Class 8: Kinetic energy work done and Class 8: Kinetic energy, work done, and conservative force
Three pillars of Newtonian mechanics p Newton’s L N aw of Motion M Newtonian Mechanics Conservati ion of Energy Conservat ion of Momentum m
Ki Kinetic Energy ti E 1 T = 2 2 T mv 2
W Work done k d 1 ` B Work done by F B v ∫ ∫ = = ⋅ W W F F d d s s → A A B B A B A v v ∫ ∫ ⋅ = ⋅ F d s - F d s ds F F A B A ⇒ → = W W → A B B A for the same path.
Conservative Force If F is a function of position: F(r) B If W A → B is path 1 ` independent, F is i d d i conservative. If W A → B is path d dependent, F is non ‐ d F i A conservative.
Conservative Force If F(r) is conservative 1. W A → A = 0 v ∇ × = 2. F 0 A
Two Important Equations for Vector Analysis Two Important Equations for Vector Analysis v ∇ ⋅ ∇ × = 1 . ( F ) 0 ∇ × ∇ Φ = 2. ( ) 0
Potential Energy If F(r) is conservative, there exist a scalar f function U(r) so that i U( ) h v = ∇ ∇ F F - U U U i U is called the potential energy at point r. ll d h i l i Note that U is not unique in satisfying N h U i i i i f i above equation. It requires “boundary condition” to fix it. diti ” t fi it
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