ON A DYNAMICAL TOP. 251 



DEFINITION. The Angular Momentum of a particle about an axis is mea- 

 sured by the product of the mass of the particle, its velocity resolved in the normal 

 plane, and the perpendicular from the axis on the direction of motion. 



* The angular momentum of any system about an axis is the algebraical 

 sum of the angular momenta of its parts. 



As the rate of change of the linear momentum of a particle measures the 

 moving force which acts on it, so the rate of change of angular momentum 

 measures the moment of that force about an axis. 



All actions between the parts of a system, being pairs of equal and opposite 

 forces, produce equal and opposite changes in the angular momentum of those 

 parts. Hence the whole angular momentum of the system is not affected by 

 these actions and re-actions. / 



* When a system of invariable form revolves about an axis, the angular 

 velocity of every part is the same, and the angular momentum about the axis is 

 the product of the angular velocity and the moment of inertia about that axis. 



* It is only in particular cases, however, that the whole angular momentum 

 can be estimated in this way. In general, the axis of angular momentum differs 

 from the axis of rotation, so that there will be a residual angular momentum 

 about an axis perpendicular to that of rotation, unless that axis has one of three 

 positions, called the principal axes of the body. 



By referring everything to these three axes, the theory is greatly simplified. 

 The moment of inertia about one of these axes is greater than that about any 

 other axis through the same point, and that about one of the others is a mini- 

 mum. These two are at right angles, and the third axis is perpendicular to 

 their plane, and is called the mean axis. 



* Let A, B, C be the moments of inertia about the principal axes through 

 the centre of gravity, taken in order of magnitude, and let ^ w 2 &> 3 be the 

 angular velocities about them, then the angular momenta will be A(o lt Bu 

 and C<a 3 . 



Angular momenta may be compounded like forces or velocities, by the 

 law of the " parallelogram," and since these three are at right angles to each 

 other, their resultant is 



+ CV = H (1), . ,, 



and this must be constant, both in magnitude and direction in space, since no 

 external forces act on the body. 



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