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7. WORK AND ENERGY.
Key words: Work, Kinetic Energy, Work – Energy Theorem,
Potential Energy, Gravitational Potential Energy, Elastic
Potential Energy, Hooke’s Law, Spring Force, Force Constant or
Spring Constant, Restoring Force, Elastic Potential Energy,
Conservative and Nonconservative Forces, General Form of the
Work – Energy Theorem, Nonconservative or Dissipative Force,
General Form of the Work – Energy Principle, Total Mechanical
Energy, Principle of Conservation of Mechanical Energy, Law of
Conservation of Energy, Power.
In previous chapters we considered the dynamics of a particle. We
capable now to find how position of a particle varies with time for
the special case of a constant force. The problem is much more
difficult, however, when the force acting of a particle and thus its
acceleration are not constant. In this case, the Newton’s laws of
motion actually become the differential equations and their
solution becomes not so simple as before. But more important is
following problem. Using this approach that is called force
approach, we need to know force at any instant of time and at any
position. Therefore we need a lot of information, and this
information could not be always available. In subsequent two
chapters, we discuss an alternative analysis of the motion of an
object in terms of quantities that we will introduce – in terms of
energy and momentum. The significance of these quantities is that
they are conserved. This gives us a deeper insight into nature of the
world and effective way to solve problems. In previous chapters,
we considered the motion of a single particle or two connected
particles. In subsequent chapters, we will try to extend our
consideration to systems of particles. The conservation laws of
energy and momentum are especially valuable in dealing with
systems of many particles, in which a detailed consideration of the
forces involved would be difficult or impossible. These laws are
applicable to a wide range of phenomena, including the atomic and
subatomic world, where Newton’s laws of motion do not apply.
Energy is great importance because it is useful not only in the
study of motion, but in all areas of physics and in other sciences as
well. We will start with the concept of work that is in close relation
with the concept of energy
7.1 Work
Work is defined in Physics as follows: When object undergoes a
displacement d with magnitude d along a straight line, while a constant
force F with magnitude F, making an angle θ with d, acts on the object, the
work done by the object is
W = F d cosθ
(7-1)
The work is a scalar quantity. Unit of the work in the SI system is joule (J)
called after famous English scientist James Joule who made a large
contribution into discovery of the law of conservation of energy.
1 joule = 1 J = 1 N m = 1 (kg m²) / s²
(7-2)
When several forces F1, F2, and so on are simultaneously applied to the
object, the total work done can be written as follows
Wtot = W1 + W2 + …= F1 d cosθ1 + F2 d cosθ2 +…= Ftot d cosθtot
(7-3)
Where Ftot is the total force (the vector sum of all forces applied to the object,
θtot is the angle between this force and displacement. From definition of the
work (7-1), we can deduce that when projection of the force vector on the
direction of motion directed in the same direction as the displacement
vector, work is positive: W > 0. When it is directed in the opposite direction,
the work is negative, W < 0. For example the work done by the frictional
force is always negative Wf = 0, because the frictional force always directed
in the direction opposite to the direction of displacement. When force acted
perpendicularly to the displacement, the work done by this force is zero. For
example, when object is horizontally displaced along flat surface, the work
done by the normal force WN = 0, and the work done by the gravity force Wg
= 0, because during the horizontal displacement of an object along flat
surface both of these forces are directed perpendicularly to the displacement.
However, when object is displaced along incline, WN = 0, but Wg ≠ 0.
