1、7. Conservation of Energy能量守恆1. Conservative & Non-conservative Forces守恆和非守恆力2. Potential Energy位能3. Conservation of Mechanical Energy機械能的守恆4. Potential Energy Curves位能曲線How many different energy conversions take place as the Yellowstone River plunges over Yellowstone Falls?當黃石河在黃石瀑布下墮時，經歷了幾種能量轉換。P.
2、E. K.E., sound, & heat位能 動能，聲音，和熱Work against conservative force (path independent),對守恆力作功 (與路徑無關) ，e.g., gravity, & static E 如重力和靜電場:Work can be stored potential energy. 功可存 位能Work against non-conservative force (path dependent),對非守恆力作功 (隨路徑而定) ，e.g., friction & E+B(t) 如摩擦和 E+B(t) :For friction, wo
3、rk is dissipated heat & entropy.摩擦: 功有消耗 熱和熵7.1. Conservative & Non-conservative Forces守恆和非守恆力F is conservative if 如果每條密閉路徑 C 都是0CdFrfor every closed path C.F 便是守恆的。BABAWdFris path-independent與路徑無關i.e.,0CdFrBABAWdFris path-dependent由路徑决定WABWBA + WAB = 0 WAB = WBA= WAB F is non-conservative if there
4、is a closed path C such that如果有一條密閉路徑 C 是F 便是非守恆的。WBAWABMathematicaExample: Work done on climber by gravity重力對爬山者所作功Going up 上山時 : W1 = ( m g ) h = m g hGoing down 下山時 : W2 = ( m g ) ( h) = m g hRound trip 來回 : W = W1 + W2 = 0Horizontal displacement requires no work.水平方向的位移不需作功。Gravity is conservati
5、ve 重力是守恆的.m g Fjh rjh rjm g FjClimber does (+) work.爬山者作 (正) 功。(+) work done on climber.爬山者被作 (正) 功。Example: Work done on trunk by friction摩擦對櫃子所作功Going right 往右時 : W1 = ( m g ) L = m g LGoing left 往左時 : W2 = ( m g ) ( L) = m g LRound trip 來回 : W = W1 + W2 = 2 m g L 0Friction is non-conservative 摩擦是
6、非守恆的.L rifm g FiL rifm gFiMover always does (+) work against friction.搬櫃子的人永遠都要對摩擦作 (正) 功。GOT IT 懂嗎? 7.1.If it takes the same amount of work to push a trunk across a rough floor as it does to lift a weight to the same distance straight upward.如果把一個櫃子推過一段粗糙樓板地所需的功，和把一個砝碼朝上舉起到同樣距離所需的功一樣。How do the amo
7、unts of work compare if the trunk & weight are moved along curved paths between the same starting & end points?如果起和終點不變，但櫃子和砝碼的路徑都改成曲線，兩者所需的功比較起來會怎樣?Ans. Work is greater for the trunk.答： 櫃子需要的功較大。7.2. Potential Energy位能Conservative force 守恆力 :Potential energy = stored work位能 = 所存的功= ( work done by f
8、orce ) = (力作的功)BABABAUWd FrNote: only difference of potential energy matters. 注 : 祇有位能差算數。1-D case: 一維時： 21xxUF x dx Constant F :不變的力 F :21UF xx Gravitational Potential Energy 重力位能Um g h Horizontal component of path does not contribute. 路徑的水平分量無貢獻Vertical lift:垂直上提mghUmg y m gExample 7.1. Riding the
9、 ElevatorA 55 kg engineer takes an elevator from her office on the 33rd floor to the 59th floor.一個 55 kg 的工程師乘電梯從她辦公室的 33 樓到 59 樓。Later, she descends to street level. 稍後，她下去一樓。If she takes her office as the zero of potential energy, 如果她把她的辦公室當成位能的原點，and if the distance between floors is 3.5 m,而且樓層的高
