小班讲义高一物理第8讲牛顿运动定律的应用1—瞬时问题

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1、龙 文 教 育精品小班课程辅导讲义讲义编号 2013秋季 辅导科目： 物理 年级： 高一课 题第8讲：牛顿运动定律的应用1瞬时问题、弹簧问题教学目标1、理解牛顿运动定律的瞬时性，矢量性等。2、会运用牛顿运动定律解决物体与弹簧相连时的运动过程分析。教学重点、难点1掌握弹簧弹力和刚体弹力特点，会求任意时刻的力与加速度；2掌握弹簧弹力的特点，能合理应用牛顿运动定律解决有关弹簧问题，会讨论有弹簧的情况下力和加速度的变化，会讨论有弹簧的情况下物体的运动情况教学内容活动一、 完成问题，熟悉牛顿第二定律的应用例题1.如图所示，一轻质弹簧一端与墙相连，另一端与一物体接触（不相连），当弹簧在O点位置时弹簧没有形

2、变，现用力将物体压缩至A点，然后放手。物体向右运动至C点而静止， 图中AB段光滑，BC段粗糙。回答下列问题1. 分析物块在AO,OB,BC段各做什么运动？2. 试用v-t图像表示物块的运动。3. 物块在哪个位置加速度最大？若物块质量为m，劲度系数为k，OA长为x，则物体具有的最大加速度为？活动二、思考下列问题，学会分析涉及弹簧的物体运动过程例题2。一物体从某一高度由静止开始自由落下，落在直立固定的轻弹簧上，如图所示，在A点物体开始与弹簧接触，一直到C点时物体的速度为零。分析运动过程，回答问题（1）小球从初始位置O到C的运动过程中，可以分为哪几个阶段？（2）分析各个阶段的受力，并写出合力的表达式

3、？（3）定性分析物体从O到C的运动情况？（4）你能用v-t图像形象的表示出物体从O到C的运动过程吗？（5）定量计算，若物体的质量为1kg，弹簧的劲度系数k=500N/m，AC的距离是5cm，求物体速度最大时弹簧的形变量是多少？及物体到达C点时的加速度？活动三、适度拓展，进一步理解掌握涉及弹簧的物体在运动过程中的定量分析例题3.如图所示，竖直放置的劲度系数的轻弹簧上有一质量不计的轻盘，盘内放着一个质量的物体，一开始m处于静止状态，现给物体施加一个竖直向上的力F，使物体从静止开始向上一直做匀加速直线运动。已知开始内F是变力，在后F是恒力，取问题讨论（1）物块与轻盘什么时候分离？物块在与轻盘分离之前

4、受到哪些力？分析分离前拉力F应该如何变化？（2）在开始的0.2s内物块上升的距离是多少？上升阶段加速度是多大（3）拉力F在什么时候最小，什么时候最大？求出F最小值和最大值经典例题图3-5-1例1：如图3-5-1所示，一根轻质弹簧上端固定，下端挂一质量为M的平托盘，在盘中有一质量为m的物体，当盘静止时，弹簧的长度比自然长度伸长了L，现向下拉盘使弹簧再伸长L后停止，然后松手，设弹簧总处于弹性限度内，则刚松开手时，盘对物体的支持力大小等于多少？分析与解：松手后，求支持力研究m，受力情况与运动情况分析如图3-5-2所示，则 mgNa（m+M）gfa图3-5-2图3-5-3求支持力也就是求加速度，而m与

5、M的加速度相同，所以我们可以研究整体，受力情况与运动情况分析如图3-5-3所示，其中弹簧弹力变化需要形变发生变化，形变发生变化需要时间，在瞬间还可认为没变，形变量为所以，将开始平衡时的条件代入，得， ABCO1O2图3-5-4例2：质量为3m的箱子C，顶部悬挂质量为2m的小球B，小球B的下方通过一轻弹簧与质量仍为m的小球A相连，箱子C用轻绳O1O2悬于天花板上而处于平衡状态，如图3-5-4所示现剪断轻绳O1O2，在剪断轻绳的瞬间，小球A、B和箱子C的加速度分别为多少？分析与解：研究A球，受到重力和弹簧的弹力作用，因为弹簧弹力的大小与弹簧的形变有关，形变需要时间，在瞬间（一个相当短的时间），可以