7.2 Kinetic Energy.
We know that when force acts on an object appears acceleration (2nd
Newton’s Law of motion), so the velocity of object changes. Now we will
try to relate this change in velocity to the work done when the total
horizontal force Ftot is applied to the object with the mass m and displaced it
in the horizontal direction by displacement d. We will choose as positive Xaxis the direction of displacement. Work done during the displacement can
be written as follows:
Wtot = Ftot d = m a d
(7-4)
Because force is constant acceleration a will be also constant. All time it will
have the same magnitude and direction. Therefore we can use the results of
1D kinematics for motion with constant acceleration. If object initially was
moving with the velocity (speed) v1, and at the end it will have the speed v2,
than
v2² = v1² + 2 a d Æ a = (½v2² -- ½v1²) d
(7-5)
Substituting (7-5) into (7-4), we will get
Ftot d = ½ m v2² -- ½ m v1²
(7-6)
It could be shown that relationship (7.6) is hold when force acting on an
object is not constant. Take a look on a relationship (7-6). Left side of this
relationship depends on the work done by the force during all path of motion
of the object, but the right side depends only on state of the object at initial
and final points disregarding what is between them. You can feel that from
right side we have change in some important physical quantity. This quantity
is called Kinetic Energy
K = ½ m v²
(7-7)
Energy is maybe the most important concept in science at whole. Each
branch of Physics has its own type of energy. It is difficult to give short
strict definition of energy that is good for all cases. In Mechanics, we can
define energy in the traditional way as “the ability to do work.” A moving
object can do work on another object it strikes. A moving object exerts the
force on a second object, which undergoes a displacement caused, by the
force. An object in motion has the ability to do work and thus can be said to
have energy. This is Kinetic Energy defined by the equation (7-7).
The Kinetic Energy as well as work is a scalar quantity and so have no
direction associated with them. This often makes them easier to work with
than vector quantities such as acceleration and force. Kinetic Energy unlike
work is always positive. The unit to measure kinetic energy in SI system is
the same as for the work – joule.
7.3 Work – Energy Theorem.
We can rewrite now equation (7-7) and formulate
Work – Energy Theorem:
During any displacement of a particle, the total work done by the net force
on it is equal to its change in kinetic energy:
Wtot = ½ m v2² -- ½ m v1² = K2 – K1 = ΔK
(7-8)
If Wtot > 0, then the object’s kinetic energy increases and v2 > v1. If Wtot < 0,
then the object’s kinetic energy decreases and v2 < v1. That is, a net force
exerted on an object opposite to the object’s direction of motion decreases
its speed and its kinetic energy. The total kinetic energy of a group of objects
Ktot is the sum of the kinetic energies of the individual objects K1, K2, K3,
and so on.
Ktot = K1 + K2 + K3 + ……
(7-9)
The Work – Energy Theorem is useful in solving problems.
EXAMPLE 7.1. A space probe of mass m = 5.00 • 10^4 kg is traveling at a
speed of v1 = 1.10 10^4 m/s through deep space. Then the engine exerts a
constant force of 4.00 10^5 N on the probe. The force is parallel to the
displacement. As a result, the probe moves in a straight line for a
displacement d = 2.50 10^6 m. Determine: (a) the work done on the probe;
(b) the final kinetic energy of the probe; (c) the final speed of the probe.
m = 5.00•10^4 kg
v1 = 1.10•10^4 m/s
F = 4.00•10^5 N
d = 2.50•10^6 m
θ=0
___________________
(a) Wtot -- ?
(b) K2 -- ?
(c) v2 -- ?
(a)Wtot = Ftot d cos θ = 1.00•10^12 J
(b)Wtot = K2 – K1 = ΔK Æ K2 = K1 + Wtot = 4.03•10^12 J
________
(c) K2 = ½ m v2² Æ v2 = √ 2 K2 / m = 1.27 •10^4 m/s
7.4. Potential Energy.
Sometimes object has ability to do work or potential to do work without
motion. Actually consider an object that is elevated about the ground to
some height y = h (suppose that level of the ground corresponds y = 0). If
the object will be released, it will be in free fall motion and will be able to
perform some work when, for example, strike a pole and drives it into
ground. This object has potential to do work because it is the part of a
system of objects interacting with each other by force depending on the
position. In considered case this force is the gravitational force. The similar
situation will occur when a spring will be stretched to some extent x from
the natural equilibrium position. If we attach some object to the end of the
spring and the spring will be released, the object will be moved and work
will be done. In this case, the parts of the spring interact with each other with
the spring (elastic) force that depends on the position of different parts of the
system of interacting objects. In both cases, we deal with ability to do work,
thus with the energy that we call the Potential Energy. In our course we
will consider two sorts of the potential energy corresponding two types of
forces involved: gravitational potential energy and spring (elastic) potential
energy.
