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Bilingual Mechanics
Chapter 4
Energy and Work
制作 张昆实
制作 张昆实
Yangtze University
Yangtze University
Chapter 4 Energy and Work
4-1 What Is Physics?
4-2 What Is Energy?
4-3 Kinetic Energy
4-4 Work
4-5 Work and Kinetic Energy
4-6 Work Done by Force
4-7 Power
Chapter 4 Energy and Work
4-8 Work and Potential Energy
4-9 Path Independence of Conservative
Forces
4-10 Determining Potential Energy Values
4-11 Conservation of Mechanical Energy
4-12 Work Done on a System by an
External Force
4-13 Conservation of Energy
4-1 What Is Physics
★ One of the fundamental goals of physics is to
investigate the topic “energy”, which is
obviously important .
★ One job of physics is to identify the different
types of energy in the world.
gravitational
potential energy
elastic
mechanic energy
kinetic energy
★ In this chapter we learn what is physics through
the study of energy and work.
4-2 What Is Energy?
★ potential energy is the energy detemined
by the configuration of a system of
objects that exert forces on one another.
★ kinetic energy is the energy a body has
because of its motion.
.
★ energy may be transformed from one
form to another, but it cannot be created
or destroyed, i.e. the total energy of a
system is constant.
4-3 Kinetic Energy
★Newton’s laws of motion allow us to
analyze many kinds of motion. However,
the analysis is often complicated,
requiring details about the motion.
★There is another technique for analyzing
motion , which involves energy.
★ Kinetic energy K is energy associated
with the state of motion of an object.
K  mv
2
2
SI unit: joule(J), 1J  1kg  m / s
1
2
2
(4-1)
(4-2)
4-4
Work
work W is energy transferred to or from
an object by means of a force acting on
the object. Energy transferred to the
object is positive work, and energy
transferred from the object is negative
work.
★ You acceletate an object
its kinatic
energy increaced
the work done by
your force is positive;
★ You deceletate an object
its kinatic
energy decreaced
the work done by
your force is negative;
4-5 Work and Kinetic Energy
Finding an Expression for Work
★ A bead can slide along a frictionless
horizontal wire ( x axis), A constant force
F ，directed at an angle  to the wire,
accelerates the bead along the wire.
Fx  max
(4-3)
v0
ax
v  v0  2ax d (4-4)
F
Solve for ax , yields Bead

Fx
2
2
1
1
(4-5)
2 mv  2 mv0  Fx d
start
d
W  Fx d
(4-6)
2
2
v
wire
x
end
4-5 Work and Kinetic Energy
★ To calculate the work done on an object
by a force during a displacement, we use
only the force component along the object’s
displacement. The force component perpendicular to the displacement does zero work.
Since
General form
Fx  F cos 
W  Fx d  Fd cos 
Scalar (dot) product
W  Fd cos   F  d
(4-7)
(4-8)
Cautions: (1) constant force (magnitude
and direction) (2) particle-like object.
4-5 Work and Kinetic Energy
★ Signs for work
0    90 cos   0
W  0 positive work
90    180 cos   0 W  0 negative work
  90 cos   0 W  0 does’t do work
A force does positive work when it has a
vector component in the same direction as
the displacement, and it does negative
work when it has a vector component in the
opposite direction. It does zero work when
it has no such vector component.
4-5 Work and Kinetic Energy
★ Units for work
The unit for work is the same as the unit
for energy
1J  kg  m / s  1N  m
2
2
(4-9)
★ net work done by several forces
When two or more forces act on an object,
their net work done on the object is the
sum of the works done by the individual
forces.
4-5 Work and Kinetic Energy
★ Work-kinetic Energy Theorem
1
2
mv  mv0  Fx d
2
1
2
2
K  K f  K i  W
change in the kinetic
energy of a particle
=
(4-10)
net work done on
the particle
K f  Ki  W
kinetic energy after
The net work is done
Work-kinetic Energy Theorem
for particles
(4-5)
(4-11)
=
kinetic energy
Before the net work
+
the net
Work done
4-6 Work Done by Force
A tomato is thrown upward, the work done by the
grivatational force : Wg  mgd cos 
(4-12)
In the rising process
Wg  mgd cos180  mgd
(4-13)
minus sign: the grivatational force transfers
energy (mgd) from the object’s kinetic energy,
consistant with the slowing of the object.
In the falling down process
Wg  mgd cos0  mgd
(4-14)
plus sign: the grivatational force transfers
energy (mgd) to the object’s kinetic energy,
consistant with the speeding up of the object.
4-6 Work Done by Force
★ Work done in lifting and lowering an object
Lifting an object. Displacement: upward
Lifting force: positive work;
transfers energy to the object;
Gravitational force: negative
work; transfes energy from
the object.
Lowering an object. Displacement: downward
Lifting force: negative work; transfers energy
from the object;
Gravitational force: positive work; transfes
energy to the object.
4-6 Work Done by Force
The change K in the kinetic energy due to these
two energy transfers is
K  K f  K i  Wa  Wg(4-15)
K f  Ki ( 0)
Wa  Wg  0
Wa  Wg
Wa  mgd cos 

