西南交大：《大学物理》（双语版）CHAPTER 8 Work, Energy and The CWe Theorem

UNIVERSITY PHYSICS I CHAPTER 8 Work, Energy and The CWe Theorem §8.1 Work done by variable force 1. Why we introduce the concepts of work and energy? F total=ma CWE theorem CWE theorem Conservation of energy (More universal) 2. About the systems that we discuss Ignore the size, internal structure, internal motion, deformations, and thermal effects

1 1. Why we introduce the concepts of work and energy? F ma CWE theorem r r total = CWE theorem Conservation of energy (More universal) 2. About the systems that we discuss Ignore the size, internal structure, internal motion, deformations, and thermal effects. §8.1 Work done by variable force

88.1 Work done by variable force 3. The work done by a constant force W=F.Scos6=F·S 4. The work done by variable force differential work b dw s F dr F.dr. cos 0 Fcos ads F 88.1 Work done by variable force Define the differential work done by any force is dW= fdr The work done by a particular force eis w=dw=Fdr In Cartesian coordinate system W=(Fi+E,j+F) dr d b dxi+dvj+dzk) (F dr+ f dy+F,dz) F

2 W F S F S v v = ⋅ ⋅ cosθ = ⋅ θ F v s v F v 3. The work done by a constant force F s F r W F r cos d d cos d d θ θ = = ⋅ ⋅ = ⋅ v v v differential work a b o F v r v d ds r v r′ v θ F r 4. The work done by variable force §8.1 Work done by variable force §8.1 Work done by variable force Define the differential work done by any force is W F r r r d = ⋅d The work done by a particular force is F r ∫ ∫ = = ⋅ f i W F r r r W d d In Cartesian coordinate system: ( d d d ) )ˆ d ˆ d ˆ (d ) ˆ ˆ ˆ W ( F x F y F z xi yj zk F i F j F k y z r r x y z r r x f i f i = + + + + = + + ⋅ ∫ ∫ r r r r a b o F v r v d ds r v r′ v θ F r

88.1 Work done by variable force w=l(F dr+ F dy+ F, dz) Fd+∫F中+F Work done by a constant force: =x1++k x l+yr/+3 W=∫Fx+「F+Fdz F(x-x)+F,(r-yi)+F(z-i) W=F·=F(7-) 88.1 Work done by variable force The properties of the work Work is a scalar quantity; A0 @the work done by a force depends on the path followed by the system w=f&dr=f2 AF=(fni).(si)=-frs H=「m7·d=7:,=()(-s)=s Welsdnath =w+w=-fis-fiS=-2fRS

3 ∫ ∫ ∫ ∫ = + + = + + f i f i f i f i z z z y y y x x x y z r r x F x F y F z F x F y F z d d d W ( d d d ) r r Work done by a constant force: r x i y j z k r x i y j z k f f f f i i i i ˆ ˆ ˆ ˆ ˆ ˆ = + + = + + r r ( ) ( ) ( ) W d d d x f i y f i z f i z z z y y y x x x F x x F y y F z z F x F y F z f i f i f i = − + − + − = + + ∫ ∫ ∫ ( ) f i W F r F r r r r r r r = ⋅∆ = ⋅ − §8.1 Work done by variable force The properties of the work: 1work is a scalar quantity; 2the work done by a force depends on the path followed by the system; §8.1 Work done by variable force W f r f r f i si f s k k k B A = k ⋅ = ⋅ = − ⋅ = − ∫ )ˆ ) ( ˆ d ( r r r r ∆ W f r f r f i si f s k k k A B = k ⋅ = ⋅ = ⋅ − = − ∫ )ˆ ) ( ˆ d ( r r r r ∆ A B s x y W W W f s f s f s k k k 2 clsd path = + ′ = − − = − A 0

88.1 Work done by variable force @the work done by a force depends on the choice of the reference frame. Elevator: W=0, Earth:W≠0 @the work done by a pair of forces is not always zero. M N +形 N 0 W,+W<0 88.1 Work done by variable force 5. The work done by the total force The work done by the total force acting on the system is the algebraic, scalar sum of the work done by the individual forces total ∫m(丙++ F1·d+F2dr+ W,+W,+

