Lesson 8: Work, Energy, Power (Sections ) Chapter 6 Conservation of Energy

Transcription

1 Lesson 8: Wok, negy, Powe (Sectons ) Chapte 6 Conseaton o negy Today we begn wth a ey useul concept negy. We wll encounte many amla tems that now hae ey specc dentons n physcs. Conseaton o enegy Wok Potental Powe In some cases, t can be agued that these tems hae a physcs denton that s smla to ts eeyday usage. The Law o Conseaton o negy The total enegy n the unese s unchanged by any physcal pocess: total enegy beoe = total enegy ate om page 9: In odnay language, conseng enegy means tyng not to waste useul enegy esouces. In the scentc meanng o conseaton, enegy s always conseed no matte what happens. Conseaton o enegy s one o the ew unesal pncples o physcs. No ecepton has ee been ound. It apples to physcal, chemcal, and bologcal systems. Also om page 9: Some poblems can be soled usng ethe enegy conseaton o Newton s second law, so t always pays to consde both methods. Usually the enegy method s ease. Usng Newton s second law noles ecto methods snce oces ae ecto quanttes. In most poblems, enegy noles scala quanttes, whch ae much ease to deal wth (and moe amla). When decdng whch o these two appoaches to use to sole a poblem, ty usng enegy conseaton st. Kangaoos ae mentoned at the begnnng o ths chapte. oms o negy Page 93: At the most undamental leel thee ae thee knds o enegy. Knetc enegy enegy due to moton. Potental enegy enegy due to nteacton a. Gatatonal potental enegy nteacton between the ath and a mass b. lastc potental enegy nteacton between a spng and a mass 3. Rest enegy ntenal enegy to a body Lesson 8, page

2 Lesson 8: Wok, negy, Powe (Sectons ) negy s measued n Joules (J). Wok Suppose a oce causes an object to moe a dstance paallel to The wok done by a constant oce s dened as DO NOT MMORIZ! W Suppose a constant oce causes an object to moe along not paallel to Lesson 8, page

3 Lesson 8: Wok, negy, Powe (Sectons ) whee s the angle between and W cos. MMORIZ. Wok s a scala quantty and can be poste, negate, o zeo. Poste: between 0 o and 90 o Negate: between 90 o and 80 o Zeo: = 90 o o Usually tenson and nomal oce do no wok. The ecepton s (c) below. The wok done by seeal oces can be ound om the net oce W total W W W cos Wok and Knetc negy Choosng the as along net, (usng = cos ) We had an equaton om Chapte W net total net ma a N Lesson 8, page 3

4 Lesson 8: Wok, negy, Powe (Sectons ) a ( ) Substtutng nto Wtotal W total m( Snce the net oce s n the -decton, ay and az ae both zeo. Only the -component o the elocty changes ) ( y z ) ( y z ) and W total The tanslatonal knetc enegy s dened as The wok-knetc enegy theoem s m( K m W total K Whle ths epesson s oundatonal to ths chapte, do not memoze. We shall dee a moe useul om. ) Gatatonal Potental negy () The weght can do wok. Toss a ball up and t slows down. In ou new language, ts knetc enegy deceases. The knetc enegy s coneted nto anothe om o enegy we call gatatonal potental enegy. The change n gatatonal potental enegy U ga W ga In tems o poston U ga mgy Ths equaton s tue een the object does not tael n a staght lne. Lesson 8, page 4

5 Lesson 8: Wok, negy, Powe (Sectons ) The gatatonal potental enegy s U ga mgy The nal om o ou elaton s W nc K U Ths s t. You need to know t. We hae anothe enty nto ou cause and eect table. Anothe useul om s K U W W nc nc K U K K U U K U Wnc s the wok done by nonconseate oces. Nonconseate oces do not hae a potental enegy. A good eample s cton. The mechancal enegy s mech K U Conseaton o Mechancal negy When nonconseate oces do no wok, mechancal enegy s conseed: mech, mech, The zeo o potental enegy s abtay. Choose whatee s conenent, usually the gound. The wok done by a conseate oce s ndependent o the path taken. Poblem: A 0.0-kg ball s thown at 5.0 m/s om a 0 m towe. What s ts speed when t s 5.0 m aboe the gound? What s ts speed when t hts the gound? Soluton: Use the conseaton o mechancal enegy. mgy gy m ech, U K m U m ech, gy K mgy m Lesson 8, page 5

