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CHAPTER 14 Kinetics of a particle:Work and Energy 14.1 The Work of a Force 1. Definition of Work A force F does work in a particle only when the particle undergoes a displacement in the direction of the force. r Here dr ds r F =force acts on the particle dr = r -r =displacement of the particle ds = dr =angle between F and dr positive dU negtive 0 du F cos ds F * dr 00 900 0 0 90 180 0 90 .dr 0 F S cos Unit of Work (SI) 1 N*m=1 joule (J) 1 ft*lb= F S cos 2. Work of a Variable Force F F S F S1 F cos r1 S2 r2 S2 S1 ds dU Fdr F cos ds 2 U1 2 dU 1 r1 s1 Fdr F ( s ) cos ds r2 s2 S 3. Work of a Constant Force moving along a straight live U 1 2 Fc S1 y z s1 ds r xi yj zk dr dxi dyj dzk w wk (position) (displacement) Fdr S1 y1 S2 s2 F cos S 2 S1 S2 4. Work of a Weight S F cos ds F cos Fc cos W s 1 S s2 r1 r2 y2 r2 U1 2 r1 wdr r2 wk dxi dyj dzk r1 z1 wdz w z2 z2 z1 dz w z 2 z1 positive Z 0 wZ negtive Z 0 5. Work of a spring forces Consider a linear spring force Fs ks (1) Work done on the spring S sd Fs S1 S2 Fs Fs ks S1 S2 s2 S s2 U12 Fsds ksds s1 s1 12 ks22 12 ks12 12 k ( s 2 s 2 ) (2)Work of a spring done on the particle A particle (or only) attached to a spring F.B.D of spring and particle spring force on the particle Fs ks work of a spring on the particle is U1 2 2 Fs dr ksds 1 ks 1 2 2 s2 s1 12 k ( s2 s1 ) 0 2 2 14.2 Principle of Work and Energy (PWE) 1. P.W.E The particle’s initial kinetic energy plus the work done by all the forces acting on the particle as it moves from its initial to its final position is equal to the particle’s final kinetic energy. T1 U12 T2 or 12 mv12 U12 12 mv2 2 z 2 1 Here T 2 mv kinetic energy P( FR )t U12 Work done by forces. r 2. Derivation FR F external force Equation of motion of particle FR ma mat ut manun ( FR ) t ut ( FR ) n un FR F t n ( Fe ) n y x Initial frame Work done on particle P by external force FR is FR dr F dr ma dr F dr Fds cos ( F cos u )ds a at ut an u n dr dsut d 2s ( F )t mat m dt a dr at ds F dr ( F ) t ds mat ds (1)Applying the kinetic equation at ds Vdv to the above equation yields Fv dr Ft ds mvdv (2)Integrate both sides to yield v 1 r1 r2 2 2 2 2 1 1 U mv mv mv r F dr r mvdv 12 2 v1 2 2 2 1 T2 T1 2 1 or T1 U12 T2 3. Remark (1) PWE represents an integrated form of equation of motion F t mat s2 v2 dv Ft mat Ft mv ds s Ft ds v mvdv at vdv ds F ds mvdv 1 1 2 1 U12 12 mv2 12 mv T2 T1 t 2 (2) PWE provides a convenient substitution for F t mat When solving kinetic problems involving force , velocity and n displacement. Ex: r From P.W.E we have T1 U12 T2 1 2 mv1 mgr sin mv2 2 V2 2 gr sin 1 2 T ? 2 mg t 14.3 Principle of Wok and Energy for a Si System of Particles principle of Work and Energy for Particle i ri 2 ri 2 1 2 2 1 2 mi vi1 Fx dri f x dri 2 mi vi 2 ri1 ri1 i Fi n t fi Fi =resultant external force on ith particle fi n f ij =resultant internal force on ith particle Since work and energy are scalars both work and kinetic energy applied to each particle of the system may be added together algebraically. r r 2 2 i2 i2 1 1 m v F d r f d r m v So that 2 i i 2 r i i r i i 2 i i 2 i1 i1 or T U T j 1 j i 1 1 2 2 T T U Here =System’s initial kinetic energy 2 =System’s final kinetic energy 12 =Work done by all external and internal forces 1 Note: r i2 (1) ri1 f i dri 0 ,since the paths over which corresponding particles travel will be different. (1) f ij i f ji Si (2) (Non rigid Body) Elastic..plastic… i j j Rigid body j S j ri 2 f i dri 0 (2) ,if the particles are contained within ri1 the boundary of a translating rigid body , or particles connected by inextensible cables. i 14.4 Power and Efficiency 1. Power The amount of work per unit of time. U dv Pav or P t dt dv F dr work F dr dr P F F v dt dt unit of power 1N m s 1J s 1W 2. Efficiency 效率 poweroutput powerinput energyoutput energyinput 14.5 Conservative Forces and Potential Energy 1. Conservative force The force moves the particle form one point to another point to produce work which is independent of the path followed by the particle. 1 F dr 2 F dr 3 F dr F Conservative force 1 F 2 3 (1) Work done by weight U Wy (2) Work done by the spring force on a particle S w w V ( ks2 ks ) 1 2 Spring force S1 y y0 S2 2 1 2 2 1 F (保守力) 2. Potential Energy 位能 P A measure of the amount of work a conservation force will datum do when it moves from a given position to the datum or a Vg Wy w 推回基準作 reference plane. P y 的功 F datum (1) Gravitational potential energy y Vg Wy (y=positive upward) Vg Wy 作負功 (2) Elastic Potential Energy Ve 12 ks2 always positive. The spring force has the capacity for always positive work on the particle. Unstretched S Ve ? S push datum pull 3. Potential Function V V Vg Ve The work done by conservation forces(W and Fs) in moving the particle from point ( x1 , y1 , z1 ) to point ( x2 , y2 , z2 ) Is V12 V1 V2 Ex: (U ) Wy W ( S S ) 1 2 w 2 (Unstretched position) 2 V1 Vg1 Ve1 WS1 ks1 1 2 datum S1 2 V1 V2 W ( S 2 S1 ) 12 ks1 12 ks2 2 (U12 ) w (V12 ) Fs U12 w 2 V2 Vg 2 Ve2 WS 2 ks2 1 2 S2 v2 1 (V12 ) Fs ( 12 ks2 12 ks1 ) l0 v1 2 2 14.6 Conservation of Energy The principle of work and energy is rewritten as T1 (U12 ) cons (U12 ) noncons T2 (U12 ) cons V1 V2 Work done by conservative forces (U12 ) noncons Work done by nonconservative forces T1 V1 (U12 ) noncons T2 V2 If (U12 ) noncons 0 , then T1 V1 T2 V2 Conservation of energy The sum of the particle’s kinetic and potential energy remains constant during the motion . Conservation of energy for a system of particles is T V T V 1 1 2 2 end