Lecture 3 work and energy 2015

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Lecture WORK and ENERGY OUTLINE • • • • Work and Kinetic Energy The Work-Kinetic Energy Theorem Power Conservative Force-Nonconservative Force • Potential Energy • Mechanical Energy • Conservation of Mechanical Energy 4.1 Work and Kinetic Energy The work done by a constant force F on the object when it moves a straight distance s is: F=const  W  Fs cos  F s   s In general case, the work is not constant, the path is a curve The work done by force F when the object moves a very small displacement ds (we can consider F constant and ds a straigh  linet:  F  ds (1) (2)   dv dW  F ds  m ds  mv dv dt The work done by force F when the object moves from position (1) to (2) is:  v2      W   F ds   mv dv  mv22  mv12  2 v1 We define: Kinetic Energy: K mv The total work done on a particle is equal to the change in its kinetic energy 1 W  K  mv22  mv12 Work-Kinetic Energy Theorem 2 4.2 Power work _ done Power  per _ unit _ of _ time  dW F ds   P   F v dt dt t2 W   dW   Pdt t1 if P  const  W  Pt 2 W  mv2  mv1 2 2 mv2  mv1 W 2 t   P P Conservative Nonconservative Force Definition: A force is Conservative if the work done by the force is independent on the path, it is dependent only on the initial and final position - Gravity and spring force are conservative forces,while kinetic friction is not Work done by Gravitation Force: (1) r1 Fgrv dr  m ds Work done by the grav force F on object m when it moves a displacement ds:  Mm dW  F ds   F ds cos    G dr    r dr (2)Work done by the grav force F on object m when it moves from (1) to (2) r r2 M G  6.67 10 11 N m2 / kg Mm Mm Mm W    G dr  G G r r2 r1 r1 Universal Graviattional Cconstant + Work done is independent of the path, but of the initial and final position gravitation force is conservative + we define a scalar quantity called gravitational potential energy of two object sseparated by a distance r : Mm U ( r )  G r C If we choose U=0 when r=, we have C=0, If we choose U=0 on the surface of Earth: C=GMm/R We can write: W  U1  U  U Conservative Forces The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle The work done by a conservative force on a particle moving through any closed path is zero (A closed path is one in which the beginning and end points are identical.) the work Wc done by a conservative force on an object as the object moves from one position to another is equal to the initial value of the potential energy minus the final value (2) (a)  W12   Fc .ds  (1a ) F (b) (1) ds (1)(2)   Fc ds  U1  U  U (1b )  W   Fc .ds  (C ) Mechanical Energy    ma  Fc  Fnc • If an object is exerted by Conservative Force Fc and Nonconservative Force Fnc, • from the Work-Kinetic Energy Theorem : • Fc is conservative: WFc  U  U1  U (2) • From (1) and (2): K  K1  U1  U  WFnc K  K  K1  WFc  WFnc (1) ( K  U )  ( K1  U1 )  WFnc • Mechanical Energy: E=K+U E  E2  E1  WFnc Conservation of Mechanical Energy The change in Mechnaical energy of an object is equal to the work done by nonconservative force on the object as it takes a path form position (1) to (2) E  E2  E1  WFnc If Fnc= or WFnc =0 =>E=0: E=const Conservation of Mechanical Energy Conservative Force & Potential Energy        F  Fx i  Fy j  Fz k ds  dxi  dyj  dzk  dW  Fc ds  Fx dx  Fy dy  Fz dz (1) F _ is _ conservative  U U U  dW  dU   dx  dy  dz (2) y z   x U U U Fx   ; Fy   ; Fz   x y z  F   gradU Gradient Operator       grad  i  j k x y z Lực bảo tòan   p   p   p  F   grad p  ( ex  ey  ez ) x y z Dao động xung quanh vị trí cân bền Khai triển Taylor hàm xung quanh vị trí cân bằng:  p  p  p ( x)   p ( xe )  ( x  xe )  ( x  xe ) x x  x x x  x e     e 0 vị trí cân  p ( x)   p ( xe )  k ( x  xe ) 2 Fx   p / x  k ( x  xe ) mx  k ( x  xe ) k bền k>0 ko bền k