大学物理-保守力和势能

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.
ba acbW0 elastriccefo dko wo
WWorakccabnbxexac
stkoxreddxanxbdrkexcdoxverOed. xc
a
b
c
12k(xa2 xc2)12k(xc2 xb2) 12k(xa2 xb2) <0
Conservative and nonconservative forces:
有没有势能？
超重、失重
势能的形式是什么？
关 于 势 能：
(1) 势能总是与保守力相联系。存在若干种保守力时， 就可引进若干种势能。
(2) 势能的绝对数值与零势能位形的选取有关，但势能 的差与之无关。不同保守力对应的势能，其零势能 位形的选取可以不同。
(3) (3) 势能既然与各质点间相互作用的保守力相联系， 因而为体系所共有。
Fz dz
(2)
mxdx mydy mzdz (1)
(2)
(1)
d
(
1 2
m x2
1 2
m y2
1 2
m z2 )
(2) (1)
d
பைடு நூலகம்
(
1 2
mv
2
)
(2)
dT
(1)
Kinetic energy T=1/2mv2 is also the ability to do
work 运动质点速度改变而所作出的功
Wab 12k(xa2xb2)Va Vb
V GMm r
Vmgh?
V
1 2
k
x2c??
引力可以通过加速度被“创造出来”，和被“消灭 掉”：势能是否能被“创造”或“消
灭”?
非惯性参考系：惯性力的势能问题
平动参考系：附加重力势能
g'ga
V'm'h gmg m h ah
转动参考系：
惯性离心力 Fm2r
(3) Work done by resultant of all forces equals
ww 1w 2w n
合力的功为各分力的功的代数和。
Case study 1: Work done by gravitation 引G力ra的vi功tation两al个w质or点k之: 间在万有引力作用下相对运动时 ， 以 引inM力d所e方p在向en处与d为矢en原径t点方of,向tMh相e指反p向。amtmh的在o方Mf 向t的h为e万矢b有o径引dry力的;的正作方用向下。从m受a 的点 运de动p到enbd点s，o万nl有y 引on力t的he功s:tarting and ending points
(3) 1D Energy diagram 一维势能曲线
Stable equilibrium 稳定平F衡？ dU 0 Unstab平le 衡eq条udi件lixbrium
U(x)
U(x) 不稳定平衡？
x The elastic potential energy for a spring
弹簧振子的势能曲线
Solution I:
In Frame M
M
m
b
a
W a bG r2M d rm GM (1 am b 1)
Solution II:
r1
r2
r1 ' r2 '
Re. center of mass
M: r1 m: r2
M: F 12G r12 2M e r1m 2(rG 1r2 M )2e rm 12
保守力 与势能
南京大学物理学系 王思慧
Summary on work 功的性质
(1) Generally speaking, work is dependent on the path.
功是过程量，一般与路径有关。
(2) Work is a scalar with magnitude and signs. 功是标量，但有正负。
The work done by the net force on a particle equals the change in the particle’s kinetic energy
动能定理: 运动质点的动能的增加等于其它物体对它所
做的功.
(
2
)
F
d
r
(1)
(2)
(1) Fx dx
Fy dy
Potential energy:
W con (V2V1).
IfWtotWcon thenT2 T1 (V2 V1).
T2V2T1V1.
Mechanical energy is conserved when only conservative forces do work.
Mechanical energy:
l2
w12k2l2
Kinetic energy (动能)
Energy associated with the motion is T=1/2mv2 A moving object has the ability to do work A scalar quantity
T=1/2mv2
Work-energy theorem: T2T1W.
Solution:
k1
k2 f
(1) f =k1x1=k2x2 x1+x2=x
l
l
w1 0 f dx 0 k1x1dx
l 0
k1
k2x k1 k2
dx
1 2
k1k2 k1 k2
l2
(2) f =k2x2=k2x
w 20 l fd x0 lk2xx d1 2k2l2
1 k1k2 2k1k2
m
m
体作质功心G 占系主里m mm要，M M 内 地(力位1a的。b1)功与质量成r1反0比Mm 。m 对a,小r1' 质M 量m 物mb
引力的功只m与M物1体1系统的初始和最终相对位置 有W 关2，G 与路M m 径无M关(a。b)
W 1W 2Gm(a 1M b 1)W W1/W2m/M
Case study 2: Work done by elastic forces
Energy associated with the position of a system.
