The kinetic energy for a particle is given by the following scalar equation:

Where:

For a rigid body experiencing planar (two-dimensional) motion, the kinetic energy is given by the following general scalar equation:

Where:

The figure below illustrates the general case where a rigid body is experiencing planar (two-dimensional) motion.

Note in equation (2) that the kinetic energy of the rigid body consists of two parts: The kinetic energy due to the velocity of the center of mass, and the kinetic energy due to rotation.

If the rigid body is rotating about a fixed point

Where:

The figure below illustrates the case where a rigid body is experiencing planar (two-dimensional) motion, and is rotating about a fixed point

For general three-dimensional motion, the kinetic energy of a rigid body is given by the following general scalar equation:

Where:

The moments of inertia

The moment of inertia terms are given by

The product of inertia terms are given by

Note that for equation (4) the local

Also note in equation (4) that the kinetic energy of the rigid body consists of two parts: The kinetic energy due to the velocity of the center of mass, and the kinetic energy due to rotation.

If the rigid body has a fixed point

The variables are the same here as for equation (4). The only difference is that the local

Note that in equations (2)-(5), if we assume that the rigid body has negligible dimensions (where the inertia terms are close to zero), then the equations reduce to the kinetic energy equation for a particle — equation (1).

The scalar equations (2)-(5) can be used when solving problems involving energy calculations. For planar (two-dimensional) motion use equation (2) or (3). For general three-dimensional motion use equation (4) or (5).

The equations for planar (two-dimensional) motion follow naturally from the equations for general three-dimensional motion, so it is only necessary to derive the kinetic energy equations for three-dimensional motion.

To derive the previous expressions for the kinetic energy of a rigid body we must apply the equation for kinetic energy of a small mass element in the rigid body, and then sum it over the entire rigid body. This problem is set up as shown in the figure below.

For a small mass element

Where:

Now, from the equations for general motion of a rigid body:

Where:

Substituting equation (7) into equation (6) we get

It remains to carry out the vector dot product followed by vector cross product multiplications and then sum

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