课前预读:　《费曼物理学讲义》I　:Chpt.　13、14　《新概念物理教程:力学》　:第三章　第13　讲　能量与功　对于地球表面的自由落体,我们知道其机械能是守恒的　1　2　\ud835\udc5a\ud835\udc632　+　\ud835\udc5a\ud835\udc54ℎ　=　\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc60\ud835\udc61　其中动能\ud835\udc47　=　1　2　\ud835\udc5a\ud835\udc632　,势能\ud835\udc48　=　\ud835\udc5a\ud835\udc54ℎ,因此能量守恒表达为　\ud835\udc47　+　\ud835\udc48　=　\ud835\udc50\ud835\udc5c\ud835\udc5b\ud835\udc60\ud835\udc61　这个规律可以认为是实验定律。另一方面,这一守恒律也是可以通过牛顿定律　导出的。　动能的变化率为　\ud835\udc51\ud835\udc47　\ud835\udc51\ud835\udc61　=　\ud835\udc51　\ud835\udc51\ud835\udc61　(　1　2　\ud835\udc5a\ud835\udc632　)　=　\ud835\udc5a\ud835\udc63　\ud835\udc51\ud835\udc63　\ud835\udc51\ud835\udc61　根据牛顿方程　\ud835\udc51\ud835\udc63　\ud835\udc51\ud835\udc61　=　\ud835\udc39　\ud835\udc5a　,可以得到　\ud835\udc51\ud835\udc47　\ud835\udc51\ud835\udc61　=　\ud835\udc39\ud835\udc63　由于\ud835\udc39　=　−\ud835\udc5a\ud835\udc54,速度\ud835\udc63　=　\ud835\udc51ℎ　\ud835\udc51\ud835\udc61　,则有　\ud835\udc51\ud835\udc47　\ud835\udc51\ud835\udc61　=　−\ud835\udc5a\ud835\udc54　\ud835\udc51ℎ　\ud835\udc51\ud835\udc61　又可以写作　\ud835\udc51(\ud835\udc47　+　\ud835\udc5a\ud835\udc54ℎ)　\ud835\udc51\ud835\udc61　=　\ud835\udc51(\ud835\udc47　+　\ud835\udc48)　\ud835\udc51\ud835\udc61　=　0　由此得到能量守恒。　考虑一般的　3　维运动的质点,其动能为　\ud835\udc47　=　1　2　\ud835\udc5a(\ud835\udc63\ud835\udc65　2　+　\ud835\udc63\ud835\udc66　2　+　\ud835\udc63\ud835\udc67　3　)　对动能求导得　\ud835\udc51\ud835\udc47　\ud835\udc51\ud835\udc61　=　\ud835\udc39　⃗　∙　\ud835\udc63　⃗　=　\ud835\udc39　\ud835\udc65\ud835\udc63\ud835\udc65　+　\ud835\udc39　\ud835\udc66\ud835\udc63\ud835\udc66　+　\ud835\udc39　\ud835\udc67\ud835\udc63\ud835\udc67　因此动能的变化率为力和速度的标量积,称之为功率。　\ud835\udc43　=　\ud835\udc39　⃗　⋅　\ud835\udc63　⃗　也就是说动能的变化为施加与之上的力的功率。上式两边对时间做积分,注意到　\ud835\udc63　⃗\ud835\udc51\ud835\udc61　=　\ud835\udc51\ud835\udc5f　⃗　∫　\ud835\udc43　\ud835\udc51\ud835\udc61　=　∫　\ud835\udc39　⃗　⋅　\ud835\udc51\ud835\udc5f　⃗　=　\ud835\udc4a　上式右边为力内积位移的积分,　称之为功。　需要注重的是此积分是沿着路径的积　分。用微元的写法上式又可以表述为　\ud835\udc43\ud835\udc51\ud835\udc61　=　\ud835\udc39　⃗　⋅　\ud835\udc51\ud835\udc5f　⃗　=　\ud835\udc51\ud835\udc4a　或者　\ud835\udc43　=　\ud835\udc51\ud835\udc4a　\ud835\udc51\ud835\udc61　也就是说功率是单位时间内力做的功。　常力下做的功:　如果位移为Δ\ud835\udc5f　⃗,则\ud835\udc4a　=　\ud835\udc39　⃗　⋅　Δ\ud835\udc5f　⃗　=　\ud835\udc39\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udefc|Δ\ud835\udc5f　⃗|　变化的力做的功:　当力在不断变化时,可以用微元的办法计算功,也就是说将路径分成很多的　小段,每段路上将力近似看作是不变的,于是有　\ud835\udc4a　=　∑　\ud835\udc39　⃗\ud835\udc56　⋅　Δ\ud835\udc5f　⃗\ud835\udc56　\ud835\udc56　=　∑　\ud835\udc39\ud835\udc56\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udefc\ud835\udc56|Δ\ud835\udc5f　⃗\ud835\udc56|　\ud835\udc56　当小段路径趋于零时,上式就变成了积分　\ud835\udc4a　=　∫　\ud835\udc39　⃗　⋅　\ud835\udc51\ud835\udc5f　⃗　\ud835\udc4f　\ud835\udc4e　在直角坐标系下　\ud835\udc4a　=　∫　\ud835\udc39\ud835\udc65\ud835\udc51\ud835\udc65　\ud835\udc65\ud835\udc4f　\ud835\udc65\ud835\udc4e　+　∫　\ud835\udc39　\ud835\udc66\ud835\udc51\ud835\udc66　\ud835\udc66\ud835\udc4f　\ud835\udc66\ud835\udc4e　+　∫　\ud835\udc39　\ud835\udc67\ud835\udc51\ud835\udc67　\ud835\udc67\ud835\udc4f　\ud835\udc67\ud835\udc4e　在自然坐标系下　\ud835\udc4a　=　∫　\ud835\udc39　⃗　⋅　\ud835\udc51\ud835\udc5f　⃗　=　∫　\ud835\udc39\ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udefc\ud835\udc51\ud835\udc60　\ud835\udc4f　\ud835\udc4e　\ud835\udc4f　\ud835\udc4e　受约束运动　在很多情况下