CMC-205 MECÂNICA ANALÍTICA Prof.: Ijar M. da Fonseca Divisão de Mecânica Espacial e Controle

1. CMC-205 MECÂNICA ANALÍTICA Prof.: Ijar M. da Fonseca Divisão de Mecânica Espacial e Controle Instituto Nacional de Pesquisas Espaciais – INPE São José dos Campos, S.P., Brasil

2. Fundamentals of Newtonian Mechanics

3. First Law : If there are no forces acting upon a particle, the particle will move in a straight line with constant velocity. • A particle is an idealization of a material body whose dimensions are very small compared with the distance to other bodies and whose internal motion does not affect the motion of the body as a whole. • Mathematically it is represented by a mass point having no extension in space. • Denoting by F the force vector and by v the velocity vector measured relative to an inertial space, the first law can be stated mathematically: • If F = 0, then v = const.

4. Second Law : A particle acted upon by a force moves so that the force vector is equal to the time rate of change of the linear momentum vector. • Linear Momentum Definition:product of the mass and the velocity of the particle. Mathematically: • p = mv • Then:

5. 2 F21 1 F12 • Third Law : When two particles exert forces upon one another, the forces lie along the line joining the particles and the corresponding force vectors are the negative of each other. • This law is also known as the law of action and reaction. Denoting by F12the force exerted by particle 2 (1) upon particle 1 (2), the law can be stated. Mathematically: • where the vectors F12 and F21 are clearly collinear. • It must be noted that the first law, which is Galileo's inertial law, is a special case of the second law. • A notable exception to the third law are the electromagnetic forces between moving particles.

6. Perhaps the concepts can be further clarified by examining the motion of a particle of mass m along curve s, as illustrated in Fig. 1.1. Axes x, y, z represent an inertial space, and we shall refer to the motion with respect to such a system as absolute. The radius vector r1 denotes the absolute position of m at time t1 (t2). By definition, the absolute velocity vector is given by note that v is a vector tangent to curve 5 at any instant t. In a similar way, we can write the expression for the absolute acceleration vector of the particle for a particle, m=const and

8. For the most part, Newtonian mechanics predicts the motion of planets quite well. • One notable exception is the anomalous behavior of the perihelion of Mercury. Many attempts were made to explain the discrepancy between the observed motion of Mercury and the prediction based on Newton's law of gravitation but to no avail • The discrepancy was finally explained by Einstein's general theory of relativity.

9. m2 m1

10. Impulse and Momentum Let us multiply F bydt through by dt and integrate with respect to time between the times t1 and t2 to obtain

12. Moment of a Force and Angular Momentum

14. Conservation of Angular Momentum

15. Work and Energy

17. Conservative Field

21. Principle of the conservation of energy