6、认为弹簧弹力没变，大小等于A物体的重力mg所以，A球的加速度为零研究B球，受到重力、弹簧的弹力（瞬间大小仍为mg），但绳子中的拉力在瞬间发生变化了绳中是否有弹力，有的话变为多少？不清楚所以，我们可以假设没有拉力，B的受力情况与运动情况分析如图3-5-5所示，2mgFaBB图3-5-5则而C只受到重力，所以2mg+3mgFaBCB、C图3-5-6，所以假设不成立，绳子将拉紧，B、C的加速度相同研究B、C，受力情况与运动情况分析如图3-5-6所示，则图3-5-7例3：两个质量不计的弹簧将一金属块支在箱子的上顶板与下底板之间，箱只能沿竖直方向运动，如图3-5-7所示两弹簧原长均为0.80m，劲度系数

7、均为60N/m当箱以a=2.0m/s2的加速度匀减速上升时，上弹簧的长度为0.70m，下弹簧的长度为0.60m现若上顶板弹力是下底板弹力的四分之一，试判断箱的运动情况（g=10m/s2）mgF1F2a图3-5-8分析与解：设原长l=0.80m，l1=0.70m，l2=0.60m当箱以a=2.0m/s2的加速度匀减速上升时，研究箱受力情况与运动情况分析如图3-5-8所示，则当上顶板弹力是下底板弹力的四分之一，研究箱，由于题目中讲的是弹力，不知是拉力还是压力，所以，要分情况讨论，设上弹簧的长为l3，下弹簧的长度为l4，则l3+ l4= l1+ l2=1.30m（1）假设上弹簧拉长，下弹簧压缩，则依

8、题意有 l3=0.90m，l4=0.40m（假设成立）箱子的加速度为（2）假设上弹簧拉长，下弹簧拉长，则依题意不可能（3）假设上弹簧压缩，下弹簧拉长，则依题意有 l3=0.90m，l4=0.40m（假设不成立）（4）假设上弹簧压缩，下弹簧压缩，则依题意有 l3=0.76m，l4=0.56m（假设成立）箱子的加速度为所以，箱子可能的运动是：箱子向上加速或者是向下减速，加速度的大小为或者AFB图3-5-9例4：如图3-5-9所示，物体A静止在台秤的秤盘B上，A的质量mA=10.5kg，B的质量mB=1.5kg，弹簧的质量忽略不计，弹簧的劲度系数k=800N/m现给A施加一个竖直向上的力F，使它向上

9、做匀加速直线运动，已知力F在t=0.2s内是变力，在0.2s后是恒力.求此过程中F的最大值与最小值（g=10m/s2）分析与解：研究A、B整体，受力情况与运动情况分析如图3-5-10（甲）所示，则由（1）式可知，当向上运动时，x减小，F增加（mA+ mB）gfFamAgNFamBgfa图3-5-10甲乙丙AB再研究A物体，受力情况与运动情况分析如图3-5-10（乙）所示，则由（2）式可知，随着F的增加（x减小），N减小，当N减小到零后（N保持为零，不会小于零），F就不变了所以，开始时，x最大，F最小，最后，支持力等于零时，x最小，F最大研究整体，不加F时，加速度为零，则由（1）式得研究脱离时的

