For two-dimensional rigid body dynamics problems, the body experiences motion in one plane, due to forces acting in that plane.

A general rigid body subjected to arbitrary forces in two dimensions is shown below.

The full set of scalar equations describing the motion of the body are:

Where:

Note that, if the rigid body were rotating about a fixed point

The figure below illustrates this situation.

Where the point

Here are some examples of problems solved using two-dimensional rigid body dynamics equations:

The Physics Of A Golf Swing

Trebuchet Physics

For three-dimensional rigid body dynamics problems, the body experiences motion in all three dimensions, due to forces acting in all three dimensions. This is the most general case for a rigid body.

A general rigid body subjected to arbitrary forces in three dimensions is shown below.

The first three of the six scalar equations describing the motion of the body are force equations. They are:

Where:

Note that the subscripts

However, it is not necessary to resolve the quantities along the

To solve three-dimensional rigid body dynamics problems it is necessary to calculate six inertia terms for the rigid body, corresponding to the extra complexity of the three dimensional system. To do this, it is necessary to define a local

For two-dimensional rigid body dynamics problems there is only one inertia term to consider, and it is

For the general case (where we have an arbitrary orientation of

Where:

The six inertia terms are evaluated as follows, using integration:

The orientation of

This orientation is defined as the principal direction of

With this simplification, the moment equations become:

These are known as the

Note that, for the three moment equations and six inertia terms, their quantities

For example, a moment acting about the Y-axis must be resolved into its components along the

Note that, if the rigid body were rotating about a fixed point

The figure below illustrates this situation.

Where the point

For two-dimensional rigid body dynamics problems the angular acceleration vector is always pointing in the same direction as the angular velocity vector. However, for three-dimensional rigid body dynamics problems these vectors might be pointing in different directions, as shown below.

These vectors can be expressed as:

In two-dimensions, to find the angular acceleration you simply differentiate the magnitude of the angular velocity, with respect to time. In three-dimensions you have to account for the change in magnitude

Using calculus, the angular acceleration is calculated as follows (taking the limit as

The three force equations, and three moment equations shown here for three-dimensional rigid body dynamics problems "fully describe"

Note that the positive directions of the individual X, Y, Z axes are in the directions shown in the first figure. Similarly, the positive directions of the individual

For additional background information see:

Parallel Axis And Parallel Plane Theorem

A Closer Look At Velocity And Acceleration

Here are some examples of problems solved using three-dimensional rigid body dynamics equations:

The Physics Of Bowling

Euler's Disk Physics

Gyroscope Physics

To gain more insight into rigid body dynamics see Mechanical Waves.

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