6 Lesson 8: Wok, negy, Powe (Sectons ) g( y y ) (5m/s) (9.8m/s )(0m 5m). m/s At the bottom, y = 0, = 4.9 m/s. Notce the decton o the thow s not mentoned. No matte whch way the ball s thown, t has the same speed at the same heght! Ths s ey had to poe usng Newton s second law. Gatatonal Potental negy () o objects a om the ath, U GmM Whle ths looks ey deent om mgy, the tet (p. 08) shows they ae equalent. ample 6.8 What s the escape elocty o the ath? Soluton: Use conseaton o enegy. We cannot use U = mgy o an object a om the eath! Gmm mech, U K U m mech, K Gmm m When an object escapes om the ath, ath s gaty s not actng on t and. I the object baely escapes the ath, = 0. Gmm m 0 Gm The statng poston s on the ath s suace and = R = m. The mass o the ath s m = kg. The escape elocty s Gm (6.670 Nm / kg )( (6.370 kg) 4 kg),00m/s Ths s Lesson 8, page 6

7 Lesson 8: Wok, negy, Powe (Sectons ) m m, s 609m Wok done by a aable oce. The adantage o enegy methods s seen when dealng wth a aable oce. Suppose the oce changes wth dsplacement. An eample o such a dece s shown on the net page. How do we calculate the wok done? We dde the oeall dsplacement nto a sees o small dsplacements,. Oe each o the smalle dsplacements, the oce s almost constant. The wok done o a small dsplacement s W m s The total wok s the aea unde the cue shown aboe. Ths pocedue was used to nd the gatatonal potental enegy o an object a om the ath GMm U Hooke s Law and Ideal Spngs The oce eeted by the ache nceases as the bowstng s dawn back. Robet Hooke poposed an deal spng whee the oce s popotonal to the dsplacement k The dsplacement o the spng om the elaed poston s. The constant k, s called the spng constant. It s measued n N/m and t ges the stength o the spng. The lage k s, the ste the spng. The mnus sgn ndcates that the spng s stetched to the ght, the spng pulls back to the let and ce esa. Lesson 8, page 7

8 Lesson 8: Wok, negy, Powe (Sectons ) lastc Potental negy As the spng s pulled (o pushed) om ts elaed poston, wok s done on t. The wok done s ndependent o the path taken and accodngly, a potental enegy can be dened. The elastc potental enegy s ound to be U elastc k Note that U = 0 when = 0. Poblem 09. A spng gun (k = 8 N/m) s used to shoot a 56-g ball hozontally. Intally the spng s compessed by 8 cm. The ball loses contact wth the spng and leaes the gun when the spng s stll compessed by cm. What s the speed o the ball when t hts the gound.4 m below the gun? Soluton: Ths appeas to be a pojectle poblem. It s an enegy poblem wth two potental eneges. Take the ntal poston to be at the top and the nal poston just beoe t hts the gound, Lesson 8, page 8

9 Lesson 8: Wok, negy, Powe (Sectons ) 0 k mgy m k 0 K U K U gy (9.8m/s )(.4m) (8N/m)((0.8m) 6.04m/s k( )/ m (0.m) )/(0.056kg) Powe Oten the ate o enegy coneson s mpotant. We use the tem powe to ee to the ate o enegy coneson. Oe an etended tme, the aeage powe s P a t Powe s measued n Joules/second o watts (W). Be caeul wth W o wok and W o watts. Remembe that wok changes the mechancal enegy o the system. P t cos t cos Poblem 95. A 500-kg ca coasts n neutal down a.0º hll. The ca attans a temnal speed o 0.0 m/s. (a) How much powe must the engne dele to de the ca on a leel oad at 0.0 m/s? (b) I the mamum useul powe that can be deleed by the engne s 40.0 kw, what s the steepest hll the ca can clmb at 0.0 m/s? Soluton: At the temnal speed, the -component o the weght o the ca s opposed by a esstance. When coastng down the hll, the ee body dagam s o the -component (along the nclne) Lesson 8, page 9

10 Lesson 8: Wok, negy, Powe (Sectons ) mgsn a a ma 0 mgsn (500kg)(9.8m/s )sn 53N (a) Now the ca s mong along a hozontal oad at constant elocty. om the BD ound at the top o the net page: moto a moto ma 0 a The powe deleed by the engne when the ca moes at 0 m/s (notce the angle below s not º!) s Pmoto moto cos a cos0 (53N)(0m/s)cos0 0,300W (b) Agan a ee body dagam s helpul. Lesson 8, page 0

11 Lesson 8: Wok, negy, Powe (Sectons ) Clmbng wth constant speed, a = 0, moto mgsn ma a moto 0 a mgsn The mamum powe suppled by the engne s 40.0 kw. The powe s Ths coesponds to a 5.8º angle. P m oto cos ( a mgsn ) cos0 a mgsn P a sn mg W (53N)(0m/s) (500kg)(9.8m/s )(0m/s) 0.0 Lesson 8, page