7.4 a. Gravitational Potential Energy.
We will apply Work – Energy Theorem for the object with mass m that
elevated from position y1 above the origin to a higher height y2.
Displacement in this case will be d = y2 -- y1 > 0. We suppose that the object
is located close to the surface of the earth and acceleration of gravity is
constant g = const. In this case, the gravitational force acting on the object is
constant with the magnitude Fg = mg. This force always is directed down.
The displacement d is positive and directed up the angle θ between the
displacement and the force is 180^. The Work – Energy Theorem for this
case can be written as follows:
Wg = Fg d cosθ = mg (y2 -- y1) (--1) = -- (m g y2 -- m g y1)-------(7-10)
Again we encounter in the situation when from the left side of equation we
have work done by the force acting along the all path of object motion. But,
from right side we have difference between physical quantity that is
characteristic of state of the object in final and initial position. In previous
case they were the final and initial velocities, now they are final and initial
positions. This physical quantity is called Gravitational Potential Energy.
We can define the Gravitational Potential Energy Ug as follows:
When an object with mass m is a vertical distance y above the origin of
coordinates, in a uniform gravitational field g = const, the gravitational
potential energy of the system is
Ug = mgy
(7-11)
Work done by the gravitational force (7-10) now can be written:
Wg =-- (Ug2 – Ug1) = -- Δ Ug
(7-12)
Remember that for any physical quantity Δ always means final value minus
initial value. The negative sign in the expression (7-12) is essential. When
the object moves up, the gravitational force does negative work, but the
potential gravitational energy increases.
7.4 a. Elastic Potential Energy.
In consideration of elastic potential energy, we will use the same approach
as for the gravitational potential energy. But before we need to get an
expression for the elastic (spring). Suppose we apply a force with magnitude
Fspr at each end of a spring and stretch this spring an amount x beyond its
unstretched length. Experiment show, that if elongation is not to great, the
spring obeys Hooke’s Law: The Spring Force Fspr is proportional to the
distance by which the spring is stretched or compressed
Fspr = -- kx
(7-13)
Where k is characteristic of the spring it is a positive proportionality
constant called the Force Constant or Spring Constant. It is measured in
units N / m. The negative sign in (7-13) means that the spring force always
directed oppositely to the direction of displacement. This is why spring force
sometimes called Restoring Force. The Hooke’s Law is an approximate
relation. It is good for metallic springs when deformations are small
compared with the overall length of the spring. Solving problems with
springs, we always suppose that they obey Hooke’s Law.
