保守力與勢(shì)能s

1、保守力與勢(shì)能1Summary on work 功的性質(zhì)(1) Generally speaking, work is dependent on the path. 功是過程量，一般與路徑有關(guān)。(2) Work is a scalar with magnitude and signs. 功是標(biāo)量，但有正負(fù)。(3) Work done by resultant of all forces equals 合力的功為各分力的功的代數(shù)和。2Case study 1: Work done by gravitation 引力的功兩個(gè)質(zhì)點(diǎn)之間在萬有引力作用下相對(duì)運(yùn)動(dòng)時(shí) ，以M所在處為原點(diǎn), M指向m的方向?yàn)槭?/p>

2、徑r的正方向。m受的引力方向與矢徑方向相反。m在M的萬有引力的作用下從a 點(diǎn)運(yùn)動(dòng)到b點(diǎn)，萬有引力的功:Gravitational work: independent of the path of the body; depends only on the starting and ending pointsBut if Mm, what is the work?3Example: 質(zhì)量為M、m的兩球原來相距為a，在萬有引力作用下逐漸靠近至相距為b，求在此過程中引力所作的功。abSolution I: In Frame MMm4abCr1r2Solution II: Re. center of

3、massM: m: M:m:5各個(gè)力的功與參考系有關(guān)，但一對(duì)力的功與參考系的選擇無關(guān)。質(zhì)心系里，內(nèi)力的功與質(zhì)量成反比。對(duì)小質(zhì)量物體作功占主要地位。引力的功只與物體系統(tǒng)的初始和最終相對(duì)位置有關(guān)，與路徑無關(guān)。6Case study 2: Work done by elastic forces kOabkOacbOakkOabkOa0Work done by elastic forces: independent of the path of the body; depends only on the starting and ending pointsRule of signs:Work can

4、be stored and recovered.7Conservative and nonconservative forces:Work done by a conservative force 保守力的功：(1) Reversible, “work” can be stored in a “BANK”; (2) Independent of the path of the body; (3) Zero work for closed path.ABL1L2等價(jià)8Energy is defined as the ability to do work.Work is defined as th

5、e transfer of energy.9Potential energy(勢(shì)能) Energy associated with the position of a system.Stored in a system, later recovered.與相互作用物體的位置有關(guān)的能量。 Work by conservative forces potential energy勢(shì)能的增量等于保守力所做功的負(fù)值.10 (1)保守力做功 勢(shì)能Gravitational potential energyElastic potential energy11引力可以通過加速度被“創(chuàng)造出來”，和被“消滅掉”：

6、勢(shì)能是否能被“創(chuàng)造”或“消滅”? 12超重、失重平動(dòng)參考系：附加重力勢(shì)能轉(zhuǎn)動(dòng)參考系：慣性離心力有沒有勢(shì)能？勢(shì)能的形式是什么？非慣性參考系：慣性力的勢(shì)能問題13關(guān) 于 勢(shì) 能：勢(shì)能總是與保守力相聯(lián)系。存在若干種保守力時(shí)，就可引進(jìn)若干種勢(shì)能。勢(shì)能的絕對(duì)數(shù)值與零勢(shì)能位形的選取有關(guān)，但勢(shì)能的差與之無關(guān)。不同保守力對(duì)應(yīng)的勢(shì)能，其零勢(shì)能位形的選取可以不同。(3) 勢(shì)能既然與各質(zhì)點(diǎn)間相互作用的保守力相聯(lián)系，因而為體系所共有。(4) 與勢(shì)能相聯(lián)系的是保守力對(duì)質(zhì)點(diǎn)系所作的總功，與參考系無關(guān)。14Why Sam can not lift the box? How can he make it?15 (2)勢(shì)能 保

7、守力梯度：16Example: find gravity from gravitational potential V=mgySolution:17(3) 1D Energy diagram 一維勢(shì)能曲線xU(x)The elastic potential energy for a spring彈簧振子的勢(shì)能曲線U(x)xOA hypothetical potential energy function假想的勢(shì)能曲線平衡條件Stable equilibrium 穩(wěn)定平衡？Unstable equilibrium不穩(wěn)定平衡？18Stable equilibrium 穩(wěn)定平衡條件：U(x)xOA

8、hypothetical potential energy function假想的勢(shì)能曲線19Summary on potential energyWork done by a conservative force can be represented in terms of a potential energyPotential energy is a shared property of the system, not one particle.(3) Components of force are In vector form 20Exercise: 質(zhì)量為m的物體放在光滑的水平面上,