4 3the work done by a force depends on the choice of the reference frame. Elevator:W=0, Earth: W≠0 y W g r v r + ′ = 0 W N W N N r ′ v c N r v m f c r ′ s s ′ M + ′ < 0 W f W f f r m §8.1 Work done by variable force 4the work done by a pair of forces is not always zero. 5. The work done by the total force The work done by the total force acting on the system is the algebraic, scalar sum of the work done by the individual forces. §8.1 Work done by variable force L L r r r r r L r r r r = + + = ⋅ + ⋅ + = ⋅ = + + ⋅ ∫ ∫ ∫ ∫ 1 2 1 2 total total 1 2 d d d ( ) d W W F r F r W F r F F r f i f i f i

88.1 Work done by variable force 6. The the geometric interpretation W=Fdx+”F+Fdz The work done on the system by the force component as the system moves from r tor is the area under the curves of the graph of that force components versus the corresponding coordinate. 88.1 Work done by variable force Positive or negative work(area) depends on both the sign of the force and the direction of Ar Initial position Final position Initial position Ar=(x-x)i Ar=(x-x)i Parallel to 1 Parallel to-i

5 §8.1 Work done by variable force 6. The the geometric interpretation ∫ ∫ ∫ = + + f i f i f i z z z y y y x x x W F dx F dy F dz The work done on the system by the force component as the system moves from to is the area under the curves of the graph of that force components versus the corresponding coordinate. ir r f r r Positive or negative work (area) depends on both the sign of the force and the direction of . r r ∆ §8.1 Work done by variable force

88.1 Work done by variable force Example 1: A spring hangs vertically in its relaxed state. a block of mass m is attached to the spring, but the block is held in place so tha the spring at first does not stretch. Now the hand holding block is slowly lowered, so that the block descends at constant speed until it reaches the point at which it hangs at equilibrium with the hand removed. At this point the spring is measured to have stretched a distance d from its previous relaxed length. Find the work done on the block in this process by(a)gravity,(b)the spring, (c) the hand. 88.1 Work done by variable force Solution: (a)gravity--a constant force tF W,=F2 4F=-mgj(0-d)j=mgd F, (b)the spring-a variable force mg F=-kyj kd=mg k g Ws= Fs dyj=l-kvj dyj272 d 2 (c)the hand-a variable force F+F.+F=0 F-F =刚yj+mgj 6

6 Example 1: A spring hangs vertically in its relaxed state. A block of mass m is attached to the spring, but the block is held in place so that the spring at first does not stretch. Now the hand holding block is slowly lowered, so that the block descends at constant speed until it reaches the point at which it hangs at equilibrium with the hand removed. At this point the spring is measured to have stretched a distance d from its previous relaxed length. Find the work done on the block in this process by (a)gravity, (b)the spring, (c) the hand. §8.1 Work done by variable force Solution: (a)gravity—a constant force W F r mgj d j mgd g = g ⋅ = − ⋅ − =ˆ (0 ) ˆ r r ∆ (b)the spring—a variable force W F y j kyj y j kd mgd s s 2 1 2 1 ˆ d ˆ ˆ d 2 -d 0 -d 0 = ⋅ = − ⋅ = − = − ∫ ∫ r F kyj s ˆ = − r d mg kd = mg k = (c)the hand —a variable force F F F kyj mgj F F F h s g s h g ˆ ˆ 0 = − − = + + + = r r r v r r mg r Fs r Fh y r §8.1 Work done by variable force

88.1 Work done by variable force (ky+ mg )dy mgd-gmd F mg 2 2 w+w,+w=0 Can you explain? 88.1 Work done by variable force Example 2: A small object of mass m is suspended from a string of length L the object is pulled sideways by a force F that is always horizontal, until the string finally makes an angle om with the vertical. The displacement is accomplished at a small constant speed. Find the work done by all forces that act on the object

7 mgd gmd mgd W F yj ky mg y d d h h 2 1 2 1 ( )d ˆ d 0 0 = − = − = ⋅ = + ∫ ∫ − r − + + = 0 Ws Wh Wg Can you explain? mg r Fs r Fh y r §8.1 Work done by variable force Example 2: A small object of mass m is suspended from a string of length L. the object is pulled sideways by a force F that is always horizontal, until the string finally makes an angle φm with the vertical. The displacement is accomplished at a small constant speed. Find the work done by all forces that act on the object. §8.1 Work done by variable force