Now we will apply Work – Energy theorem to this situation. We need to
calculate work done by the spring force. We could not use for this purpose
just equation (7-1), because the spring force unlikely to the gravitational
force is not constant, it depends on the position x. It could be shown that if
you have a graph of a force as a function of position, then the area under the
curve between the initial and final positions represents the total work done
by the force. From the Hooke’s Law equation (7-13) we can deduce that
there is direct proportionality between the magnitude of the spring force and
position. Therefore the graph Fspr vs. x should be a straight line. The total
work done on the spring stretched to the position x by the spring force kx is
represented by the area under the straight line. This area looks as a triangle
and can be calculated as the half product of the base x and side kx, so the
area will be to equal ½ k x². Finally, if the spring is stretched a distance x1 at
the start and then the work W we have to do by applying the force F = kx on
it and stretching the spring to a greater elongation x2 can be written as
follows:
W = ½ k x2² -- ½ k x1²
(7-14)
The spring force all time has the same magnitude as applied force but
directed in opposite direction. Therefore the work done by the spring force is
Wspr = -- (½ k x2² -- ½ k x1²)
(7-15)
Following the same procedure as for gravitational work, we define Elastic
Potential Energy as follows: When spring that obeys Hooke’s law is
stretched or compressed a distance x from its undistorted state, the
associated elastic potential energy
Uel = ½ k x²
(7-16)
The SI unit to measure elastic potential energy is the same as for all other
types of energy -- joule. We can write now (7-15) as follows
Wspr = --(Uel2 – Uel1)
(7-17)
7.5. Conservative and Nonconservative Forces. General Form of the
Work – Energy Theorem:
Thus, for gravitational and spring forces it is possible to introduce
corresponding potential energies (potential energy functions). These forces
are called conservative because if only these forces act on the object, then
there is the law of conservation of energy. The work of conservative forces
is always reversible. If, for example, external force elevates an object
performing the work against gravitational force, this work will be brought
back when object will be released and falls to the initial position. The elastic
force will do the same, if a spring will be stretched and then released. As a
result, when starting and ending points are the same, the total work done by
a conservative force is zero. In our course, we will consider two
conservative forces: gravitational force and spring (elastic) force.
Not all forces are conservative. We could not introduce the potential energy
for this force. Friction force does not bring back work done against it. When
the direction of motion reverses, so does the friction force, and it does
negative work in both directions. This type of force is called a
Nonconservative or Dissipative Force. Now we will try to generalize the
work—energy principle including there along with kinetic energy the
potential energy also. We split up total work done into work done by
conservative forces Wc and work done by nonconservative forces W nc
Wtot = Wc + Wnc
(7-18)
As a result we will get
Wc + Wnc = Δ K = K2 -- K1
(7-19)
The work done by conservative forces is related to the corresponding
potential energy U by the same way it was in cases of gravitational (7-12) or
elastic potential energies (7-17): We = --(U2 – U1). Substituting this
expression into (7-19) and performing some rearrangement we will get
finally General Form of the Work – Energy Principle:
Wink = Δ K + Δ U
(7-20)
The work done by the nonconservative forces acting on an object is equal to
the total change in kinetic and potential energies.
7.6 Conservation of the Total Mechanical Energy.
When no nonconservative forces are present, then Wnc = 0, and from
(7-20) we will get
(K2 -- K1) + (U2 – U1) = Δ K + Δ U = Δ (K + U) = Δ E = 0
(7-21)
Where E is called the Total Mechanical Energy of the system. It is the sum
of the kinetic and potential energies at any moment:
E=K+U
(7-22)
If the change in some physical quantity is zero, it means that this quantity is
constant, therefore using (7-21) and (7-22) we can right
E1 = E2
K1 + U1 = K2 + U2
(7-23)
(7-24)
The total mechanical energy E remains constant as no nonconservative
forces ac. (K + U) at same initial time t1 is equal to the (K + U) at any later
time t2. We can also rewrite (7-21) as
Δ K = -- Δ U
(7-25)
That is, if the kinetic energy of a system increases, then the potential energy
must decrease by an equivalent amount to compensate, but the total
mechanical energy remains constant. We can formulate the Principle of
Conservation of Mechanical Energy:
If only conservative forces are acting, the Total Mechanical Energy of a
system in any process stays constant – it is conserved.
But, why the total mechanical energy is conserved? Laws of conservation in
Physics result from the basic properties of space and time. Particularly, the
law of conservation of the total mechanical energy stems from uniformity of
time.
