chapter 7 work and energy. 2 energy momentum integration over space integration over time scalar...
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Chapter 7
Work and Energy
2
Energy
Momentum
Integration over space
Integration over time
Scalar
Vector
Significance: They are conserved
Valuable when the forces are not detailed
Wide range including Relativity & Quantum M.
Energy is useful in all areas of physics and in other science as well
Alternative description of dynamics
Work done by constant force
3
Work done by a constant force
cosW Fd or W F d
Accomplished by action of force over a distance
F
d
When W = 0?
Example1: An object moves as ,
and there is a force acting on it,
what is the work done by this force in 0~2s?
22r ti t j
5 6F i k
4 4r i j
20W F r J
F d
Work done by varying force
4
Divide the path into short intervals
Total worki iW F l
Sum → Integral0il
b
aW F dl
During each interval, force ~ constant
i iW F l
Work
General definition of work
F
F
a
b
Work in component form
5
b
aL
W F dl F dl
This is called a line integral.
, x y zF F i F j F k dl dxi dyj dzk
Work can be written asb b b
x y za a aW F dx F dy F dz
Or:tan, dl dl F dl F dl
tan
b
aW F dl
F
F
a
b
dl
Work depends on path
6
tan
b b
a aW F dl F dl
Work depends on the specific path from a to b
a b
1
2
3
Moves on horizontal ground,
3 different paths from a to b, work by friction is different!
b
aW Ndl N S
Friction is called a nonconservative force
Work done by gravity
7
Work done by gravity is path independent.
Determined only by the initial and final position.
Gravity is called a conservative force.
All constant forces are conservative.
ha
hb
La
b
mg
o
y
x
C
G mg
b
yaW F dy
mg h
b
a
h
h mgdy
Gravity of object:
Lose weight by climbing
8
Example2: Bob (80kg) wants to lose weight by climbing mountains. How many calories he loses at least when he goes from A to B? ( 1cal=4.184J )
A
B
h=1500m
Solution: The work done by gravity is
W mgh 680 9.8 1500 1.18 10 J
5or 2.81 10 281cal kcal
So Bob loses energy 61.18 10 J
That is about 31g fat.
Stretching a spring
9
Example3: Someone slowly pulls a spring from unstretched to make the object m leave the ground, calculate the work done by this person.
m
F
k
Solution: force to stretch a spring:
0 0
x xW Fdx kxdx
F = k x
21
2kx
How to make m leave the ground?
F mg /x mg k
Work :2
21 ( )
2 2
mgW kx
k
Kinetic Energy
10
Energy: one of most important concepts in science.
There are many types of energy, it can be usually
regarded as “the ability to do work”.
Moving objects can do work → have energy
The energy of motion is called kinetic energy
21
2kE mv
Object starts from rest under constant net force:
net netW F d mad
Translational kinetic energy
221
2 2
vma mv
a
Work-energy principle
11
Work-energy principle
tan
b
aF dl
For varying force, 3 dimensional motion:b
net netaW F dl
tan
dvF m
dt
b
a
dvm dl
dt
b
a
dlm dv
dt
b
amvdv
2 21 1
2 2b a kmv mv E
The net work done on an object is equal to
the change in its kinetic energy.
Work in an elliptical motion
12
Example4: A particle m moves under the equation , where a, b, are positive constants. Determine the net work during t=0~/2.
cos sinr a ti b tj
Solution1: The net force is
2
2 22
d rF m m r m xi yj
dt
1 1
0 0
x y
x yx yW F dx F dy
2 2 2 21 1
2 2m a m b
Net work
0 2 2
0
b
am xdx m ydy
13
Example4: A particle m moves under the equation , where a, b, are positive constants. Determine the net work during t=0~/2.
cos sinr a ti b tj
Solution2: Using work-energy principle
sin cosdr
v a ti b tjdt
t=0: v0= b;
Net work equals to the change of kinetic energy:
2 2 2 2 2 20
1 1 1 1
2 2 2 2W mv mv m a m b
t=/2: v = a
Work done by friction
14
Example5: An object moves along a semi-circular wall on smooth horizontal plane. Known: v0 and µ, determine the work done by friction from A to B.
v
o
v0
.N
fµ
R
Aµ
B
Solution:
2
v dv
R dt
2vN m
RRadial forcedv
N mdt
Tangential force
dv d v dv
d dt R d
dv
dv
0v v e 2 2
0
1 1
2 2W mv mv 2 2
0
1( 1)
2mv e
Kinetic energy of spring
15
Example6: Considering the mass of spring ms (k, L),
it is attached by a mass M, determine kinetic energy of the spring when M is moving with velocity v.
x dx
M
Solution: take dx of the spring, the mass is ms·dx/L
If the spring deforms uniformly, then vdx = v·x/L2 2
23
1 1
2 2k s s
dx x xdE m v m v dx
L L L
2
0
1
2 3
Ls
k k
mE dE v
Effective mass
*Forms of energy
There are 5 main forms of energy
Mechanical
Thermal / heat
Electromagnetic
Chemical
Nuclear
(Fission & Fusion)
Chain reaction
*Nuclear bomb
“Little boy” & “Fat man”
The effects of nuclear bomb:
Heat, shock wave and radiation (γ ray and neutrons)
“If the radiance of a thousand suns were to burst into the sky, that would be like the
splendor of the Mighty One.”