9、m的兩邊分別與勁度系數(shù)(force constant,spring constant)為k1和k2的兩個(gè)彈簧相聯(lián)，若在右邊彈簧末端施以拉力f，問：（1）若以拉力非常緩慢地拉了一段距離l，它做功多少？（2）若拉到距離l后突然不動(dòng)，拉力做功又如何？fk1k2Solution: (1) f =k1x1=k2x2 x1+x2=x 21(2) f =k2x2=k2x 22Kinetic energy (動(dòng)能) Energy associated with the motion is T=1/2mv2 A moving object has the ability to do work A scalar

10、quantityT=1/2mv223Work-energy theorem: The work done by the net force on a particle equals the change in the particles kinetic energy動(dòng)能定理: 運(yùn)動(dòng)質(zhì)點(diǎn)的動(dòng)能的增加等于其它物體對(duì)它所做的功.24 Kinetic energy T=1/2mv2 is also the ability to do work 運(yùn)動(dòng)質(zhì)點(diǎn)速度改變而所作出的功 牛頓第三定律: 運(yùn)動(dòng)質(zhì)點(diǎn)的1/2mv2 值的減少正等于它所做的功. 運(yùn)動(dòng)質(zhì)點(diǎn)以力f 施于它物所作功: 25Conservation

11、 of mechanical energy 機(jī)械能守恒原理Work-energy theorem: The total work done by the net force on a particle equals the change in the particles kinetic energyPotential energy: Mechanical energy is conserved when only conservative forces do work.26 Mechanical energy:When only gravitation does work: (1) Near

12、the earths surface 質(zhì)點(diǎn)高度變化不大: (2) High above the earths surface 質(zhì)點(diǎn)高度變化很大: When only elastic force does work: 彈性力場(chǎng): 機(jī)械能守恒原理適合于由若干個(gè)物體組成的系統(tǒng)（如果系統(tǒng)內(nèi)只有保守力作功）27Work-energy theorem: 功能原理作用于質(zhì)點(diǎn)的力FAll forces Fc所作的功Wc可用勢(shì)能的減少來表示.Fd所作的功Wd不（可）用勢(shì)能的減少來表示.The work done by all external and nonconservative forces equals

13、the change in mechanical energy系統(tǒng)機(jī)械能的增量等于外力的功和非保守力內(nèi)力的功的總和。28The Law of Conservation of Energy 能量守恒定律Energy is never created or destroyed; it only changes form.在封閉系統(tǒng)內(nèi)，不論發(fā)生何種變化過程，各種形式的能量可以互相轉(zhuǎn)化，但能量的總和是恒量。功總是與能量變化或交換的過程相聯(lián)系著的，而能量代表著系統(tǒng)在一定狀態(tài)時(shí)所具有的特征，能量的量值只決定于系統(tǒng)的狀態(tài)，系統(tǒng)在一定的狀態(tài)時(shí)，就具有一定的能量。能量是系統(tǒng)狀態(tài)的單值函數(shù)。29例（P111，21

14、2）：豎直上拋的物體，最小應(yīng)具有多大的初速度V0才不再回到地球？（第二宇宙速度或逃逸速度）PRO（1）動(dòng)力學(xué)運(yùn)動(dòng)定律方法（2）動(dòng)力學(xué)運(yùn)動(dòng)定理方法0分析：在無限遠(yuǎn)處，機(jī)械能至少為0。30例（P221）：質(zhì)量為m的人造衛(wèi)星在環(huán)繞地球的圓軌道上,軌道半徑為,求衛(wèi)星的勢(shì)能動(dòng)能和機(jī)械能.(不計(jì)空氣阻力)（1）勢(shì)能（2）動(dòng)能RO31例（P213）：飛車演員從光滑的傾斜軌道自由滑下，并進(jìn)入半徑為R的豎直圓形軌道。問出發(fā)高度h0 最小應(yīng)為多少才得以通過豎直圓形軌道而不掉落下來？h0mgR隔離物體 具體分析(單側(cè)的約束運(yùn)動(dòng))建立坐標(biāo)（二維“自然”坐標(biāo)）內(nèi)稟運(yùn)動(dòng)運(yùn)動(dòng)方程32求解及分析（設(shè)圓心高度的重力勢(shì)能為0）根據(jù)機(jī)械能守恒原理演員通過豎直圓形軌道的條件N0（=）33例（P215）：半徑為R的圓環(huán)狀細(xì)管在水平面內(nèi)以勻角速繞A點(diǎn)轉(zhuǎn)動(dòng)。管的內(nèi)壁是光滑的。求解質(zhì)點(diǎn)M在其相對(duì)平衡位置附近作小振動(dòng)的周期，及約束反力。2mvONABMm2r34求解及分析在平衡點(diǎn)B周圍作