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Tham khảo tài liệu 'advanced engineering dynamics 2010 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14 Newtonian mechanics 1.12 Coriolis s theorem It is often advantageous to use reference axes which are moving with respect to inertial axes. In Fig. 1.10 the x y z axes are translating and rotating with an angular velocity co with respect to the xyz axes. The position vector OP is r R r OO O P 1.39 Differentiating equation 1.39 using equation 1.13 gives the velocity r 7 v coXr 1.40 where v is the velocity as seen from the moving axes. Differentiating again f R a co X r CO X v co X v co X r R a cõ X r 2co X v co X co X r 1.41 where a is the acceleration as seen from the moving axes. Using Newton s second law F mr m R a cô X r 2co X v co X co X r 1.42 Expanding the triple vector product and rearranging gives F - mR - má X r - 2wco X v - zn co co r - co2 ma 1.43 This is known as Coriolis s theorem. The terms on the left hand side of equation 1.43 comprise one real force F and four fictitious forces. The second term is the inertia force due to the acceleration of the origin O the third is due to the angular acceleration of the axes the fourth is known as the Coriolis force and the last term is the centrifugal force. The centrifugal force through p is normal to and directed away from the 0 axis as can be verified by forming the scalar product with Cừ. The Coriolis force is normal to both the relative velocity vector v and to 0. Fig. 1.10 Newton s laws for a group of particles 15 1.13 Newton s laws for a group of particles Consider a group of n particles three of which are shown in Fig. 1.11 where the zth particle has a mass mị and is at a position defined by r relative to an inertial frame of reference. The force on the particle is the vector sum of the forces due to each other particle in the group and the resultant of the external forces. If fj is the force on particle i due to particle J and F is the resultant force due to bodies external to the group then summing over all particles except for j i we have for the ith particle s fj Fi mfi 1.44 We now form the sum .