Week 6, Chapter 7 Sect 15


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1 Week 6, Chapter 7 Sect 15 Work and Knetc Energy Lecture Quz The frctonal force of the floor on a large sutcase s least when the sutcase s A.pushed by a force parallel to the floor. B.dragged by a force parallel to the floor. C.pulled by a force drected at an angle above the floor. D.pushed by a force drected at an angle nto the floor. E.turned on ts sde and pushed by a force parallel to the floor. Introducton to Energy The concept of energy s one of the most mportant topcs n scence and engneerng Every physcal process that occurs n the Unverse nvolves energy and energy transfers or transformatons Energy s not easly defned Energy Approach to Problems The energy approach to descrbng moton s partcularly useful when Newton s Laws are dffcult or mpossble to use An approach wll nvolve changng from a partcle model to a system model Ths can be etended to bologcal organsms, technologcal systems and engneerng stuatons Systems A system s a small porton of the Unverse We wll gnore the detals of the rest of the Unverse A crtcal skll s to dentfy the system Vald System Eamples A vald system may be a sngle object or partcle be a collecton of objects or partcles be a regon of space vary n sze and shape 1
2 Work The work, W, done on a system by an agent eertng a constant force on the system s the product of the magntude F of the force, the magntude r of the dsplacement of the pont of applcaton of the force, and cos where s the angle between the force and the dsplacement vectors Work, cont. W = F r cos The dsplacement s that of the pont of applcaton of the force A force does no work on the object f the force does not move through a dsplacement The work done by a force on a movng object s zero when the force appled s perpendcular to the dsplacement of ts pont of applcaton Work Eample The normal force and the gravtatonal force do no work on the object cos = cos 90 = 0 The force F s the only force that does work on the object More About Work The system and the agent n the envronment dong the work must both be determned The part of the envronment nteractng drectly wth the system does work on the system Work by the envronment on the system Eample: Work done by a hammer (nteracton from envronment) on a nal (system) The sgn of the work depends on the drecton of the force relatve to the dsplacement Work s postve when projecton of F onto r s n the same drecton as the dsplacement Work s negatve when the projecton s n the opposte drecton Unts of Work Work s a scalar quantty The unt of work s a joule (J) 1 joule = 1 newton. 1 meter J = N m Work Is An Energy Transfer Ths s mportant for a system approach to solvng a problem If the work s done on a system and t s postve, energy s transferred to the system If the work done on the system s negatve, energy s transferred from the system 2
3 Work Is An Energy Transfer, cont If a system nteracts wth ts envronment, ths nteracton can be descrbed as a transfer of energy across the system boundary Ths wll result n a change n the amount of energy stored n the system Clcker Queston Assumng that the earth s movng around the sun wth a unform crcular moton. The work done by the gravtaton force on earth s: A. Postve B. Negatve C. Zero D. Impossble to determne Scalar Product of Two Vectors The scalar product of two vectors s wrtten as AB It s also called the dot product AB A B cos s the angle between A and B Appled to work, ths means W Frcos Fr Scalar Product, cont The scalar product s commutatve A B B A The scalar product obeys the dstrbutve law of multplcaton A BC A BA C Dot Products of Unt Vectors ˆˆ ˆˆ ˆ ˆ jjkk1 jˆˆ kˆˆ jk0 Usng component form wth vectors: A A ˆ ˆ ˆ Ayj Azk BB ˆ ˆj kˆ By Bz AB A B A B A B y y z z Work Done by a Varyng Force Assume that durng a very small dsplacement,, F s constant For that dsplacement, W ~ F For all of the ntervals, f W F 3
4 Work Done by a Varyng Force, cont f lm f 0F F d Therefore, W f F d The work done s equal to the area under the curve between and f Work Done By Multple Forces If more than one force acts on a system and the system can be modeled as a partcle, the total work done on the system s the work done by the net force W W F d net f In the general case of a net force whose magntude and drecton may vary f W Wnet Fdr Work Done by Multple Forces, cont. If the system cannot be modeled as a partcle, then the total work s equal to the algebrac sum of the work done by the ndvdual forces W net W by ndvdual forces Remember work s a scalar, so ths s the algebrac sum Work Done By A Sprng A model of a common physcal system for whch the force vares wth poston The block s on a horzontal, frctonless surface Observe the moton of the block wth varous values of the sprng constant Hooke s Law The force eerted by the sprng s F s =  k s the poston of the block wth respect to the equlbrum poston ( = 0) k s called the sprng constant or force constant and measures the stffness of the sprng Ths s called Hooke s Law Hooke s Law, cont. When s postve (sprng s stretched), F s negatve When s 0 (at the equlbrum poston), F s 0 When s negatve (sprng s compressed), F s postve 4
5 Hooke s Law, fnal The force eerted by the sprng s always drected opposte to the dsplacement from equlbrum The sprng force s sometmes called the restorng force If the block s released t wll oscllate back and forth between and Clcker Queston The work done by a sprng to be compressed from =0 to =a s: W 0 = ka 2 /2. What s the work done by the same sprng to be compressed from =a to =2a? A. W 0 B. 2W 0 C. 3W 0 D. 4W 0 E. 5W 0 Work Done by a Sprng Identfy the block as the system Calculate the work as the block moves from =  ma to f = 0 W f F d 0 k d 1 2 ma ma 2 k s The total work done as the block moves from ma to ma s zero Work Done by a Sprng, cont. Assume the block undergoes an arbtrary dsplacement from = to = f The work done by the sprng on the block s f Ws kd k k f 2 2 If the moton ends where t begns, W = 0 Sprng wth an Appled Force Suppose an eternal agent, F app, stretches the sprng The appled force s equal and opposte to the sprng force F app = F s = (k) = k Work done by F app s equal to ½ k 2 ma The work done by the appled force s f Wapp kd kf k 2 2 Knetc Energy Knetc Energy s the energy of a partcle due to ts moton K = ½ mv 2 K s the knetc energy m s the mass of the partcle v s the speed of the partcle A change n knetc energy s one possble result of dong work to transfer energy nto a system 5
6 Knetc Energy, cont Calculatng the work: f vf v f W F d mad W mvdv 1 1 W mv mv 2 2 W K K K net f 2 2 f Clcker Queston In a symmetrc projectle moton, the ntal speed s fed. When the object lands, the knetc energy would be mamum, f the angle of ths moton from horzontal s: A. 0 B. 90 degrees C. 30 degrees D. 45 degrees E. Does not matter WorkKnetc Energy Theorem The WorkKnetc Energy Theorem states W = K f K = K When work s done on a system and the only change n the system s n ts speed, the work done by the net force equals the change n knetc energy of the system. The speed of the system ncreases f the work done on t s postve The speed of the system decreases f the net work s negatve Also vald for changes n rotatonal speed WorkKnetc Energy Theorem Eample The normal and gravtatonal forces do no work snce they are perpendcular to the drecton of the dsplacement W = F W = K = ½ mv f20 Lecture Quz When a car goes around a crcular curve on a level road, A. no frctonal force s needed because the car smply follows the road. B. the frctonal force of the road on the car ncreases when the car s speed decreases. C. the frctonal force of the road on the car ncreases when the car s speed ncreases. D. the frctonal force of the road on the car ncreases when the car moves to the outsde of the curve. E. there s no net frctonal force because the road and the car eert equal and opposte forces on each other. 6
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