7.3 SPATIAL DYNAMICS

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The basic principles of planar dynamics are expanded up for 3D spatial problems. The added dimension adds some complexity that should be addressed.

7.3.1 Moments of Inertia About Arbitrary Axes

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Moments of Inertia are normally found for a single axis of rotation. When the object is rotating about another axis, we must recalculate the moments of inertia.

If we take the moments of inertia for the original axes, and project these values onto new vectors, we can get new values,

7.3.2 Euler's Equations of Motion

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We can use Euler's equations of motion to determine moments produced by angular velocities and accelerations.

These can be used to examine rotating three dimensional masses. Consider the following,

7.3.3 Impulses and Momentum

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Momentum is a convenient alternative to energy in analysis of systems.

7.3.3.1 - Linear Momentum

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momentum is defined as,

If no external forces are applied, momentum remains constant (is conserved). In this case L is a constant.

An impulse is a force applied that will change momentum.

7.3.3.2 - Angular Momentum

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Angular momentum is for rotating objects. The rotation about some center tends to make these equations a bit more complicated than linear momentum.

We can start to find this as a velocity times a distance of rotation, and this will lead to the eventual relationships,

These equations show the angular momentum H, along with other familiar terms.