Lecture_02

Lecture_02 - PHYS 342 Fall 2010 Lecture 02: Particles and...

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HYS 342 PHYS 342 Fall 2010 ecture 02: Particles and Waves Lecture 02: Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 02 1
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Equations of Motion for Particles Newton: Predict the future trajectory of a particle ! 2 2 () d r mF r dt 2
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Equations of Motion for Particles Linear Motion Rotational Motion Displacement d = v o t+½ at 2 = o t+±½± t 2 Inertia m I=mr 2 Velocity v = v o + at = o + t Newton’s 2 nd Law Momentum F = m a p = m v = r x p =I α onservation of f xt =0 en f xt =0 en F = dp/dt = d /dt Newton’s 2 nd Law Conservation of momentum If F ext 0, then p = constant If ext 0, then = constant Kinetic Energy (K) ½ mv 2 ½ (mr 2 ) 2 = 2 /2I 3
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Equations of Motion for Particles – cont. Linear Motion Rotational Motion ork = •dx = Potential Energy, V V(x)=-W Work W Fdx W= ∫τ d θ V( θ )=-W If Force is conservative (not a function of time), then Potential Energy for linear motion is: gy f m () ref P P Vx F x d x  x ref P If Torque is conservative (not a function of time), then Potential Energy for rotational motion is: o Q Q Vd   o Q 4
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Important Consequences: Constants of Motion Work-Energy Theorem: W net = K f –K i If no work, then K=constant of motion If no external forces acting at point of contact, then momentum=constant (Newton’s 3 rd Law) mm m ( L) Virial Theorem: < K > time aver. =-½ < V > time aver. oether’s heorem (1915): Any conservation law Noether s Theorem (1915): Any conservation law implies an underlying symmetry: Conservation of energy implies time translation symmetry nvariance of EOM under the transformation ) t T (invariance of EOM under the transformation Conservation of linear momentum implies linear translation symmetry (invariance of EOM under the transformation ) rr R  o tt Conservation of angular momentum implies rotational translation symmetry (invariance of EOM under the transformation ) 5 o
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Lecture_02 - PHYS 342 Fall 2010 Lecture 02: Particles and...

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