Kinetics of a Particle: work and energy

Transcription

1 Kinetic of a Particle: work and energy Work ha been done by a force on a particle only when the particle undergoe a diplacement in the direction of the force du F d co du Fdr Unit of work: Joule J= Nm Poitive work: force acting on a particle ha the ame ene of diplacement. Negative work: force acting on a particle ha the oppoite ene of diplacement.

2 Work of a variable force r U d F d co 1 F r r1 1

3 Work of a Contant force moving along a traight line U1 Fc co d 1 U1 F co c ( 1)

4 Work of a Weight Work done by a weight on the particle U1 Fdr ( Wj) ( dxidyjdzk) y y1 r r 1 Wdy W( y y) 1 U W y 1

5 Work of a Spring Force Work done by particle on the pring U F d k d k 1 k 1

6 Work of a Spring Force Work done by the pring on the particle 1 1 U 1 ( k k1)

7 Principle of Work and Energy Kinetic energy U T 1 1 v t t v F d ma d mv dv 1 1 Fd t mv mv 1 1 U1 mv mv1 mv U1 mv m0 mv T Total work mut be done on the particle to bring it from the ret to a velocity tate v T U T 1 1

8 Example (Principle of work and Energy)

9 Example (Principle of work and Energy)

10 Power and Efficiency Power Unit Efficiency du P dt du Fdr dr P F Fv dt dt dt 1W 1 J / 1 Nm/ 1 hp 550 ftlb / = power output power input energy output energy input

11 Conervative Force and Potential Energy Conervative force: When the work done by a force in moving a particle from one point to another i independent of the path followed by the particle, then thi force i called a conervative force. Example of conervative force: (1) work done by the weight (gravitational force) of a particle () work done by a pring force on a particle Non conervative force: friction force

12 Example of conervative force:weight Work done by a weight on the particle U Fdr ( Wj) ( dxidyjdzk) 1 y y1 r r 1 Wdy W( y y) 1 U W y 1 Work done by a weight i only dependent on the poition y1 and y and independent of the path between poition y1 and y, thu the weight i a conervative force.

13 Example of conervative force: Spring force Work done by particle on the pring U F d k d k ( ) 1 Work done by a pring force i only dependent on the poition 1 and and independent of the path between poition 1 and, thu the pring force i a conervative force.

14 Gravitational Potential Energy Gravitational potential Energy: V Wy mgy g Work done againt the gravitational field to elevate the particle a ditance y above ome arbitrary ref. plane Vg 0 V V V mg( y y ) g U 1 1 Work done on the particle due to gravitational force V 1g g

15 Elatic Potential Energy V e Elatic potential Energy : the work done on the pring to deform it. 1 Ve F 0 d k d k 0 1 Ve k ( 1 ) Work done on the body by pring force U V 1pring e

16 Conervative Force Field and Potential Function F i a function of the coordinate. The work done by F during a diplacement du Fdr ( F if jf k) ( dxidyj dzk) x y z FdxFdyFdz x y z dr The total work done along it path from poition 1 to poition U Fdr F dxf dy Fdz x y z If Fdr i an exact differential dv of ome calar function V of the coordinate, then V U dv V V V1 1 1 Work depend only on the end point of the motion Independent of the path followed

17 Conervative Force Field and Potential Function If V V V Fdr FxdxFydy Fdz z ( dx dy dz) dv x y z Then, work depend only on the end point of the motion F V x, F V y, F V z x y z V V V V V V F i j k i j kv x y z x y z F V where V i potential function, del V i gradient of the potential function The force with thi characteritic i aid to be conervative F V

18 Conervation of Dynamic Energy Principle of work and energy U 1 T 1 ( 1) U1 ( U1 ) noncon U1 pring U1 g T ( U ) ( V ) ( V ) T noncon e g U T V V noncon e g If there are only conervative force, then 1 ( U ) noncon 0 T V V e T V V T V V 1 e1 g1 e g g 0 Law of conervation of dynamic energy

19 Conervation of Dynamic Energy ( U ) 0 1 noncon. T1 V1 T V E T1V 10Wh Wh v v a ( yy ) 0 c 0 h 0( )( ) h 1 W ( ) g v g h gh EVT W gh Wh

20 Work and Energy (example 1)

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22 Work and Energy (example )

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