Chapter 4 Work and Energy
The concept of energy is one of the most important in the world of science. In everyday usage, the term energy has to do with the cost of fuel for transportation and heating, electricity for lights and appliances, and the foods we consume. However, these ideas do not really define energy. They tell us only that fuels are needed to do a job and that those fuels provide us with something we call energy.
Energy is present in the Universe in a variety of forms, including mechanical energy, chemical energy, electromagnetic energy, heat energy, and nuclear energy. Although energy can be transformed from one form to another, the total amount of energy in the Universe remains the same. If an isolated system loses energy in some form, then by the principle of conservation of energy, the system must gain an equal amount of energy in other form. The transformation of energy from one form into another is an essential part of the study of physics, chemistry, biology, geology, and astronomy.
In this chapter we are concerned only with mechanical energy. We introduce the concept of kinetic energy, which is defined as the energy associated with motion, and the concept of potential energy, the energy associated with position. We shall see that the ideas of work and energy can be used in place of
Newton’s law to solve certain problems.
4.1 Work and Power
1. Work W done by a constant force is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement.
ϕcos FS S F W =⋅=
Where the force makes an angle of ϕ with displacement .S
The SI unit of work is the joule (J), named for James Prescott Joule, an English scientist of the 1800s. It is derived directly from the units for mass and velocity:
1joule=1J=1(kg) (m/s 2) (m)=1 kg m 2/s 2
2. Work done by a variable force
(1). The increment of work dW done on the particle by F during the displacement d r is
r d F dW ⋅=, where Force F is function of its position.
(2). The work W done by F while the particle moves from an initial position a to a final position b is then
⎰⎰⋅==b a b a r d F dW W
(3). We use the components of
F and r d to express the force
and displacement, then we have
dz
F dy F dx F k dz j dy i dx k F j F i F r d F W z y b a x z y b a x b a ++=++⋅++=⋅=⎰⎰⎰
)()( 3. Work done by multiple forces : If there are several forces act on a particle, we can replace F in above equation with the net force ∑F , where +++=∑321F F F F , where j F are the
individual forces. Then
+++=⋅+++=⋅=⎰⎰321321)(W W W r d F F F r d F W b a
b
a 4. Power
(1). The rate at which work is done by a force is said to be the power due to the force . If an amount of work W is done in a time interval
t ∆ by a force, then the average power due to the force is t W
P ∆=.
(2). The instantaneous power P is the instantaneous rate of doing work, which can be written as
v F dt
r d F dt dW P ⋅=⋅==. (3). The SI unit of power is the joule per second . This unit is used so often that it has a special name, the watt (W), after James Watt, who greatly improved the rate at which steam engines could do work.
4.2 Kinetic Energy and Work-Kinetic Energy Theorem
Energy is a scalar quantity that is associated with a state of one
or more object. The term state here has its common meaning: it is the condition of an object.
1. Kinetic energy K is associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy. For an object of mass m and whose speed v is well below the speed of light , we define kinetic energy as 22
1mv K = The SI unit of kinetic energy is the same as work —joule.
A convenient unit of energy for dealing with atoms or with subatomic particles is the electron-volt (eV).
1 electron-volt = 1 eV =1.60 x 10 –19 J.
2. Work-kinetic energy theorem : If a force changes the speed of an object, it also changes the kinetic energy of the object. If the kinetic energy is the only type of energy of the object being changed by the force, then the change in kinetic energy is equal to the work W done by the force:
W K K K i f =-=∆ Here i K is the initial kinetic energy (=202
1mv ) and f K is the kinetic energy (221mv ) after the work is done.
3. We can prove work-kinetic energy theorem as follow:
00202222222121212
1)(21k f v v b a z y x z z y y x x b a b a b a K K mv mv v m mvdv W dv v v v d dv v dv v dv v v d v v d v m dt v dt
v d m s d f W -=-====++=++=⋅⋅=⋅⋅=⋅=⎰⎰⎰⎰
(Numerator and denominator of a fraction)
4.3 Work done by weight and by a spring force
1. Work done by weight :
[]a b a b b a b a m gh m gh h h m g dh m g dW W m gdh
m gdl m gdl l d g m dW --=--=-==-=--==⋅=⎰⎰)()cos(cos απα
We can find that the work done by weight on a particle between two points does not depend on the path taken by the particle . Or no matter what path we choose to move the particle, the work done by its weight is the same . In other word if we move a particle around a closed path, the work done by weight on the particle is zero .
2. Work done on a
particle-like object by a
particular type of variable
force, namely, spring
force —the force exerted
by a spring.