88.1 Work done by variable force Solution: From the force diagram x component: F-Tsin=0 y component: T cos o-mg=0 Eliminate T we find F=mg tang H=∫F·dF=∫rd mg tanddx=. mg tandd( Lsin g) o mgL sindo=mgL (1-cos gm) 88.1 Work done by variable force Wn=「TdF=0 Wn=[.-mgj·dF L-Lcos中m mgd mg L-Lcos ou) w=W+W+W=0I 8

8 Solution: From the force diagram F = mg tanφ Eliminate T, we find sin d (1 - cos ) tan d tan d( sin ) d d m 0 φ φ φ φ φ φ φ mgL mgL mg x mg L W F r F x m f i f i f i f i F = = = = = ⋅ = ∫ ∫ ∫ ∫ ∫ r r x component: y component: cos 0 sin 0 − = − = T mg F T φ φ §8.1 Work done by variable force ( cos ) d d ˆ d 0 cos 0 m L L f i g T mg L L mg y W mgj r W T r m φ φ = − − = − = − ⋅ = ⋅ = ∫ ∫ ∫ − r r r = + + = 0 W WF WT Wg §8.1 Work done by variable force r r d

8.1 Work done by variable force Example 3: As shown in figure, M=2kg, k=200Nm S=0.2m,g≈10ms.A student pull the string very slowly, ignore the friction between the pulley and the string, as well as the masses of the pulley and string. At initial time, the spring is in the relaxed state. Find the work done by the force F 88.1 Work done by variable force Solution: When the string is pulled down 0.2 m, what will be the height of m? Kro=Mg-)xo=0.Im and S=0.2m the height of the m will be o 1m kx(0<xso1m)a variable force F=Ak o=Mg(0.1<x<0.2m)a constant force then w kx dx+ Mg dx 0.1 =k210.1 mgr 0.2 0

9 M F k S Example 3: As shown in figure, M=2kg , k =200N·m-1 , S=0.2m , g≈10m·s-2 . A student pull the string very slowly, ignore the friction between the pulley and the string , as well as the masses of the pulley and string . At initial time, the spring is in the relaxed state. Find the work done by the force . F r §8.1 Work done by variable force Solution: Q kx0 = Mg → x0 = 0.1m and S = 0.2m ∴ the height of the M will be 0.1m When the string is pulled down 0.2 m, what will be the height of M? F= k x (0<x≤0.1m) a variable force k x0 =Mg (0.1<x≤0.2m) a constant force | | 3 ( ) J d d 0 .2 0 .1 0 .1 0 2 2 1 0 .2 0 .1 0 .1 0 = + = = + ∫ ∫ kx Mgx then W F kx x Mg x §8.1 Work done by variable force

88.2 Conservative force and the potential energy 1. What is conservative force If a work done by the force on any system as it moves between two points is independent of the path following by the system between the two points or if the work done by the force on any system around any closed path is zero. Then the force is called conservative force a(2) FF.dr=0 (b) 88.2 Conservative force and the potential energy If{F·dr≠0, then the force is called nonconservative force IffF.dr=0, then the force is called zero-work force 2. Some examples of conservative forces local gravitational force F=-mgj --a constant force W=F·△F 10

10 §8.2 Conservative force and the potential energy 1. What is conservative force If a work done by the force on any system as it moves between two points is independent of the path following by the system between the two points; or if the work done by the force on any system around any closed path is zero. Then the force is called conservative force. ⋅d = 0 ∫ F r r r ∫ ∫ ⋅ = ⋅ b a b a F r F r (1) (2) d d r r r r b ⋅d ≠ 0 ∫ F r r r If , then the force is called nonconservative force. If , then the force is called zero-work force. ⋅d = 0 ∫ F r r r 2. Some examples of conservative forces 1local gravitational force W F r r r = ⋅ ∆ F mgj ˆ = − r --a constant force §8.2 Conservative force and the potential energy