular momentum of the rigid body about the center of mass, also taken in the inertial frame N; and $^^R$ is the angular velocity of the rigid body R relative to the inertial frame N.

References

* Hanno Essén:
Average Angular Velocity
' (1992), Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden.
* Samuel König (Sam. Koenigio): ''De universali principio æquilibrii & motus, in vi viva reperto, deque nexu inter vim vivam & actionem, utriusque minimo, dissertatio''

.
* Paul A. Tipler and Gene Mosca (2003), Physics for Scientists and Engineers (Paper): Volume 1A: Mechanics (Physics for Scientists and Engineers), W. H. Freeman Ed.,

Works Cited

Mechanics
Physics theorems

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