(4-16)
  180
 0
(in lifting and lowering ; K f  K i ) (4-17)
The angle between Fg and d
Up:
Wa  mgd
. Down:
Wa  mgd
4-6 Work Done by Force
The spring force
Fig(a): A block is attached to a
spring in equilibrium (neither
compessed nor stretched).
X axis along the spring
relaxed state
Fig(b): when stretched to right
The spring pulls the block to the
Left (restoring force)
Fig(c): when compressed to left
The spring pulls the block to the
right (restoring force)
The spring force
(variable force)
K: spring constant
stretched
compressed
F  kd
(Hooke’s law) (4-18)
F  kx
(Hooke’s law) (4-19)
4-6 Work Done by Force
The work done by a spring force
assumptions: spring
massless; ideal
spring(obeys Hooke’s law); contact frictionless.
Ws    Fxj x
x  0
'
F

F
Calculus metheod
(4-20)
Sumintegration
xf
xf
xi
xi
Ws    Fx dx   kxdx
1 2 1 2
Ws  kxi  kx f
2
2
( work by a spring force )
(4-21)
(4-22)
(4-23)
o
x
P
F
dW
O
xf
dx
xi x
x
4-6 Work Done by Force
The work done by a spring force
Work Ws is positive if the block ends up closer to
the relaxed position (x=0) than it was initially. It is
negative if the block ends up father away from x=0.
It is zero if the block ends up at the same distance
from x=0.
1 2 1 2
Ws  kxi  kx f
2
2
(4-23)
If
xi  0
and
xf  x
1 2
Ws   kx
2
'
F

F
o
x
P
then
( work by a spring force )
(4-24)
x
4-6 Work Done by Force
The work done by a spring force
Suppose we keep applying a force Fa on the
block, our force does work Wa , the spring force
does work Ws on the block.
The change K in
the kinetic energy
of the block

F
K  K f  Ki  Wa  Ws
o
x
(4-25)
If the block is stationary before and after the
displacement, then K  K  0
f
i
Wa  Ws
(4-26)
Fa
P
x
4-6 Work Done by Force
The work done by a spring force
Wa  Ws
(4-26)
If a block that is attached to a spring is stationary
before and after a displacement, then the work
done on it by the applied force displacing it is the
negative of the work done on it by the spring
force.

F
o
Sample problem 4-2 : P93
x
Fa
P
x
4-6 Work Done by Force
One-dimensional Analysis
the same method as the calculation of the
work done by a spring force (calculus).
W j  Fj ,avg x
(4-27)
W  W j  Fj ,avg x
(4-28)
W  lim Fj ,avg x
(4-29)
x 0
xf
W   F ( x)dx
xi
( Work:
variable force )
(4-30)
4-6 Work Done by Force
Three-dimensional Analysis
A three-dimensional force acts on a body
(4-31)
F  Fx i  Fy j  Fz k
Simplifications: Fx only depends on x and so on
Incremental displacement
dr  dxi  dyj  dzk
The work
(4-32)
dW done by F during
dr
dW  F  dr  Fx dx  Fy dy  Fz dz
is
(4-33)
rf
xf
yf
zf
ri
xi
yi
zi
W   dW   Fx dx   Fy dy   Fz dz
(4-34)
4-6 Work Done by Force
work-kinetic energy theorem with a variable
force
Let us prove:
xf
xf
xi
xi
W   F ( x)dx   madx
(4-35)
dv
dv dx
dv
madx  m dx  m
dx  m vdx  mvdv
dt
dx dt
dx
(4-36)
(4-37)
(4-38)
Substituting Eq. 4-38 into Eq. 4-35:
xf
vf
vf
xi
vi
vi
W   madx   mvdv  m  vdv
 mv f  mvi
1
2
2
1
2
W  K f  K i  K
2
(4-39)
work-kinetic energy theorem
2-7
Power
Power : The power due to a force is the rate
at which that force does work on an object.
work down by a force: W
in a time interval : t
the average power
Instantaneous power
Unit: 1watt = 1 W = 1 J / s
the instantaneous power
can be expresscd in terms
of the force and the particle’s velocity :
W