10、度是 3.5 m ，whats her potential energy 在(a) in her office.她的辦公室，(b) on the 59th floor. 59 樓，和(c) at street level ?一樓時， 她的位能為何？(a)330U(b)595933Um g hh2559.8/3.55933kgm sm49 kJ(c)1133Um g hh2559.8/3.51 33kgm sm60 kJ 33591Application: Pumped Storage 應用: 抽水式存能Northfield Mountain Pumped Storage Project, MA
11、, USA美國麻省北田山抽水式儲能計劃Excess electric energy stored by pumping water to higher ground.(see Prob. 29)過剩的電能以抽水到高地的方式儲存 (參閱習題29)Elastic Potential Energy 彈性位能 21xxUF x dx Ideal spring:理想彈簧210 xxk xxdx 212012xxkxx x00Uat xx 020012xxU xk xxdxk xxparabolic拋物線x x0 x = x0 x x0壓縮壓縮 平衡平衡伸長伸長U is always positive U
12、 永遠都是正的Fkx 0k xx x0 = equilibrium position平衡位置222102112kxxxxx22201012kxxxxLetSetting x0 = 0 : 212U xk x212kx222112kxx Example 7.2. Springs vs Gasoline彈簧對汽油A cars suspension consists of springs with overall effective k = 120 kN/m.一輛汽車的避震器內的彈簧組合的有効 k = 120 kN/m.How much these springs need be compresse
13、d to store the same amount of energy as in 1 gram of gasoline?這些彈簧要壓縮多少，才能儲存與 1 公克汽油相同的能量？From Appendix C: Energy contents of gasoline = 44 MJ / kg.從附錄 C ：汽油的能含量 = 44 MJ / kg。33211044 10/120/2kgkJ kgkN m x221115xm0.86xm86cmSprings cant compete with gasoline as energy source.彈簧在能源上無法與汽油相比。Example 7.3
14、. Climbing Rope爬山索Climbing ropes are springy to cushion falls.爬山索都有彈性，以減輕下墜時的損傷。Consider rope with F = k x + b x2 , where k = 223 N/m, b = 4.10 N/m2.茲有一索，其 F = k x + b x2 ，且 k = 223 N/m, b = 4.10 N/m2.Find the potential energy when its stretched 2.62 m, taking U = 0 at x = 0.求把它拉長 2.62 m 時的位能，設 x =
15、0 時 U = 0 。20 xUk xb xdx 2301123xk xb x 231123k xb x23211223/2.624.10/2.6223N mmN mm741Jwhich is 3% less than U = k x2 = 765 J.比 U = k x2 = 765 J 少 3% 7.3. Conservation of Mechanical Energy機械能的守恆netKWcncWWncUW ncEKUWMechanical energy 機械能:EKULaw of Conservation of Mechanical Energy 機械能守恆定律: 0EKUif 若0
16、ncW( no non-conservative forces )無非守恆力constantEKUExample 7.4. Tranquilizing an Elephant麻醉大象A biologist uses a spring-loaded gun to shoot tranquilizer darts into an elephant.一生物學家用彈簧槍把麻醉鏢射入一頭大象。The guns spring has k = 940 N/m, & is compressed x = 25 cm before firing a 38-g dart.槍的彈簧 k = 940 N/m，壓縮了 x
17、 = 25 cm 後射出 38-g 的鏢。Assuming the gun points horizontally, at what speed does the dart leave the gun?設槍朝水平方向，鏢離槍口時的速率為何？Initial state 初態 :20012EUkxFinal state終態 :212EKm vkvxm0EE23940/25 1038 10N mvmkg39/m sProblem is harder to solve by 2nd law.此題以第二定律來解會比較難。Example 7.5. Spring & Gravity彈簧比重力A 50-g b
18、lock is placed against a spring at the bottom of a frictionless slope.在一個無摩擦力的斜坡底，一 50-g 的方塊緊靠着一個彈簧。The spring has k = 140 N/m and is compressed 11 cm.彈簧的 k = 140 N/m ，且已壓縮了 11 cm 。When the block is released, how high up the slope does it rise?放開方塊後，它能沿斜坡升到多高？Initial state 初態 :200012EUk xFinal state