10、B盘，此时F最大，受力情况与运动情况分析如图3-5-10（丙）所示，则且解（3）（4）式得 ，a=6m/s2，将N=0、a=6m/s2代入（2），则得F的最大值研究 t =0时，（1）式中x即为x1，此时，F为最小值【思路点拨】一般产生弹力的物体可分为两类，一类是弹簧（尤其是软件包弹簧）它的形变发生改变的过程需时较长在瞬间（时间很短）内形变量可认为是不变的；另一类是钢丝（包括细丝）等属于刚体，它的形变发生改变过程极短（比瞬间还短），在极短的时间内弹力能发生很大的变化所以，遇到研究弹力变化的问题时，先要把研究的对象等效为软弹簧模型或刚体模型，然后再进行分析计算在名题精析例1中，松手的瞬间，弹簧中

11、的弹力变化很小，所以被忽略了，而盘与物体间的弹力变化较大，不能被忽略例2中，同样如此，弹簧中的弹力变化被忽略了，但绳中弹力的变化是要考虑的，在例1、例2中我们求出了瞬时力和瞬时加速度在例3中，我们知道弹簧中的弹力可能拉力，也可能是压力，题中没说清的话，就要讨论，这样才不至于漏解，同时值得注意的是，在例3中，2根弹簧的实际长度之间是有关系的，这个关系也经常要用到在例4中，结合弹簧弹力特点，讨论了物体运动过程中物理量（力）的变化问题，从而找出了临界值，求出了最大值与最小值 第 3 次课后作业学生姓名： 1静止在光滑水平面上的物体受到水平恒定拉力F作用，则下列说法正确的是（ ）AF刚作用的瞬间，物体

12、立即获得速度和加速度BF刚作用的瞬间，物体立即获得加速度，但速度仍为零C经过一段时间，物体速度增加，加速度也增加D当撤去外力F时，加速度立即变为零，物体做匀速直线运动图3-5-112如图3-5-11所示，质量为M的框架放在水平地面上，一轻质弹簧上端固定在框架上，一端栓着一个质量为m的小球，小球上下振动时，框架始终没有跳起，当框架对地面压力为零的瞬间，小球的加速度大小为（ ）Ag B（M-m）g/m C0 D（M+m）g/mMN30°ba图3-5-123如图3-5-12所示，倾角为30°的光滑杆上套有一个小球和两根轻质弹簧a、b，两弹簧的一端各与小球相连，另一端分别用销钉M、

13、N固定于杆上，小球处于静止状态，设拔去销钉M瞬间，小球的加速度大小为6m/s2，若不拔去销钉M，而拔去销钉N瞬间，小球的加速度是（g取10m/s2）（ ）A11m/s2，沿杆向上 B11m/s2，沿杆向下C1m/s2， 沿杆向上 D1m/s2， 沿杆向下abc图3-5-134如图3-5-13所示，小球质量为m，被三根相同的轻质弹簧a、b、c拉住，c竖直向下，a、b、c三者都夹120°角小球平衡时a、b、c弹力大小之比为331，设重力加速度为g，则在剪断弹簧C瞬间小球加速度的大小及方向可能为（ ）Ag/2，竖直向下 Bg/2，竖直向上Cg/4，竖直向下 Dg/4，竖直向上5如图3-5-

14、14所示，质量为m的物体A放置在质量为M的物体B上，B与弹簧相连，它们一起在光滑水平面上作简谐振动，振动过程中A、B之间无相对运动设弹簧的劲度系数为k当物体离开平衡位置的位移为x时，A、B间摩擦力的大小等于（ ）BA图3-5-14A0 Bkx Cm kx /M D m kx /(Mm)图3-5-156如图3-5-15所示，轻质弹簧上面固定一块质量不计的薄板，竖立在水平面上，在薄板上放一重物，用手将重物向下压缩到一定程度后，突然将手撤去，则重物将被弹簧弹射出去，则在弹射过程中（重物与弹簧脱离之前）（ ）A重物受的支持力先减小后增大 B弹簧的弹力先减小后增大C重物的加速度先减小后增大 D重物先加速

