La mécanique analytique pour les étudiants de physique et génie . fois plus élevé lorsqu'il- ofelevation l'angle est de 60° que quand il est à 30°, mais porteront le Bame distance le long ahorizontal plan. 10. Une roue d'émeri éclate en petits morceaux tout en rendant 100 tions par seconde. Si le rayon de la roue est de 10 cm. trouver farthitance atteint par l'un de ses morceaux. Mouvement d'une particule SOUS L'ACTION D'UNE FORCE VARIABLE. 110. I. mouvement circulaire uniforme. - Étudier la motion particules oof projetée dans un tube circulaire, fig. 68, le innersurface perfectlysmooth de qui est. Soit m la masse de la particule, v

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Analytical mechanics for students of physics and engineering . times as high when it- angle ofelevation is 60° as when it is 30°, but will carry the Bame distance along ahorizontal plane. 10. An emery wheel bursts into small pieces while making 100lutions per second. If the radius of the wheel is 10 cm. find farthitance reached by any of its pieces. MOTION OF A PARTICLE UNDER THE ACTION OF A VARYING FORCE. 110. I. Uniform Circular Motion. — Consider the motion ofa particle projected into a circular tube, Fig. 68, the innersurface of which is perfectlysmooth. Let m be the mass ofthe particle, v its speed of pro-jection and r the radius of thecircle formed by the tube. Theradius of the cross-section ofthe tube is supposed to be neg-ligible. Suppose the particle tobe acted upon by no forces ex-cept the reaction of the innersurface of the tube. Then, sincethe surface is smooth, the reac-tion is normal and consequently there is no force along thetangent to the path. Therefore the force equations for thetangential and the normal directions give do. Fra. 68. and a0- m— = — N, r (1) 2 where N represents the reaction of the walls of the tube uponthe particle. It is evident from equation (1) thai the mag-nitude of the velocity is constant. Therefore the particledescribes the circular path with a constant speed. Equation(2) states that the normal reaction of the surface equals the 126 ANALYTICAL MECHANICS normal kinetic reaction and is oppositely directed. There-fore the reaction of the surface is directed towards the centerof the circular path. Equations (1) and (2) are independent of the specialmethod used to keep the particle in a circular path. If, for instance, the particle were connected to the center bymeans of an inextensible string and then projected in adirection perpendicular to the string the results would havebeen the same. The force which constrains the particle to move in thecircular path is called the central force. This force may bethe reaction of a surf

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