⎥⎦
⎤⎢⎣⎡--=-=-==-=⋅=-=⎰⎰222212121'a b x x x x b a kx kx x k kxdx dW W kxdx
s d F dW law s Hook kx F b a b a 3. Conservative force and Non-conservative force : If the work
done by the force is independent of the path the particle moves,
the force is a conservative force ; otherwise a non-conservative force . Weight and spring-force are conservative forces; friction is a non-conservative force.
4.4 Potential energy
Potential energy U is energy that can be associated with the configuration (or arrangement) of a system of objects that exert a force on one another. If the configuration of the system changes, then the potential energy of the system also changes.
1. We know that work done by a conservative force has nothing to do with the path the particle taken. So we can introduce a quantity which is the function of the state of the system to indicate this kind of nature for a conservative force. We call it potential .
2. Gravitational Potential Energy
(1). The work done by weight can be expressed as: []U mgh mgh W a b ∆-=--==, where U ∆ is the change in the gravitational potential energy. Since the work done by weight has definite magnitude from an initial position to a final position, so only a change
U ∆ in gravitational potential energy is physically important .
(2). However, to simplify a calculation or a discussion, we can
say that a certain gravitational potential U is associated with any given configuration of the system, with the particle at a given height h. To do so, we rewrite the above equation as:
)(i i h h mg U U -=-
Then we take i U to be the gravitational potential energy of the
system when it is in a reference configuration, with the particle at a reference point
i h . Usually, we set 0=i U and 0=i h , then we have mgh U =. So the gravitational potential energy associated with a particle-Earth system depends on the height h of the particle relative to the reference position of h i =0, not the horizontal position.
3. Elastic Potential Energy
(1). Similar to a particle-Earth system, the work done by the spring force can be rewritten as U kx kx W a b ∆-=⎥⎦⎤⎢⎣⎡--=)21()21(22.
(2). To associated a potential energy U with any given configuration of the system, with the block at position x, we set the reference point for the block as x i =0, which is always at the equilibrium position of the block. And we set the corresponding elastic potential energy of the system as U i =0. Thus we have 22
1)(kx x U =. Attention: (1). Potential energy belongs to the whole system.
(2). The magnitude of potential energy depends on
the choice of the reference point.
4.5 Work-Energy Theorem and Conservation of Mechanical Energy
1. Work-kinetic energy theorem for one particle: We have W K K K i f =-=∆
2. Work-Energy Theorem: Suppose we have particles of N in the system we discussed and we use work-kinetic energy theorem for each particle, then we have total amount of N equations like those:
ji
jf j i
f i
f K K W K K W K K W -=-=-=222111
We can divide the forces exerted on every particle into external forces and internal forces. And the internal forces can also be classified as conservative internal forces and non-conservative forces. Summing the two side of above equations, we will have: i f
N j ji jf N j j K K K K W -=-=∑∑==11)(
Or to be exactly,
K W W W noncon in con in ext ∆=++--
We also know that the work done by the conservative internal
force can be written as the minus difference of potential energy. So we get the Work-Energy Theorem :
E U K W K W W con in noncon in ext ∆=∆+∆=-∆=+--
The work done by external forces and non-conservative internal forces in a given system is exactly equal to the difference of its Mechanical Energy .
3. Conservation of Mechanical Energy : When only conservative forces act within a system, the kinetic energy and potential energy can change. However, their sum, the mechanical energy E of the system, does not change . 0=∆+∆=∆U K E
4.6 Reading a Potential energy curve
Consider a particle that is part of a system in which a conservative force acts. Suppose that the particle is constrained to move along an x axis while the conservative force does work on it.
1. Finding the Force Analytically : For one-dimensional motion, the work W done by a conservative force that acts on a particle as the particle moves, and the potential energy have the relation as follow
dx x F dW x U )()(-=-=∆
So can get the force from the potential energy
tion ensionalmo one dx x dU x F dim )
()(--=
We can, for example, check this result by putting
221)(kx x U =, which is the elastic potential
energy function for a spring
force. Above equation yields
kx x F -=)( as expected.
2. The Potential Energy
Curve : The following figure
is a plot of a potential energy
function U(x) for a system in
which a particle is in
one-dimensional motion
while a conservative force
F(x) does work on it. We can
easily find F(x) by taking the
slope of the U(x) curve at
various points. Fig. (b) is a plot of F(x) found in this way.
3. Tuning Point : Since there is only conservative force acting on the particle, the system will remain conservation of its mechanical energy. So we have
)()(x U E x K -=. Since kinetic energy
)(x K is not less than zero. As the particle moves from 2x
to 1x (Fig. a), when the particle reaches 1x , its kinetic
energy is zero, meanwhile the force on the particle is positive. It means the particle does not remain at
x but instead begins to
1
move back to the right. Hence
x is a tuning point, a place
1
where K=0 and the particle changes direction of its motion.
4.Equilibrium Points:
Neutral equilibrium
Unstable equilibrium
Stable equilibrium。