The Principle of Conservation of Mechanical Energy is very useful in
problems in which there are no conservative forces. Below we will specify
expression (7-24) for different possible situations. If there is no any mention
of spring forces in the problem, but there is change of elevation of object,
then equation (7-24) can be written as follows:
½ m v1² + mgy1 = ½ m v2² + mgy2
(7-26)
If there is no mention of changes of object’s elevation, but there are changes
in position of the spring’s end, there is only change in the elastic potential
energy. Then equation (7-24) can be written as follows:
½ m v 1² + ½ k x1² = ½ m v2² + ½ k x2²
(7-27)
If both the gravitational and elastic potential energies are changing, then
equation (7-24) can be written as follows:
½ m v1² + mgy1 + ½ k x1² = ½ m v2² + ngy2 + ½ k x2²
(7-28)
EXAMPLFE 7.2. Only gravitational potential energy is involved.
A roller coaster, shown on Fig. 7.3 is pulled up to point A when is released
from rest. Suppose that frictional is so small that could be neglected.
Calculate the speed at points (a) B, (b) C, (d) D.
hA = 30 m
vA = 0
hB = 0
hC = 25 m
hD = 12 m
(a) vB -- ?
(b) vC -- ?
(c) vD --?
A
C
D
hA
hC
B
hD
Fig. 7.1 Example 7-2.
We choose y = 0 at point B (usually it is most convenient choice as zero-level
the lowest level in a problem). With no friction, energy is conserved.
The initial (and constant) energy is
E = EA = mghA + (1/2)mvA2
= m(9.8 m/s2)(30 m) + 0 = (294 J/kg)m .
(a) At point B we have
E = mghB + (1/2)mvB2;
(294 J/kg)m = m(9.8 m/s2)(0) + (1/2)mvB2,
which gives vB = 24 m/s
(b) At point C we have
E = mghC + (1/2)mvC2;
(294 J/kg)m = m(9.8 m/s2)(25 m) + (1/2)mvC2,
which gives vC = 9.9 m/s
(c) At point D we have
E = mghD + (1/2)mvD2;
(294 J/kg)m = m(9.8 m/s2)(12 m) + (1/2)mvD2,
which gives vD = 19 m/s
EXAMPLFE 7.3. Gravitational and Elastic potential energies are
involved. A vertical spring, whose spring constant is 900 N/m and mass can
be ignored, is attached to a table and compressed down by 0.150 m. (a) What
upward speed can it give to a 0.300 kg ball when released? (b) How high
above its original position when spring compressed will the ball fly?
v=0
h
+y
v
x
y=0
Fig. 7.2. Example 7-3.
We choose the potential energy to be zero at the compressed
position (y = 0).
(a) For the motion from the release point to where the
ball leaves the spring, we use energy conservation:
Ki + Ugravi + Uspringi = Kf + Ugravf + Uspringf
0 + 0 + (1/2)kx2 = !mv2 + mgx + 0;
(1/2)(900 N/m)(0.150 m)2 =
(1/2)(0.300 kg)v2 + (0.300 kg)(9.80 m/s2)(0.150 m),
which gives v = 8.03 m/s
(b) For the motion from the release point to the highest
point, we use energy conservation:
Ki + Ugravi + Uspringi = Kf + Ugravf + Uspringf;
0 + 0 + (1/2)kx2 = 0 + mgh + 0;
0 + 0 + (1/2)(900 N/m)(0.150 m)2 = (0.300 kg)(9.80 m/s2)h,
which gives h = 3.44 m.
But what should we do if the nonconservative forces are working in the
problem? We should use the General Form of the Work – Energy Principle.
(7-20). In problems, we usually face frictional force Ff as nonconservative
force. In this case, we will have Wnc = -- Ff d, because frictional force
always directed in opposite direction with respect to displacement d. Then
after some rearragement we can write the General Form of the Work –
Energy Principle in the form:
K1 + U1 = K2 + U2 + Ff d
(7-29)
EXAMPLE 7.4. Nonconservative (frictional force) is involved. Suppose
that the roller coaster considered in the Exemple 7-2 (Fig. 7-1) passes point
A with a speed of 1.70 m/s and friction force is not zero and is equal to one
fifth of its weight. With what speed will the roller coaster reach point B? The
distance traveled is 45.0 m.
hA = 30 m
vA = 1.70 m/s
hB = 0
Ff = 1/5 m g
vB -- ?