15、后减速FAB图3-5-167如图3-5-16所示，质量相同的木块A、B用轻绳连接静止于光滑的水平面上，弹簧处于自然状态现用水平恒力F推A，则从开始到弹簧第一次被压缩到最短的过程中（ ）A两木块速度相同时，aA=aB B两木块速度相同时，加速度aA<aBC两木块加速度相同时，速度vA<vB D两木块加速度相同时，速度vA>vBl1l2a b l1l2图3-5-178如图3-5-17（a）所示，一质量为m的物体系于长度分别为l1、l2的两根细线上，l1的一端悬挂在天花板上，与竖直方向夹角为，l2水平拉直，物体处于平衡状态现将l2线剪断，求剪断瞬时物体的加速度（l）下面是某同学对该

16、题的一种解法：解：设l1线上拉力为T1，线上拉力为T2，重力为mg，物体在三力作用下保持平衡T1cosmg， T1sinT2， T2mgtg剪断线的瞬间，T2突然消失，物体即在T2反方向获得加速度因为mg tgma，所以加速度ag tg，方向在T2反方向你认为这个结果正确吗？请对该解法作出评价并说明理由（2）若将图3-5-17（a）中的细线l1改为长度相同、质量不计的轻弹簧，如图3-5-17（b）所示，其他条件不变，求解的步骤和结果与（l）完全相同，即 ag tg，你认为这个结果正确吗？请说明理由ABF图3-5-189如图3-5-18所示，两矩形物块A和B叠放在竖直的弹簧上，已知mA=2m，m

17、B=m今用一竖直向下的大小为F的力压物块A（未超过弹簧的弹性限度），在突然撤去力F的瞬时A对B的支持力多大？ 10有三个质量分别为5M、3M和2M的小球A、B和C，其中B球带正电Q，其余两球不带电用足够长的不会伸长的绝缘线连接均置于竖直向下的匀强电场中，场强大小为E如图3-5-19所示释放A球，让三球由静止落下，下落一小段时间（球不相碰），此时三球的加速度大小分别为多少？ABCE图3-5-19+Q11如图3-5-20所示，木块A与B用一轻弹簧相连，竖直放在C板上，A、B、C的质量分别为m、2m、3m（1）若将三者静止放在水平地面上，当沿水平方向迅速抽出C的瞬时，A、B的加速度分别多大？方向如何

18、？（2）若用手托住C板，为使C板能即刻与B分离，则手至少要对C施加向下多大的力？ABC图3-5-2012如图3-5-21所示，小车放在水平面上，在水平外力的作用下沿水平向右做匀加速直线运动，放在小车上的两个物体A、B的质量分别为mA=1kg，mB=0.5kg它们与小车表面的动摩擦因数分别是A=0.4，B=0.2，连接两个物体的轻质弹簧的劲度系数k=10N/m设最大静摩擦力等于滑动摩擦力求当小车的加速度a=2.5m/s2时，A、B所受的摩擦力和弹簧的形变量（g=10m/s2）传感器a传感器bv图3-5-22AB图3-5-2113用如图3-5-22所示的装置可以测量汽车在水平路面上做匀加速直线运动

19、的加速度该装置是在矩形箱子的前、后壁上各安装一个由力敏电阻组成的压力传感器用两根相同的轻弹簧夹着一个质量为2.0kg的滑块，滑块可无摩擦滑动，两弹簧的另一端分别压在传感器a、b上，其压力大小可直接从传感器的液晶显示屏上读出现将装置沿运动方向固定在汽车上，传感器b在前，传感器a在后汽车静止时，传感器a、b的示数均为10N（取g=10m/s2） （1）若示感器a的示数为14N，求此时汽车的加速度大小和方向（2）当汽车以怎样的加速度运动时，传感器a的示数为零图3-5-2314将金属块用压缩的弹簧卡在一个矩形的箱中，如图3-5-23所示，在箱的上顶板和下顶板安有压力传感器，箱可以沿直轨道运动当箱以a=

