rigid body

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A rigid body is an idealization of a solid body in which the object is not allowed to undergo deformation. All points in the body maintain their distances and angles with respect to all other points, and the density of the object is not allowed to vary with time. The dynamics of a rigid body is completely specified by by the motion of its center of mass and of the relative motion of all points in the body with respect to the center of mass. A point p\, on the body has position \mathbf{R}_p\, with respect to some global origin. Then its position relative to the center of mass \mathbf{R}_{com}\,, is

\mathbf{r}_p = \mathbf{R}_p - \mathbf{R}_{com}\,,

which can be rewritten as

\mathbf{R}_p = \mathbf{R}_{com} + \mathbf{r}_p\,.

Taking the time derivative,

\mathbf{V}_p = \mathbf{V}_{com} + \mathbf{v}_p\,,

where \mathbf{V}_{com}\, and \mathbf{v}_p\, are the velocities of the center of mass and of the point p\, relative to the center of mass, respectively.

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Momentum

The total momentum is of a rigid body is equal to the sum of the momenta of all of its components.

\mathbf{P}\, = \sum_{i=1}^N m_i \times \mathbf{V}_i\,,
= M \mathbf{V}_{com} + \sum_{i=1}^N m_i \mathbf{v}_i\,.

Angular momentum

The angular momentum is equal to the sum of the angular momenta of all the components. Recall

\mathbf{L}\, = \sum_{i=1}^N \mathbf{R}_i \times \mathbf{P}_i\,,
= \mathbf{L}_{com} + \sum_{i=1}^N \mathbf{L}_{i, around\mbox{ }com}\,.

It is therefore one of our objectives to determine the angular momentum with respect to \mathbf{R}_{com}\,.

Kinetic energy

The kinetic energy is equal to the sum of the kinetic energies of all the components.

T\, = \sum_{i=1}^N \frac{1}{2} m_i V_i^2\,,
= \frac{1}{2} M V_{com}^2 + \sum_{i=1}^N \frac{1}{2} m_i v_i^2\,.

Euler angles

back to Rigid body dynamics
on to moment of inertia tensor
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