49
ENERGY
FE5
OBJECTIVES
Aims
From this chapter you will acquire a basic understanding of the concepts of work and energy.
You will not understand everything about them, because energy is an idea which has no adequate
simple definition. It is also a concept which pervades all of science, so you can expect to learn
more and more about energy throughout this course.
Minimum learning goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain, interpret and use the terms:
work [mechanical work], energy, joule, power, average power, watt, kinetic energy,
conservative force, non-conservative force, potential energy, gravitational potential energy,
mechanical energy, internal force, external force, system, isolated system.
2.
State and apply formulas for work done by constant forces on objects moving along straight
paths.
3.
Calculate work and potential energy from force-displacement graphs. Sketch and plot
graphs of potential energy against position.
4.
Calculate gravitational potential energy for objects near the Earth's surface.
5.
State the conditions for the total mechanical energy of a system to be conserved.
6.
Solve simple problems involving goals 2 to 5 above.
PRE-LECTURE
5-1
INTRODUCTION
Newton's three laws of motion can, in principle, be used to study the motion of any complex
system with the following important exceptions:(i) systems of atomic dimensions or smaller, e.g. atoms in a molecule, electrons in an atom,
protons and neutrons in an atomic nucleus (here a new form of mechanics called quantum
mechanics must be used);
(ii) systems containing particles moving at speeds near the speed of light (here Einstein's
relativistic mechanics, a modification of newtonian mechanics, must be used).
Physicists do not fully understand how to treat systems of atomic dimensions moving at
speeds near that of light.
Newton's laws are a valid approximation for systems of particles which are much larger than
atoms and move at speeds small compared to that of light. Even in these cases, however, it may
not be feasible to analyse the motion using Newton's laws because of mathematical complexity
associated with large numbers of particles. Physicists use the concept of energy to treat such
systems. Energy also plays a central role in quantum mechanics.
Historically, the concept of energy was first applied to mechanical systems like colliding
balls and the pendulum. It was observed that the sum of quantities called ‘kinetic’ and ‘potential’
energies was a constant. Later the concept was generalised to include other forms - nuclear,
FE5: Energy
chemical, radiation (light), electrical, heat and sound energy. The divisions between these forms
are somewhat arbitrary. For example, thermal energy is now understood to be nothing more than
mechanical energy of the atoms and molecules of the system.
Energy is conserved but may be transformed from one form to another, e.g. mechanical
energy to heat, etc.
This chapter is one of the key parts of this course. It describes the two forms of mechanical
energy (kinetic and potential) and how energy is transferred to systems by means of a process
called mechanical work. Under certain circumstances, the total mechanical energy (the sum of
the kinetic and potential energies) of a system remains constant in time. This result can be used to
study the motions of particles in very small systems such as nuclei, atoms and molecules, as well
as in large scale systems.
The SI unit of energy and work is the joule (J); l J = 1 N.m = l kg.m2.s-2. To give some
idea of the size of a joule: it is the energy required to raise an object weighing 1 newton (a mass of
about 0.1 kg on Earth) e.g. an apple, by 1 metre.
Power means rate of transfer of energy. In purely mechanical systems this is equal to the
rate at which work is done by some force. The SI unit of power is the watt (W); l W = l J.s-l.
For example a 100 W light globe transfers 100 J of electrical energy to heat and radiation in 1 s.
And lifting an apple weighing 1 N a height of 1 m in 1 s corresponds to an average power of
1 W.
The buoyant force exerted on objects by the air will be neglected throughout this chapter
since it is very small compared to the weight of the objects considered. (See, for example,
question 3.10 in chapter FE3.)
LECTURE
5-2
ENERGY TRANSFERS IN THE SOLAR ENERGY CYCLE
With the exception of some nuclear fuels, geothermal and tidal power, the sun is the ultimate
source of energy within the solar system. Hydrogen nuclei within the sun combine to release
energy as heat which is transferred to radiation near the surface of the sun. Some of this radiation
is converted to chemical and thermal energy at the Earth.
Radiation is absorbed directly by molecules in the atmosphere and at the Earth's surface and
appears as thermal energy. This energy is partially re-emitted as radiation; the remainder is
transferred to mechanical energy, for example in atmospheric motions and in the water cycle.
The leaves of plants use radiant energy to convert carbon dioxide and water to carbohydrates
by photosynthesis. This energy undergoes further chemical conversion in animal life and can also
be recovered by burning fossil fuels such as coal and oil. (If you are interested, see the diagram in
the interlude following this chapter and Scientific American, September 1970, for further details.)
