# Do rotating bodies have angular momentum

### 7.3 The angular momentum - equation of motion of a rigid body

The angular momentum of a discrete rigid body is related to a body-fixed center of gravityAs general

holds, follows immediately

If you write this expression for any component (e.g. ), we get

or in vector notation

We know from linear algebra that the vectors and generally not rectified when the matrix is not diagonal (that being diagonal is not enough for that . It must be additional parallel to one of the main axes (see 7.4) (see also example B7.7). This means that the angular momentum of a rotating body will generally point in a different direction than the axis of rotation!

**B7.1 Rotating dumbbell**

For the speeds of the two masses applies

and for the angular momentum

stands up vertically and . Here but not perpendicular to stands, must and have an angle to each other. On top of that, is not constant over time, but is on a cone around the axis of rotation with the angular velocity emotional. Here but depends on the time is . We know from the theorem of angular momentum that a change in angular momentum over time can only be caused by a torque

On the other hand, it holds for an inertia tensor that is constant over time

so that we get an equation of motion for the rotating body (based on a body-fixed center of gravity)

(Note the analogy to Newton's equation of motion .) There are no external torques acting and thus constant (provided that is constant over time).

*M. Keim, H.J. Ludde*

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