20、2m/s2的加速度做竖直向上的匀减速运动时，上顶板的传感器显示的压力为6.0N，下顶板的传感器显示的压力为10.0N，取g=10m/s2（1）若上顶板传感器的示数是下顶板示数的一半，试判断箱的运动情况 （2）要使上顶板传感器的示数为0，箱沿竖直方向的运动可能是怎样的？maM图3-5-2415如图3-5-24所示，弹簧的劲度系数为k=600N/m，下端挂一个质量未知的物体m，上端固定在天花板上，物体下面用质量为M=1kg的托盘托住，托盘在外力F的作用下使弹簧恰好等于原长，然后使托盘竖直向下做加速度a大小未知的（a<g）匀加速运动，直到托盘与物体分离已知在此过程中外力F的最小值Fmin=6N

21、，最大值为18N求：托盘与物体刚分离的瞬间物体m的速度为多大? （g取10m/s2）A图3-5-2516如图3-5-25所示，在倾角为的光滑斜面上端系一劲度系数为k的轻弹簧，弹簧下端连有一质量为m的小球，球被一垂直于斜面的挡板A挡住，此时弹簧没有形变若手持挡板A以加速度a（a<gsin）沿斜面匀加速下滑，求：（1）从挡板开始运动到球与挡板分离所经历的时间；（2）从挡板开始运动到球速达到最大，球所经过的最小路程30°FBA图3-5-2617如图3-5-26所示，质量mA=10kg的物块A与质量mB=2kg的物块B放在倾角=30°的光滑斜面上处于静止状态，轻质弹簧一端与物

22、块B连接，另一端与固定挡板连接，弹簧的劲度系数k=400N/m，现给物块A施加一个平行斜面向上的力F，使物块A沿斜面向上做匀加速运动已知F在前0.2s内为变力，0.2s后为恒力，g=10m/s，求： （1）力F的最大值与最小值（2）力F由最小值达最大值过程中物块A所上升的高度【学后反思】第1题第11题的练习目的是，了解弹性绳与刚性绳的区别，能求出瞬时作用力和瞬时加速度2第12题第14题的练习目的是，了解弹簧弹力的特点，求力与加速度3第15题第17的练习目的是，了解弹簧弹力的特点，讨论在弹簧弹力的参与下物体的运动【参考答案】1BD 2D 3BC 4BC 5D 6CD 7BD 8略9 2mg+ 1

23、0， 11（1）0；1.5g，竖直向下，（2）F1.5mg 122.75N ，1N ，2.5m 13（1）a1=4.0m/s2，方向向右（或向前）（2）a210m/s2，方向向左（或向后） 14（1）静止或向上、向下匀速运动（2）m向上加速或向下减速，加速度大小大于等于10m/s2 15v=0.4m/s 16（1）（2） 17（1）Fmax=100N，Fmin=60N；（2）0.05mWinger Tuivasa-Sheck, who scored two tries in the Kiwis' 20-18 semi-final win over England, has been p

24、assed fit after a lower-leg injury, while Slater has been named at full-back but is still recovering from a knee injury aggravated against USA.Both sides boast 100% records heading into the encounter but Australia have not conceded a try since Josh Charnley's effort in their first pool match aga

25、inst England on the opening day.Aussie winger Jarryd Hayne is the competition's top try scorer with nine, closely followed by Tuivasa-Sheck with eight.But it is recently named Rugby League International Federation player of the year Sonny Bill Williams who has attracted the most interest in the

26、tournament so far.The Kiwi - with a tournament high 17 offloads - has the chance of becoming the first player to win the World Cup in both rugby league and rugby union after triumphing with the All Blacks in 2011."I'd give every award back in a heartbeat just to get across the line this wee

27、kend," said Williams.The (lack of) air up there Watch mCayman Islands-based Webb, the head of Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's Premier League match at Chelsea on Sunday."I am going to

28、 be at the match tomorrow and I have asked to meet Yaya Toure," he told BBC Sport."For me it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behavio

