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Welcome to the presentation on moments so just if you were wondering I have already covered moments you just may not have recognized it because I covered it in mechanical advantage in torque but I do realize that when I covered it in mechanical advantage in torque I think I may be over complicated it and if anything I didn't cover some of the most basic moment of force problems that you see in.
The torque on a current-carrying coil, as in a DC motor, can be related to the characteristics of the coil by the 'magnetic moment' or 'magnetic dipole moment'. The torque exerted by the magnetic force (including both sides of the coil) is given by
The coil characteristics can be grouped as called the magnetic moment of the loop, and the torque written as The direction of the magnetic moment is perpendicular to the current loop in the right-hand-rule direction, the direction of the normal to the loop in the illustration. Considering torque as a vector quantity, this can be written as the vector product |
Since this torque acts perpendicular to the magnetic moment, then it can cause the magnetic moment to precess around the magnetic field at a characteristic frequency called the Larmor frequency.
If you exerted the necessary torque to overcome the magnetic torque and rotate the loop from angle zero to 180 degrees, you would do an amount of rotational work given by the integral
The position where the magnetic moment is opposite to the magnetic field is said to have a higher magnetic potential energy.
Momentum
Since the moment of inertia of an ordinary object involves a continuous distribution of mass at a continually varying distance from any rotation axis, the calculation of moments of inertia generally involves calculus, the discipline of mathematics which can handle such continuous variables. Since the moment of inertia of a point mass is defined by
then the moment of inertia contribution by an infinitesmal mass element dm has the same form. This kind of mass element is called a differential element of mass and its moment of inertia is given by
Note that the differential element of moment of inertia dI must always be defined with respect to a specific rotation axis. The sum over all these mass elements is called an integral over the mass.
Usually, the mass element dm will be expressed in terms of the geometry of the object, so that the integration can be carried out over the object as a whole (for example, over a long uniform rod).
Funny Moments
Having called this a general form, it is probably appropriate to point out that it is a general form only for axes which may be called 'principal axes', a term which includes all axes of symmetry of objects. The concept of moment of inertia for general objects about arbitrary axes is a much more complicated subject. The moment of inertia in such cases takes the form of a mathematical tensor quantity which requires nine components to completely define it.
Moment Chicago
Memento
| Examples of integration to get moment of inertia. | Straight rod |
| Cylinder | |
| Sphere |