19、終態 :EUm g h202k xhm g0EE2232140/11 102 50 109.8/N mmhkgm s1.7 mGOT IT? 7.3. Bowling Ball保齡球A bowling ball is tied to the end of a long rope and suspended from the ceiling.一個保齡球由一根長繩子吊在天花板下。A student holds the ball to her nose, then releases it from rest. 一個學生把球捧在她的鼻子前，然後放手。Should she duck as it swin
20、gs back?它擺回來時她要不要往後退？Example 7.6. Sliding Block滑動的質塊A block of mass m is launched from a spring of constant k that is compressed a distance x0.一質量為 m 的質塊，由一條長度壓縮了 x0 ，常數為 k 的彈簧彈出。The block then slides on a horizontal surface of frictional coefficient .質塊然後在一摩擦系數為 的水平面上滑動。How far does the block slide
21、 before coming to rest? 質塊要滑多遠才能停下來？Initial state 始態：200012EUk xWork done against friction 對摩擦作的功:ncfWfx 202k xxm g 100EEm gx Final state 終態:0E Launch 發射時 :21112EKm vConservation of energy :能量守恆：10ncEW7.4. Potential Energy Curves位能曲線Frictionless roller-coaster track 無摩擦的雲霄飛車軌道How fast must a car be
22、coasting at point A if its to reach point D ?車在A點要走多快才能達到D點？ACEUCriterion 條件:212AACm vm g hm g h2ACAvghhturning points轉折點能量高，車可能量高，車可到任何地方到任何地方車限於此兩點之間車限於此兩點之間車限於此車限於此potential barrier位能屏障potential well位能井EKU0KEUExample 7.7. H2 氫分子Near the bottom of the potential well of H2 在 H2 的位能井底附近 ， U = U0 + a
23、 ( x x0 )2 ,where 其中 U0 = 0.760 aJ, a = 286 aJ / nm2 , x0 = 0.0741 nm. ( 1 aJ = 1018 J )What range of atomic separation is allowed if the total energy is 0.717 aJ ? 如果總能量是 0.717 aJ，原子間的距離在甚麼範圍？Turning points轉折點：EU200Ua xx00EUxxa 20.7170.760286/aJaJaJ nm 0.0123nm 0.08640.0618xnm tonm00.074xnmForce &
24、Potential Energy力與位能Force slope of potential curve 力 位能曲線的坡度U FrFx ( x along direction of F ) x 沿 F 的方向UFx 0limxUFx dUdx dU FrWhen the curve is rising to the right, force is to the left.曲線向曲線向右右爬升時，力朝爬升時，力朝左左。At peaks & valleys, theres no force.在峯頂和谷底處都沒有力。在峯頂和谷底處都沒有力。Force is greater where the curv
25、e is steeper.力在曲線陡處較大。力在曲線陡處較大。U F,UUUxyz Gradient梯度GOT IT 懂嗎?. 7.4.Below is the potential energy curve for an electron in a microelectronic device.下面是電子在一微電子元件內的位能曲線(a) Find the point where the force on the electron is greatest.求電子受力最大之點。(b) Find the rightmost position possible if the electron has
26、total energy E1 .電子的總能量為 E1 時，它可達的最右邊位置為何？(c) Find the leftmost position possible if the electron has total energy E2 & starts out to the right of D.若電子的總能量為 E2 ，且開始時在 D 的右方，它可達的最左邊位置為何？(d) Find a point where the force on the electron is zero. 找出電子受力為零之點。(e) Find a point where the force on the electron points to the left.找出電子受力朝左之點。In some cases, there may be multiple answers.有些情形的答案不止一個。(E)(B)(A,D)(C)(B,E)