t
(4-40)
dW
P
dt
(4-41)
Pavg
P  dW dt  F cos  dx dt
P  Fv cos   F  v
(4-46)
2-8 Work and Potential Energy
W  K f  K i  K
work-kinetic energy theorem (4 -10)
Discuss the relation:
W ~U
Exp. A tomato is thrown upward
Rising:
Fg does W, leads K tomato
Energy transferred from the tomato,
Where does it go?
To
Increase the gravitational potential
energy of the tomato-earth system!
(the seperation is increased ! )
Earth
Falling: Fg does W, leads K tomato
Energy transferred
from the gravitational potential energy of the tomato-earth
system to the kinetic energy of the tomato !
2-8 Work and Potential Energy
Work and Potential Energy
Discuss the relation:
W ~U
For either rise or fall, the change
U in gravitational potential energy
is defined to equal the negative
work done on the tomato by the
gravitational force.
U  W
Earth
(4-47)
This equation also applies to a block-spring
system (P 98)
4-9 Path Independence of Conservative Forces
Conservative and Nonconservative Forces
Key elements:
1. A system (two or more objects);
2. A force acts between a object and the rest part
of the system;
3. when configuration changes, the force does
work W transffering the kinetic energy of the
1
object into some other form of energy.
4. Reversing the configuration changes, the force
reverses the energy transfer, doing work W2 .
IF W1  W2 is always true,
The other form of energy is a potential energy
The force is a conservative force!
4-9 Path Independence of Conservative Forces
Conservative and Nonconservative Forces
Nonconservative Forces: a force that is not
conservative
Exp. : (1) the kinetic frictional force :
A block is sliding on a rough surface, the kinetic
frictional force does negative work
Transfer kinetic energy
thermal energy
So the thermal energy is not a potential energy!
The frictional force
Nonconservative Forces !
(2) the drag force
4-9 Path Independence of Conservative Forces
The closed-path test : to determine
whether a force is conservative or
nonconservative.
The net work done by a conservative
force on a particle moving around every
closed path is zero.
The work done by a conservative force
on a particle moving between two
points does not depend on the path
taken by the particle.
4-9 Path Independence of Conservative Forces
The work done by a conservative force
on a particle moving between two
points does not depend on the path
taken by the particle.
Wab ,1
b
Wab ,1
b
1
1
2
Wab ,2
a
Wab,1  Wba ,2  0
2
a
Wba ,2
Wab ,1  Wba ,2  Wab ,2
(4-48)
4-10 Determining Potential Energy Values
Find the relation between a conservative
force and the associated potential energy :
The work done by a variable conservative
force on a particle (see Eq. 4-35):
xf
W   F ( x)dx
(4-51)
xi
xf
U  W    F ( x)dx
xi
(4-52)
Eq.4-52 is the general relation we sought.
4-10 Determining Potential Energy Values
Gravitational potential energy :
A particle is moving vertically along a y axis
From
xf
U  W    F ( x)dx
xi
yf
yf
yi
yi
U    (mg )dy  mg 
(4-52)
dy  mg  y 
U  mg ( y f  yi )  mg y
(4-53)
U  U i  mg ( y  yi )
(4-54)
U ( y )  mgy
( yi  0, and
Ui  0)
Gravitational
potential energy
(4-55)
yf
yi
4-10 Determining Potential Energy Values
Elastic potential energy :
A block-spring system is vibrating, the
spring force F  kx does work on the block.
xf
xf
U    (kx)dx  k  xdx  k  x 
xi
xi
xi
U  kx  kx
2
f
1
2
1
2
U  0  kx  0
1
2
2
U ( x)  kx
1
2
2
2
i
1
2
2
xf
(4-56)
( xi  0, and
Ui  0)
(4-57)
4-11 Conservation of Mechanical Energy
Mechanical energy : THe Mechanical energy
of a system is the sum of its potential energy and
the kinetic energy of the objects within it:
Emec  K  U
( Mechanical energy ) (4-58)
A conservative force does work W on the object
changing the object’s kinetic energy
(4-59)
K  W
The change U in potential energy
U  W
Combining (4-59) , (4-60):
Rewriting:
(4-60)
K  U
K2  K1  (U 2  U1 )
(4-61)
(4-62)
4-11 Conservation of Mechanical Energy
Rewriting:
Rearranging:
K2  K1  (U 2  U1 )
K2  U 2  K1  U1
(4-62)
(4-63)
( conservation of Mechanical energy )
The sum of K and U for
any state of a system
=
The sum of K and U for any
other state of the system
The principle of conservation of mechanical energy ：
In a isolated system where only conservative
forces cause energy changes, the kinetic energy
and potential energy can change, but their sum,
the mechanical energy of the system, cannot
change.
4-11 Conservation of Mechanical Energy
From Eq. 4-61 :
K  U
Emec  K  U  0
(4-64)
When the mechanical energy of a system is
conserved we can relate the sum of kinetic
energy and potential energy at one instant
to that at another instant without considering
the intermediate motion and without finding
the work done by the forces involved.
The principle of
conservation of
mechanical energy
Newton’s laws
of motion
4-11 Conservation of Mechanical Energy
A pendulum
bob swings
back and forth.
K
U
Emec = constant
( the pendulum-Earth system )
4-11 Conservation of Mechanical Energy
★ Finding the force Analytically
Know F ( x )
Know U
xf
Find U U   xi F ( x)dx
Find F ( x )
(4-52)
U ( x)  W   F ( x)x
From Eq.(4-47) :
Solving for F ( x ) and passing to the differential limit :
dU ( x)
F ( x)  
dx
( one-dimensional motion )
(4-68)
★ Check Eq.(4-68) :
2
1
U
(
x
)