29、ur of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's complaint. In a statement th

30、e Russian side added: "We found no racist insults from fans of CSKA." Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunda.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - win

31、ds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers),

32、 drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet ( Then, I would

33、 assume, he will slowly step out onto something resembling an Olympic diving platform.They blew it in 2008 when they got caught cold in the final and they will not make the same mistake against the Kiwis in Manchester.Five years ago they cruised through to the final and so far history has repeated i

34、tself here - the last try they conceded was scored by England's Josh Charnley in the opening game of the tournament.That could be classed as a weakness, a team under-cooked - but I have been impressed by the Kangaroos' focus in their games since then.They have been concentrating on the sort

35、of stuff that wins you tough, even contests - strong defence, especially on their own goal-line, completing sets and a good kick-chase. They've been great at all the unglamorous stuff that often goes unnoticed in the stands but not by your team-mates.It is as though their entire tournament has b

36、een preparation for the final.In Johnathan Thurston, Cooper Cronk, Cameron Smith and either Billy Slater or Greg Inglis at full-back they have a spine that is unmatched in rugby league. They have played in so many high-pressure games - a priceless asset going into Saturday.The Kiwis are a lot less e

37、xperienced but winning a dramatic match like their semi-final against England will do wonders for their confidence.They defeated Australia in the Four Nations final in 2010 and the last World Cup, and know they can rise to the big occasion.Winger Tuivasa-Sheck, who scored two tries in the Kiwis'

38、 20-18 semi-final win over England, has been passed fit after a lower-leg injury, while Slater has been named at full-back but is still recovering from a knee injury aggravated against USA.Both sides boast 100% records heading into the encounter but Australia have not conceded a try since Josh Charn

39、ley's effort in their first pool match against England on the opening day.Aussie winger Jarryd Hayne is the competition's top try scorer with nine, closely followed by Tuivasa-Sheck with eight.But it is recently named Rugby League International Federation player of the year Sonny Bill Willia

40、ms who has attracted the most interest in the tournament so far.The Kiwi - with a tournament high 17 offloads - has the chance of becoming the first player to win the World Cup in both rugby league and rugby union after triumphing with the All Blacks in 2011."I'd give every award back in a

41、heartbeat just to get across the line this weekend," said Williams.The (lack of) air up there Watch mCayman Islands-based Webb, the head of Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's Premier League

42、match at Chelsea on Sunday."I am going to be at the match tomorrow and I have asked to meet Yaya Toure," he told BBC Sport."For me it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings

43、against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed&quo

44、t; by Toure's complaint. In a statement the Russian side added: "We found no racist insults from fans of CSKA." Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunda.The weather plays an important role in this mission. Starting at the

45、 ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip

46、 of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.The balloon will slowly drift to the

47、 edge of space at 120,000 feet ( Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.They blew it in 2008 when they got caught cold in the final and they will not make the same mistake against the Kiwis in Manchester.Five years ago they cruised through

48、to the final and so far history has repeated itself here - the last try they conceded was scored by England's Josh Charnley in the opening game of the tournament.That could be classed as a weakness, a team under-cooked - but I have been impressed by the Kangaroos' focus in their games since

49、then.They have been concentrating on the sort of stuff that wins you tough, even contests - strong defence, especially on their own goal-line, completing sets and a good kick-chase. They've been great at all the unglamorous stuff that often goes unnoticed in the stands but not by your team-mates

50、.It is as though their entire tournament has been preparation for the final.In Johnathan Thurston, Cooper Cronk, Cameron Smith and either Billy Slater or Greg Inglis at full-back they have a spine that is unmatched in rugby league. They have played in so many high-pressure games - a priceless asset going into Saturday.The Kiwis are a lot less experienced but winning a dramatic match like their semi-final against England will do wonders for their confidence.They defeated Australia in the Four Nations final in 2010 and the last World Cup, and know they can rise to the big occasion.第 13 页 共 13 页