rk Dne by a Cntant Frce 6.-6.4 Energy & rk F N m jule () J rk Dne by a Cntant Frce Example Pullng a Sutcae-n-heel Fnd the wrk dne the rce 45.0-N, the angle 50.0 degree, and the dplacement 75.0 m. 3 ( F cθ ) [( 45.0 N) c50.0 ]( 75.0 m) 70 J 4 rk Dne by a Cntant Frce ( F cθ ) c0 c90 0 c80 5 Example Acceleratng a Crate A truck acceleratng at a rate +.50 m/. The ma the crate 0 kg and de nt lp. The magntude the dplacement 65 m. hat the ttal wrk dne n the crate by all the rce actng n t? Slutn: The key t th prblem t be able t denty that tatc rctn,, the rce that accelerate the crate t e alng wth the truck: The nerta the crate wuld make t tend t all behnd the truck ( a t e wth n acceleratn). Th tendency caue t act n the rward drectn, and t keep the crate at ret wth repect t the truck, r t accelerate at the ame acceleratn a the truck. FBD the crate: a.5 m/ 6
Example Acceleratng a Crate (cnt d) Example Acceleratng a Crate (cnt d) The angle between the dplacement and the nrmal rce 90. a.5 m/ The angle between the dplacement and the rctnal rce 0. a.5 m/ The angle between the dplacement and the weght al 90. ence, the wrk dne due t and F N are: Nrmal rce ( F c 90 ) 0 N ( c( 90 )) 0 7 A dcued n p.7, prvde the rce needed t e the crate wth the ame acceleratn a the truck. ence, mut equal t ma. Nte that nt µ F N here. hy? ma ( 0 kg)(.5m ) 80N 4 [( 80N) c0]( 65 m). 0 J 8 The rk-energy Therem and Knetc Energy Cnder a cntant net external rce actng n an bject. The bject dplaced a dtance,, n the ame drectn a the net rce, ΣF. The rk-energy Therem and Knetc Energy rk dne (n Jule) by a cntant rce n ma, m: m a ( ) m ( v v ) F The wrk mply ( F ) ( ma) 9 v v + ( ax) ( ax) ( ) v v Th de mut be an energy me rt nce t ha unt Jule. A t clearly are rm the peed the bject, t a 0 knd knetc energy. Dentn Knetc Energy DEFINITION The knetc energy KE an bject wth ma m and peed v gven by KE The rk-energy Therem th the dentn KE, and the equatn gven n p., ne may deduce the wrk-energy therem. TE ORK-ENERGY TEOREM hen a net external rce de wrk n an bject, the knetc energy the bject change accrdng t KE KE Ntce that v > v 0 r KE > KE > 0 v < v 0 r KE < KE < 0
Meanng Negatve rk Dne ( F c ) F 0 > 0 mean that the cmpnent F alng parallel t. Ptve wrk dne ncreae the knetc energy the ytem. ( F c ) F 80 < 0 mean that the cmpnent F alng ppte t. Negatve wrk dne lwer the knetc energy the ytem. 3 Interpretatn Frce-Dtance Curve Recall that wth v a t, v the area under a plt a v. t. Nw r wrk dne, F cθ. e can mlarly ner that the area under a plt Fcθ v. d, where d the dtance traveled. I the prblem ne dmennal that F ether parallel r antparallel t, the area under a plt F v. d. 4 Interpretatn Frce-Dtance Curve Cnder an bject that thrwn up nt the ar wth an ntal peed, v 0 and then all back t t ntal ptn. Ignre ar retance. Sketch the rce-dtance (F-d) curve r th mtn. F d F (Ntce that d dtance, a calar here.) In the up jurney, the drectn ppte t F negatve n the rt hal the F-d plt. But upn d >, the drectn becme ppte t and F n the ecnd hal the plt turn ptve. Quetn: In term, what are () the ntal KE, KE 0, the bject, () ttal wrk dne by n the bject and (3) the nal KE the bject when t return t t tartng pnt? v 0 5 F F Quetn: In term, what are () the ntal KE, KE 0, the bject, () ttal wrk dne by n the bject and (3) the nal KE the bject when t return t t tartng pnt? Anwer: () KE. Ater the end the up jurney, the bject velcty zer and KE 0. And the area under the F-d curve upn the up jurney,. S, we have 0 KE 0 KE 0. () Snce the jurney beng aked n the quetn cncern the bject tartng rm d0 t d, area under the F-d curve rm d 0 t d. Fr the abve F-d curve, the area under the curve rm d 0 t d zer. (Nte that the area negatve between 0<d< and ptve between <d<.) (3) KE. Snce 0 rm () and KE 0 rm (), we deduce that 0 KE. Or, KE KE 0. 6 d v 0 Example 3 Deep Space Example 3 Deep Space (cnt d) The ma a pace prbe 474 kg and t ntal velcty 75 m/. I a rce 56.0 mn generated by the prbe prpeller that drve t t accelerate thrugh a dplacement.4 0 9 m, what the prbe nal peed? [( F ) cθ ] rk dne by a cntant rce rk-energy therem - 9 ( 5.60 0 N) c0 (.4 0 m) ( 474 kg) v ( 474 kg)( 75m ) 7 v 805m 8