kx
1.
2
2.
U ( x)  mgx
F ( x)   d ( kx ) dx  kx
1
2
2
F ( x)   d (mgx) dx  mg
4-11 Conservation of Mechanical Energy
The Potential Energy
Curve U ( x )  x
U ( x)  K ( x)  Emec
K ( x)  Emec  U ( x)
x2
x1 ,
U ( x) K ( x)
At x1, K ( x)  0
F ( x)  0
x1
x1
right
is a turning point
dU ( x)
F ( x)  
dx
4-11 Conservation of Mechanical Energy
unstable equilibrium
K.E=0
F=0 on both sides
deflecting force !
neutral equilibrium
K.E=0
F=0
stable equilibrium
K.E=0
F=0 on both sides
restoring force !
4-12 Work Done on a System by an External Force
Work is energy transferred to or from a
system by means of an external force
acting on that system.
system
system
Positive W
In 4-5 :
Negative
W
K
W
(only )
The work-kinetic energy theorem
In 4-12:
Fextetnal E
(in other forms)
4-12 Work Done on a System by an External Force
No Friction Involved
throwing a ball upward,
Ball-Earth
your applied force
system
Emec  K  U
does work
Positive W
Earth
kinetic
potential
W  K  U
(4-71)
W  Emec
(4-72)
mechanical energy
( Work done on system, no friction involved )
4-12 Work Done on a System by an External Force
★
Friction Involved
v0 a
F
fk
Emec
v
W
Fd


K

f
d
k
x
d
F  f k  ma
v  v  2ad
2
2
0
Eth
(4-73)
BlockFloor
system
(4-75)
Fd  K  U  f k d
Fd  Emec  f k d (4-76)
Eth  f k d (4-77)
Fd  Emec  Eth (4-78)
2
2
1
1
Fd  2 mv  2 mv0  f k d W  Emec  Eth (4-79)
(4-74) （work done on a system, friction involved）
4-13 Conservation of Energy
Countless experiments have proved:
The law of conservation of energy
The total energy E of a system can
change only by amounts of energy that
are transferred to or from the system.
Total
energy
Mechanical energy
Thermal energy
Internal energy of any form
The work done on a system = the change in the
total emergy
W  E  Emec  Eth  Eint
(4-80)
4-13 Conservation of Energy
Isolated System
If a system is isolated from its environment
No energy transfers to or from it.
W=0
★ The total energy E of an isolated system
cannot change.
W  E  Emec  Eth  Eint  0
Emec  Emec，2  Emec，1
Emec，2  Emec.1  Eth  Eint  0
Emec，2  Emec，1  Eth  Eint
(4-81)
(4-82)
4-13 Conservation of Energy
Isolated System
★ The total energy E of an isolated system
cannot change.
★
Emec，2  Emec，1  Eth  Eint
★
(4-82)
In an isolated system, we can relate
the total energy at one instant to
the total energy at another instant
without considering the energies at
intermediate times.
This is a powerful tool in solving problems
about isolated system
4-13 Conservation of Energy
★ External Forces and Internal Energy Transfers
An initially stationary ice skater pushes
away from a railing and then slides over
the ice. Her kinetic energy increases
because of an external force F on her
from the rail.
However, that force does not thansfer
energy from the rail to her. Thus, the
force does no work on her. Rather, her
kinetic energy increases as a result of
internal thansfers from the biochemical
energy in her muscles .
K  K  K0  Fd cos 
K  U  Fd cos 
(4-83)
(4-84)
com
F
v0
v
d
x
4-13 Conservation of Energy
Fig. 4-26 A vehicle accelerates to the left using fourwheel drive. The road exerts four frictional forces (two of
them shown) on the bottom surfaces of the tires. Taken
together, these four forces make up the net external
force F acting on the car.
However, F does not
thansfer energy from the
road to the car and does
no work on the car.
Rather, the car’s kinetic
energy increases as a
result of internal
thansfers from the
energy stored in the fuel
F1
F2
4-13 Conservation of Energy
Power
★ Power is the rate at which work is done
by a force. (P96)
★
★ Power
is the rate at which energy is
transferred by a force from one form to
another.
average
power
instantaneous
power
Pavg
E

t
dE
P 
dt
(4-85)
(4-86)