Definition of conservative force One particle exerted by a force moves around a closed path and returns to its starting point If the total work done by the force during the round trip is zero, we call the force a conservative force, such as spring force and gravity If not the force is a nonconservative one Ch 12 Energy II: Potential energy
Ch.12 Energy II: Potential energy Definition of conservative force: One particle exerted by a force moves around a closed path and returns to its starting point. If the total work done by the force during the round trip is zero, we call the force ‘a conservative force’，such as spring force and gravity. If not, the force is a nonconservative one
Two Mathematical statements If F is a conservative force, we have A rab/+Wb 0 Fig 12-4 a Fds+|.Fds=0(121) Path1 Path2 F·ds=0 Statement 1 (b) b Fds=F·ds(123) Path 1 Path2 Path2 F·ds=|F· d Statement2 Path 1 Path2 Ch 12 Energy II: Potential energy
Ch.12 Energy II: Potential energy Two Mathematical statements: a a b 1 b 2 2 1 Fig 12-4 (a) (b) If is a conservative force, we have: → F W W 0 ab.1 + ba.2 = F d s F d s 0 a b b a + = → → → → (12-1) Path1 Path2 Statement 1 → → → → → → = − = F d s F d s F d s b a a b b a Path1 Path2 Path2 (12-3) 0 F ds = Statement 2 → → → → = F d s F d s b a b a Path1 Path2
Note: Newtons third law To every action, there is an equal and opposite reaction (1) Both the action and reaction forces belong to the system (2) The total work done by action and reaction forces is independent of the reference frame chosen(even in non-inertial frame) Prove point(2) Ch 12 Energy II: Potential energy
Ch.12 Energy II: Potential energy To every action, there is an equal and opposite reaction. Newton’s third law Note: (1) Both the action and reaction forces belong to the system. (2) The total work done by action and reaction forces is independent of the reference frame chosen (even in non-inertial frame). Prove point (2):
In s frame y W=W+2=「+ m m In s frame with velocity of vs relative to s frame X W=W+W2=f·c+f2·Chz (1-,)+|2(2-,n) ∫子+1的了的,山 =「f·+· W Ch 12 Energy II: Potential energy
Ch.12 Energy II: Potential energy 1 f 2 f Ｏ 1 r 2 r z y x m1 1 2 1 1 2 2 m2 W W W f dr f dr = + = + In S frame: S In S’ frame with velocity of relative to S frame: s s v ' W' W' W ' f dr' f dr' 1 2 1 1 2 2 = + = + ( ) ( ) 1 1 2 2 f dr v dt f dr v dt s's s's = − + − f dr f dr (f f ) v dt s's = + − + 1 1 2 2 1 2 1 2 f f = − 1 1 2 2 f dr f dr = + =W
12-2 Potential energy y 1, Definition 62 When work is done in a system(such as ball and arth) by a conservative force, the configuration of its rts changes, and so the potential energy changes from its initial u. value to its final value u We define the change in potential energy associated with the conservative force as A=-U1=-=「F(124) Ch 12 Energy II: Potential energy
Ch.12 Energy II: Potential energy 12-2 Potential energy 1.Definition When work is done in a system (such as ball and earth) by a conservative force, the configuration of its parts changes, and so the potential energy changes from its initial value to its final value . We define the change in potential energy associated with the conservative force as: Ui U f ΔU =U −U = −W = − F ds f i (12-4)