 # Summer School 2007B. Rossetto1 5. Kinematics  Piecewise constant velocity t0t0 tntn titi t i+1 x(t) x(t i ) h t x i = v(t i ). h Distance runned during.

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Summer School 2007B. Rossetto1 5. Kinematics  Piecewise constant velocity t0t0 tntn titi t i+1 x(t) x(t i ) h t x i = v(t i ). h Distance runned during the time interval v(t i ) is the slope of the segment.

Summer School 2007B. Rossetto2 5. Kinematics  Instantaneous velocity t0t0 tntn titi x(t) h t.... - definition of velocity - definition of acceleration t i +h M O

Summer School 2007B. Rossetto3 5. Kinematics  Velocity and acceleration Cartesian Polar (cf. chap.1 Coordinates, slide 7)

Summer School 2007B. Rossetto4 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:

Summer School 2007B. Rossetto5 5. Motion  Extension to variable mass systems Definition of the momentum of the system: 1 st law: principle of conservation of momentum 2 nd law: fundamental law of dynamics

Summer School 2007B. Rossetto6 5. Kinematics  Rotational dynamics 1 - Definition of angular momentum: 0 ( and must be evaluated relative to the same point 0) 2 - Fundamental theorem of rotational dynamics: Proof: is the torque of the force generating the movement

Summer School 2007B. Rossetto7 5. Kinematics  Motion under constant acceleration (parametric equation of a parabola) Double integration and projection:

Summer School 2007B. Rossetto8 5. Kinematics  Fluid friction (2 nd order differential equation with constant coefficients) Example: free fall of a particle in a viscous fluid. t v(t) 0 Limit speed : Speed as a function of time : From the second law : K : shape coefficient (body) : viscosity (fluid)

Summer School 2007B. Rossetto9 5. Kinematics  Sliding friction Frictional force characterized by :   Example: inclined plane Static coefficient > dynamic coefficient  Project the fundamental law of dynamics (2nd Newton law) onto Ox and Oy axes.

Summer School 2007B. Rossetto10 5. Kinematics  Uniform circular motion 0 0. and (implies Acceleration: from chap. I Coordinates, slide #7 Definition of uniform circular motion Definition of angular velocity and then (central) ) M Theorem:

Summer School 2007B. Rossetto11 5. Kinematics  Motion under central force (1) Example: gravitation.. O(m) P(m’) Theorem slide #5: From the 2 nd Binet law: Sketch of proof: expression of acceleration in polar coordinates: m: gravitational mass, equal to inertial mass (3 rd Newton law)

Summer School 2007B. Rossetto12 5. Kinematics  Motion under central force (2). F’ (Origin) F p. c b a M(r,) (ellipse, Origin is one of the focuses F’) A’ Solution of the differential equation..... A

Summer School 2007B. Rossetto13 5. Work and energy  Work Definition. Work of a force along a curve : P H Property. If there exists E P such that then is conservative. 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 E k =(1/2)mv 2.  Energy

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