（2）弹性力的功路医汉江业大学弹簧刚度系数k（N/m）弹性力 =k(r -lo)r0弹性力的功为A2F.drW12A1A2-k(r-lo)ro.drA1
（2）弹性力的功 弹簧刚度系数k（N/m） 弹性力 弹性力的功为 𝐹 Ԧ = −𝑘(𝑟 − 𝑙0 )𝑟 Ԧ 0 𝑊12 = න 𝐴1 𝐴2 𝐹 Ԧ ⋅ d𝑟 Ԧ = න 𝐴1 𝐴2 −𝑘(𝑟 − 𝑙0 )𝑟 Ԧ 0 ⋅ d 𝑟 Ԧ
-k(r -lo)ro .drW12A1r因o.drdr ==dr-d(r.r)d(r2rr2得W12- k(r - lo)dr一rik即W12(822-82)12一式中 81=r1- lo,82= r2- lo弹性力的功也与路径无关
因 式中 得 即 弹性力的功也与路径无关 𝑊12 = න 𝐴1 𝐴2 −𝑘(𝑟 − 𝑙0 )𝑟 Ԧ 0 ⋅ d 𝑟 Ԧ 𝑟 Ԧ 0 ⋅ d𝑟 Ԧ = 𝑟 Ԧ 𝑟 ⋅ d𝑟 Ԧ = 1 2𝑟 d(𝑟 Ԧ ⋅ 𝑟 Ԧ) = 1 2𝑟 d(𝑟 2 ) = d𝑟 𝑊12 = න 𝑟1 𝑟2 − 𝑘(𝑟 − 𝑙0 )d𝑟 𝑊12 = 𝑘 2 (𝛿1 2 − 𝛿2 2 ) 𝛿1 = 𝑟1 − 𝑙0 , 𝛿2= 𝑟2 − 𝑙0
3）定轴转动刚体上作用力的功SW = F. dr = Fds =FRdp由Mz = F,R得sW =M,do从角转动到角过程中力F的功为01W12M,dp02dpds若Mz=常量F则W12 = Mz(P2 - Φ1)山F
（3） 定轴转动刚体上作用力的功 则 若𝑀𝑧 = 常量 由 得 从角𝜑1转动到角𝜑2过程中力𝐹 Ԧ的功为 δ𝑊 = 𝐹 Ԧ ⋅ d𝑟 Ԧ = 𝐹td𝑠 = 𝐹t𝑅d𝜑 𝑀𝑧 = 𝐹t𝑅 𝛿𝑊 = 𝑀𝑧d𝜑 𝑊12 = න 𝜑2 𝜑1 𝑀𝑧d𝜑 𝑊 ) 12 = 𝑀𝑧 (𝜑2 − 𝜑1
（4）平面运动刚体上力系的功由=ic+iic两端乘dt,有dr=drc+diic作用在M点的力F的元功为oW, = F, . dr, = F, drc + F, -dric其中 F, ·dric = Ficosa .M,C ·dp = Mc(F)dp力系全部力的元功之和为Fsw-Zowdrie0MiZEMe(F)dp dre +dCdre=Fdrc+ Mcdp
作用在𝑀𝑖点的力𝐹 Ԧ 𝑖的元功为 力系全部力的元功之和为 （4） 平面运动刚体上力系的功 其中 由 𝑣 Ԧ 𝑖 = 𝑣 Ԧ 𝐶 + 𝑣 Ԧ 𝑖𝐶 两端乘dt, 有 d𝑟 Ԧ 𝑖 = d𝑟 Ԧ 𝐶 + d𝑟 Ԧ 𝑖𝐶 δ𝑊𝑖 = 𝐹 Ԧ 𝑖 ⋅ d𝑟 Ԧ 𝑖 = 𝐹 Ԧ 𝑖 ⋅ d𝑟 Ԧ 𝐶 + 𝐹 Ԧ 𝑖 ⋅ d𝑟 Ԧ 𝑖𝐶 𝐹 Ԧ 𝑖 ⋅ d𝑟 Ԧ 𝑖𝐶 = 𝐹𝑖cos𝜃 ⋅ 𝑀𝑖𝐶 ⋅ d𝜑 = 𝑀𝐶 (𝐹 Ԧ 𝑖 )d𝜑 𝛿𝑊 = 𝛿𝑊𝑖 = 𝐹 Ԧ 𝑖 ⋅ d𝑟 Ԧ 𝐶 +𝑀𝐶 (𝐹 Ԧ 𝑖 )d𝜑 = 𝐹 Ԧ 𝑅 ′ ⋅ d𝑟 Ԧ 𝐶 + 𝑀𝐶d𝜑