Stored in a system, later recovered. 与相互作用物体的位置有关的能量。
Work by conservative forces potential energy
WABA1BF drA2BF dr势 于能 保的 守增 力量 所等 做
运动质点以力f
施于它物所作功:
W' (2) fdr.
(1)
牛顿第三定律:
W' (2) fdr
f质 点 物 F 物 质点
(2) Fdr
(1)
(1)
(2)m rdr (2)mr rdt (2)mr dr
(1)
(1)
(1)
(2)
(1)
d(12
mr r )12
mv2
(2) 1
(1) 2
When only gravitation does work:
(1) Near the earth’s surface 质点高度变化不大:
12m2vmgz常数
(2) High above the earth’s surface 质点高度变化很大:
1 2m2vmg2/R r常数
When only elastic force does work: 弹性力场:
WabF drabGr2M e rdm r
e rdr dr co srdr
Wa bG r2M d rm GM (r1 a m r1 b)
But if M~m, what is the work?
Example: 质量为M、m的两球原来相距为a，在万有引力作
用下逐渐靠近至相距为b，求在此过程中引力所作的功。
FFxi Fy
j Fzk(Vx
iV y
j V z
k)
(i j k)V x y z
V gradV
梯度：gradUU
(i j k )U x y z
Example: find gravity from gravitational potential V=mgy
Solution:
Fmgy (mxg)yi(myg)y j (mzg)yk (mg)j
12m2v12kx2 常数
机械能守恒原理适合于由若干个物体组成的 系统（如果系统内只有保守力作功）
Work-energy theorem: 功能原理
作用于质点的力F
(4) (4) 与势能相联系的是保守力对质点系所作的总功， 与参考系无关。
Why Sam can not lift the box? How can he make it?
(2)势能
保守力
V Fxx
V V
Fx lim x0
. x x
( y,z不变)
F x V x,F y V y,F z V z. FV gra. dV
Summary on potential energy
(1) Work done by a conservative force can be represented in terms of a potential energy
WV0VV
(2) Potential energy is a shared property of the system, not
mv12
12
mv22.
运动质点的1/2mv2 值的减少正等于它所做的功.
Conservation of mechanical energy
机械能守恒原理
Work-energy theorem: WtotT2T1.
The total work done by the net force on a particle equals the change in the particle’s kinetic energy
VAVBV
功的负值.
IfV B 0 Th V A e W A n B
(1)保守力做功
势能
W A BV AV B V
或 V AW A OA O F dr
Gravitational potential energy
Wab GM(m r1a r1b)Va Vb
Elastic potential energy
one particle.
(3) (3) Components of force aF rex V x,Fy V y,Fz V z.
(4) In vector formF gra d (i V V jVk V)
(5)
x y z
Exercise: 质量为m的物体放在光滑的水平面上, m
的两边分别与劲度系数(force constant,spring constant)为k1和k2的两个弹簧相联，若在右边弹簧 末端施以拉力f，问：（1）若以拉力非常缓慢地拉了 一段距离l，它做功多少？（2）若拉到距离l后突然 不动，拉力做功又如何？
C b a
M1rm2r
m:
dW 1F 12dr 1G(r1M r2)m 2d1r G dW 2F 21dr 2G(r1M r2)m 2d2r G
Mm
(1
M m
)2
r12
Mm
(1 Mm)2r22
dr1 dr2
系W的各1 选个G择力(1无的M关功Mm与 。)2 参r1r10' dr考121r系有G关(1，MM但m)一2 (r对110力r11的') 功与参考
A
L1
F d r F c o d s s 0
L2
B
等
F dr 价 F dr
A 1 B
A 2B
Energy is defined as the ability to do work. Work is defined as the transfer of energy.
Potential energy(势能)
WoFrkdonkexby elastic forces:
independent of the path of the
bkody;
F' F'
deap endbs only on the starting and ending points
O ab
RW aua leb obf xsxabi gnkW s:x d01 2k x(exa 2xxtb 2)eroknracleFfd r' Fko' F 'w
O
x
A hypothetical potential
energy function
假想的势能曲线
Stable equilibrium 稳定平衡条件：
dU / dx 0 d 2U dx 2 0
U(x)
O
x
A hypothetical potential
energy function
假想的势能曲线