Example 4 Skatng dwn a hll Fnd an expren r the nal peed v a ker kng dwn a hll a hwn belw. The cecent knetc rctn µ k. Gravtatnal Ptental Energy Cnder an bject that all rm heght h 0 t heght h under. The wrk dne n the bject by : In th cae the net rce + n5 µ k v F n5 c 5 k ( F cθ ) ( ) h h v + g n5 µ kg c 5 / m 9 0 Th hw that the ntal heght h 0 bgger than the nal heght h, the bject gan knetc energy. But h 0 < h, the bject le knetc energy. Ntce that g +9.8 m/ n th equatn. +y Gravtatnal Ptental Energy Example 5 A Gymnat n a Tramplne ( ) h h Fr the tw cae hwn at rght, whch ha the hgher nal peed? (a) Cae (b) Cae (c) They are the ame. Cae Cae The gymnat leave the tramplne at an ntal heght.0 m and reache a maxmum heght 4.80 m bere allng back dwn. hat wa the ntal peed the gymnat? h 0.0 m h 4.80 m v 0 m/ v 0? Example 5 A Gymnat n a Tramplne (cnt d) ( h h ) v g v ( h h ) h 0.0 m h 4.80 m v 0 m/ v 0? and ( h h ) ( )(.0 m 4.80 m) 8.40m 9.80m 3 Dentn PE h h By allwng an bject t all under, we can btan ptve wrk dne rm the. Th ugget that there a ptental energy acated wth an bject beng ptned at a hgher alttude relatve t a lwer alttude. DEFINITION: Gravtatnal ptental energy (PE) the energy that an bject ma m ha by vrtue t ptn relatve t the urace the earth. That ptn meaured by the heght h the bject relatve t an arbtrary zer level: PE h N m jule ( J) 4
Dentn Cnervatve Frce Dentn Nncnervatve Frce DEFINITION: Vern A rce cnervatve when the wrk t de n a ng bject ndependent the path between the bject ntal and nal ptn. Vern A rce cnervatve when t de n wrk n an bject ng arund a cled path, tartng and nhng at the ame pnt. CORROLLARY rm the dentn cnervatve rce: Vern A rce nn-cnervatve when the wrk t de n a ng bject ndependent the path between the bject ntal and nal ptn. Vern A rce nn-cnervatve when the wrk t de n an bject nn-zer by ng arund a cled path, tartng and nhng at the ame pnt. 5 6 Cnervatve v. Nncnervatve Frce w t perceve that gravtatnal rce cnervatve? Vern A rce cnervatve when the wrk t de n a ng bject ndependent the path between the bject ntal and nal ptn. Gravtatnal rce ate th crtern. ( ) h h 7 8 Vern A rce cnervatve when t de n wrk n an bject ng arund a cled path, tartng and nhng at the ame pnt. Gravtatnal rce al ate th crtern. Pleae reer t ur dcun n F-d curve. Alternatvely, cnder a pendulum. The pendulum h alway h regan the ame peed when t return t the ame heght. Th hw that the wrk dne by arund a cled path zer. h h ( ) 0 m( v v ) An example nncnervatve rce An example a nncnervatve rce the knetc rctnal rce. ( F cθ ) c80 The wrk dne by the knetc rctnal rce alway negatve. Thu, t mpble r the wrk t de n an bject by ng t arund a cled path t be zer. k k v 0 v 9 30
Cnervatve Frce and Ptental Energy A ptental energy can be dened nly when the wrk dne depend n the ntal and nal ptn the bject nly and ndependent the path taken. ence, a ptental energy (uch a gravtatnal ptental energy, PE h) can be dened nly r cnervatve rce. rk-energy Therem (Revted) In nrmal tuatn, bth cnervatve and nncnervatve rce act multaneuly n an bject, the ttal wrk dne by the net external rce can be wrtten a c + nc KE KE KE c h h PE PE PE 3 3 rk-energy Therem (Revted) Cnceptual example 6 Energe n weght ltng c + nc KE PE + nc The weght-lter de ptve wrk t the weght durng the up cycle. hy the peed the weght equal t zer at the end the up cycle? TE ORK-ENERGY TEOREM r nc nc nc KE + PE KE KE + PE PE + KE + PE KE + PE 33 The weght-lter de negatve wrk t the weght durng the dwn cycle. here ha the lt energy gne? 34 Example 7 Tw blck cnnected by a trng Tw blck wth mae m kg and m kg are cnnected by a male trng, whch paed ver a rctnle pulley. The blck wth m hang n ar. Neglect rctn. I the tw blck tart rm ret, what the nal peed the tw blck ater they have traveled a dtance 0. m? (Take g 0 m/ ) Slutn: nc + KE + PE KE + PE KE 0 J, KE (m +m )v /, PE PE m g, nc (T)(-) + T() 0 nc n m nc n m T T F N m g m g N.B.. The dtance traveled by the tw blck are the ame.. The peed the tw blck are the ame. 3. I knetc rctn k nt zer, nc k 4. F N and m g de n wrk nce they are perpendcular t the dplacement m. 0 KE m g (m +m )v / m g v [m g/(m +m )] / m/ 35