We choose y = 0 at point B. For the work-energy principle
applied to the motion from A to B, we have
WNC = ΔK + ΔU = ((1/2)mvB2 – (1/2)mvA2) + mg(hB – hA);
– 0.20mgL = ((1/2)mvB2 – (1/2)mvA2) + mg(0 – hA);
– 0.20(9.80 m/s2)(45.0 m) = (1/2)vB2 – (1/2)(1.70 m/s)2 – (9.80
m/s2)(30 m),
which gives vB = 20 m/s
7.7. Other Forms of Energy. Law of Conservation of Energy.
From (7.29) it is seen that if there is friction, the mechanical energy (sum of
kinetic and potential energies) does not remain constant but decreases.
Because frictional forces (for example, dry friction, liquid, friction, air
resistance and so on) reduce the total mechanical energy, they are called
dissipative forces. Historically, the presence of dissipative forces hindered
the formulation of a conservation of energy law until well into XIX century.
Friction usually increased the temperature of contact bodies. It was
understood that the heat actually is a thermal energy. Other forms of energy
(electric, nuclear, chemical) were studied later. Energy can be transferred.
Transfer of energy is accompanied by the performance of work. The energy
can be transformed from one type into another. But, one of the great results
in Physics and in science at whole is that no energy is gained or lost.
This is famous Law of Conservation of Energy:
The total energy is neither increased nor decreased in any process.
Energy can be transformed from one form t another, and transferred
from one body to another, but total amount remains constant.
For conservative mechanical systems, this law we derived from Newton’s
laws of motion. But in its full generality, the validity of the law of
conservation rests on experimental observations. And though Newton’s laws
have been found to fail on the atomic world, the law of conservation of
energy has been found to hold there and everywhere. It was tested and
confirmed in every experimental situation so far.
7. 8. Power.
Often we need to know how quickly work is done. We will introduce the
new physical quantity -- Power Pav that helps to answer this question. It is
the time rate at which work is done or energy is transferred. In this course,
we will use the concept of Average Power Pav that can be defined as
follows:
When a quantity of work ΔW is done during a time interval Δt, the
average power Pav, or work per unit time, is defined as
Pav = ΔW / Δt
(7-30)
Power like work and energy is scalar quantity. Unit of Power is 1 joule per
second or J / s. Power is an important physical quantity in the science and in
technology as well. Because of this the unit of Power is given a special
name, the watt (W): 1 watt = 1 W = 1 J/s. This unit is called after Scottish
engineer James Watt who made fundamental improvement to the steam
engine leading to the widespread use of steam power in mines during
industrial revolution in England. The first Watt’s engine can substitute two
horses that used to rotate water pump. James Watt stated that this engine has
power 2 horsepower. This is the other unit of power used mostly in
technology: 1 horsepower = 1hp =746 W. A car is limited by the rate at
which it can do work. This is why automobile engines usually rated in
horsepower.
It is often convenient to write powering terms of the net force applied to an
object and the speed v. This is readily done in the simplest case when the net
force F is constant and directed in the same direction as a displacement d:
Pav = ΔW / Δt = Δ (F d) / Δt = F (Δd/Δt) = F vav
(7-31)
Power units can be used to define new units of work or energy. The
kilowatt-hour (kWh) is the total work done in 1 hour (1h = 3600 s) when
the power is 1 kW (1 kW = 1000 W =1000 J/s):
1 kWh = (kWh) (h) = (1000 J/s)(3600 s) = 3.6 10^6 J
(7-32)
Give a quick look at (7-32) and you understand why electrical bills are rated
in kilowatts but not in joules.