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5. Kinematics

5. Kinematics. Piecewise constant velocity. Distance runned during the time interval. x(t). x(t i ). x i = v(t i ) . h. v(t i ) is the slope of the segment. h. t. t 0. t n. t i. t i+1. 5. Kinematics. Instantaneous velocity. x(t). x(t). M. - definition of velocity. h. t. O.

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5. Kinematics

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  1. 5. Kinematics • Piecewise constant velocity Distance runned during the time interval x(t) x(ti) xi = v(ti) . h v(ti) is the slope of the segment. h t t0 tn ti ti+1 B. Rossetto

  2. 5. Kinematics • Instantaneous velocity x(t) x(t) M - definition of velocity h t . . . . O - definition of acceleration tn t0 ti ti+h B. Rossetto

  3. 5. Kinematics • Velocity and acceleration Cartesian Polar (cf. chap.1 Coordinates, slide 7) B. Rossetto

  4. 5. Particle motion • First law of Newton (inertia principle) Define a system (particle, system of particles, solid) • Second law (principle) of Newton As a consequence : system with interaction : changes, depending on the inertial mass m: B. Rossetto

  5. 5. Motion • Extension to variable mass systems Definition of the momentum of the system: 1st law: principle of conservation of momentum 2nd law: fundamental law of dynamics B. Rossetto

  6. 5. Kinematics • Rotational dynamics 1 - Definition of angular momentum: 0 2 - Fundamental theorem of rotational dynamics: is the torque of the force generating the movement ( and must be evaluated relative to the same point 0) Proof: B. Rossetto

  7. 5. Kinematics • Motion under constant acceleration Double integration and projection: (parametric equation of a parabola) B. Rossetto

  8. 5. Kinematics • Fluid friction Example: free fall of a particle in a viscous fluid. v(t) From the second law : t K : shape coefficient (body) h : viscosity (fluid) 0 (2nd order differential equation with constant coefficients) Speed as a function of time : Limit speed : B. Rossetto

  9. 5. Kinematics • Sliding friction Example: inclined plane Frictional force characterized by a : q Static coefficient a > dynamic coefficient a Project the fundamental law of dynamics a (2nd Newton law) onto Ox and Oy axes. B. Rossetto

  10. 5. Kinematics • Uniform circular motion Definition of angular velocity M Theorem: 0 Definition of uniform circular motion and and ) (implies . 0 Acceleration: from chap. I Coordinates, slide #7 (central) then B. Rossetto

  11. 5. Kinematics • Motion under central force (1) Example: gravitation P(m’) . (3rd Newton law) m: gravitational mass, equal to inertial mass . O(m) Theorem slide #5: From the 2nd Binet law: Sketch of proof: expression of acceleration in polar coordinates: B. Rossetto

  12. 5. Kinematics • Motion under central force (2) Solution of the differential equation M(r,q) . b p a c . . . . . . A A’ F F’ (Origin) (ellipse, Origin is one of the focuses F’) B. Rossetto

  13. 5. Work and energy • Work Definition. Work of a force along a curve : Property. If there exists EP such that H is conservative. then P • Energy Potential energy. We define the potential energy of a conservative force vectorfield as a primitive: Kinetic energy. The kinetic energy of a particle of mass m and velocity v is defined as Ek=(1/2)mv2. B. Rossetto

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