Work Energy And Power 功, 能量及功率

Transcription

1 p. 1 Work Energy And Power 功, 能量及功率 黃河壺口瀑布

2 p. 2 甚麼是 能量?

3 p. 3

4 常力所作的功 ( Work Done by a Constant Force ) p. 4 F F θ F cosθ s 要有出力才有 功 勞 造成位移才有 功 勞 W = F cos θ s ( Joule, a scalar ) = F s or F Δx F : force, s : displacement, W : work 1 J = 1 N 1 m = ( kg m 2 ) / s 2

5 W = F s = ( F x i + F y j + F z k ) (Δx i + Δy j + Δz k ) = F x Δx+ F y Δy+ F z Δz p. 5 W : mechanical work ( different from chemical work ) F s W = 0 negative work F AB A F BA B s f k (friction on B) F AB = F BA, W = F s W on A by B ( negative work ) = W on B by A ( positive work )

6 p. 6 A F AB F BA B s f k (friction on B) work done by kinetic friction W f = f k s cos180 o = f k s kinetic friction can also do positive work A f k ( f k moves block A to the right also ) B

7 非常力所作的功 ( Work Done by Variable Force in 1-Dim ) p. 7 ΔW = F Δx = F x Δx F x F x1 F x2 F x3 a Δx b x W = ΔW = F x1 Δx+ F x2 Δx + + F xn Δx n = F xi Δx i =1 as Δx 0 b b W = F x dx or more generally W = F ds a a

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9 動能 ( Kinetic Energy ) p. 9 F v W cause v change cause x change cause? change in one-dim., const. force : W = F x Δx= ma Δx v 2 = v 2 o + 2a Δx a Δx = ½(v 2 v 2 o ) W = ½ mv 2 ½mv 2 o define kinetic energy ( K.E. ) : E k = ½ mv 2 ( joule, scalar ) W net = ΔE k : Work Energy Theorem ( 功能定律 ) ( Work-Energy Theorem still valid in 3-dim. space and with non-constant forces. )

10 p. 10 垂直於運動方向的力沒有作功 ( 沒有改變物體的動能 )

11 Ex. N F m = 4 kg, θ = 53 o p. 11 f θ F cosθ F = 3 N, Δx = 2 m v o = 3 m/s, μ k = 1/8 mg Find (1) ΔE k, (2) v f (1) F y = 0 N = mg Fsinθ ΔE k = W Net = W F + W f = F x s μ k N s = F cosθ s μ k ( mg Fsinθ ) s = = 32 J (2) Δ E k = ½ mv 2 f ½mv 2 o = 32 J, v o = 3 m/s v f = 5 m/s

12 位能 ( Potential Energy ) p. 12 An experiment by Galileo Galilei : A B C raise ball from C A, do work W A C, gain K.E. = W, reach v max C B, K.E. = 0 ( A and B same height ) B C A, K.E. = 0 where does K.E. go at A and B?

13 p. 13 v = 0, K.E. = 0 v f v o v o = v f potential energy ( P.E., E p ) : energy associated with the relative positions of two or more interacting particles

14 p. 14 Example : v max earth surface F ext do work ( W ) to raise the apple Changes the apple-earth relative position Changes the apple-earth system s potential ( or the apple s potential ) release the apple stored P.E. K.E. If one raise the apple at const. speed ( K.E. unchanged ) W ext = Δ ( P.E. ) = E pf -E po

15 如何計算一個系統中的位能? A B C W ext changes particles relative position ( pendulum and earth ) P.E. changed 2 K.E. ( 1/2 mv max ) positive work increase in P.E. ( E p ) p. 15 earth Only changes in P.E. are important Freedom to assign E p = 0 configuration Once the initial configuration for E p =0 is defined : The P.E. of a system is the external work needed to bring the particles from the E p =0 configuration to the given position at a const. velocity ( or speed ).

16 Gravitational Potential Energy (near the earth s surface) p. 16 y f external work done to move m from y i to y f at a const. velocity F ext y i mg earth surface ( W net = W ext + W g = 0 = ΔE k ) W ext = F ext s, F ext = mg ( mg 為此 apple/earth 雙粒子系統內力 ) W ext = F ext s, F ext = mg = mg ( y f y i ) = mgy f mgy i = ΔE p = E p ( y f ) E p ( y i ) Define E p = 0 at y i = 0 E p ( y f ) = mg y f E p (y) = mg y

17 Spring Potential Energy p. 17 Hooke s Law : F sp = - k x, k : spring constant kx f F F sp x f x Move the spring from x = 0 to x f at a constant speed : b W = F ds = ½ k x 2 a f define E p,sp = 0 at x = 0 E p,sp = ½ k x 2 f E p,sp = ½ k x 2

18 p. 18 伽利略看遠處的大東西虎克看身邊的小東西

19 p. 19

20 Spring Potential Energy p. 20 Hooke s Law : F sp = - k x, k : spring constant kx f F F sp x f x Move the spring from x = 0 to x f at a constant speed : b W = F ds = ½ k x 2 a f define E p,sp = 0 at x = 0 E p,sp = ½ k x 2 f E p,sp = ½ k x 2

21 p. 21 +x, +F sp direction F sp F sp F sp F sp = - kx

22 p. 22 Work changes the velocity of the particle ΔE k the relative positions of particles ΔE p Inversely : Energy is a measure of the particle s, or the particle system s, capability to perform work ( on others ) 甚麼是 能量?

23 能量守恆 ( Energy Conservation ) p. 23 mechanically conservative system ( 機械守恆系統 ) : no energy enters or leaves the system ( e.g. heat, radiation, work by external force, work done by internal chemical reaction ) In a mechanically conservative system : v 2 A system of 2 particles ( apple & earth) y 2 E k + E p = const. v 1 ( E k + E p ) initial = ( E k + E p ) final y 1 earth ½mv mg y 2 = ½ mv mg y 1

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25 p. 25 There are other forms of energy : e.g. thermal energy ( Actually, in microscopic scale, thermal energy may also be a kind of kinetic energy )

26 p. 26 從位能轉換為動能 動能再轉換為電能 2003 三峽大壩

27 p. 27 The ball on which track will hit the finish line earlier?

28 功率, 機械功率 ( Mechanical Power ) p. 28 mechanical power P av = ΔW / Δt ( J / s ), 1 J/s = 1 watt ( W ), 1 hp = 746 W lim ΔW/Δt = P = dw/dt Δt 0 ΔW = F Δs dw = F ds v = ds / dt P = dw / dt = F ds / dt = F v Ex. A tractor exert 3 x 10 4 N, moving at 5 m/s. What is its horsepower? P = F v = 3 x 10 4 x 5 = 1.5 x 10 5 ( W ) 200 ( hp )