5-3
MECHANICAL WORK - A MEANS OF ENERGY TRANSFER
Work is a measure of energy transfer. Work is done by a force acting on an object whenever the
object moves. If there are several forces acting on the same object then each of them does work.
(This is different from the colloquial use of the word ‘work’ - in the physics definition, no work is
done if there is no movement.)
50
51
FE5: Energy
Work done by a constant force for straight-line motion
Case (i): Force parallel to the motion
Consider first the simple case of a particle moving along a straight path, being acted on by a
constant force F, which acts in the same direction as the motion.
W = F (x l - x 0) ,
i.e. the work is equal to the product of the force and the displacement.
Case (ii): Force at an angle to the motion
In this case we consider a constant force in a direction which makes a constant angle  to the
direction of the motion. Work is done only by the component, Fx , of the force which is parallel
to the line of motion. The work done by the constant force as the particle goes from position x0 to
a point x l is:
W
=
Fx∆x = F cos(x1 - x0)
... ( 5.1) .
•
•
•
Note the following.
Case (i) is a special instance of (ii) with  = 0.
The component of F perpendicular to the line of motion does no work.
The work done by F is the same as that done by Fx alone.
•
The work done is positive if the component Fx is in the direction of motion and negative if
Fx is opposed to the direction of motion; i.e. if the force is ‘holding back’ the object, it does
negative work.
•
If the constant force component Fx is plotted as a function of the position x of the object, the
work done is represented by the area between the graph and the x axis from x0 to xl as shown in
figure 5.1.
Parallel force
component
Fx
W
x0
Figure 5.1
x
1
Position x
Work done by a constant force during a
straight-line displacement
Sign convention for graphs
In drawing graphs we arbitrarily choose a particular direction and call it positive. If Fx and the
displacement (∆x = xl - x0) are both positive or both negative, then W is positive. However if Fx
and the displacement have opposite signs (i.e. if the force is opposed to the direction of motion)
then W is negative.
52
FE5: Energy
Motion in a straight line with a variable force
Now consider the case where F is an arbitrary force, so that Fx varies with position x as the body
moves along a straight path. On a graph of Fx against x the work W is now represented by the area
between the curve and the x axis (with the above sign convention) See figure 5.2.
Parallel force component
Fx ( x )
x
+
0
x
-
Figure 5.2
1
Position
x
Work done by a variable force in straight line motion
Only the component F of the force which is parallel to the straight path does any work.
x
The procedure illustrated in figure 5.2 is equivalent to the evaluating the integral
W
x1
x
2
Fx dx
.
... (5.2)
General case
In the most general case of all, in which the motion is along a curved path, work can be defined
by considering a variable coordinate direction which is everywhere parallel (tangential) to the path
traced out by the moving object. The coordinate x is replaced by a coordinate measured along the
path, whose magnitude is equal to distance, s, along the curved path. Consider a graph of the
tangential component, Fˆ, of the force plotted against the coordinate s. Then the work done by the
force F is represented by the area under the curve, exactly as in the one-dimensional case.
5-4
TRANSFER OF ENERGY TO SYSTEMS BY MECHANICAL WORK
A force doing a positive amount of work on a system transfers energy to that system. The energy
can end up in one or more of the following forms:•
non-mechanical energy,
•
kinetic energy
or
•
potential energy
Non-mechanical energy
An example is a system consisting of a rigid object and a rough surface over which the object
moves (figure 5.3). The applied force does work on the object while the object moves at constant
velocity, and energy is transferred to thermal energy at the surfaces in contact. Note that there
must be an equal and opposite force - friction - acting on the object which balances the applied
force, since we are told that the object is moving at constant velocity.
53
FE5: Energy
Applied force
Friction
Figure 5.3
Object dragged across a rough surface
Kinetic energy
An example is a system consisting of an object and smooth (frictionless) surface on which it sits
(figure 5.4). When a horizontal force is applied to the object there is no opposing force - the
applied force increases the speed of the object and hence, also, its kinetic energy.
Applied force
Smooth surface
Figure 5.4
Object dragged across a perfectly smooth surface
The kinetic energy of an object of mass m travelling at speed v is defined to be
K
=
1
2
mv2 .
...(5.3)
Potential energy
An example is a system consisting of a rigid object, a smooth surface on which it sits and a spring
connected as shown in figure 5.5. Suppose the applied force is pulling the object to the right at
constant velocity, thereby stretching the spring. (Note that the applied force must be equal and
opposite to the force exerted by the spring on the object since the velocity of the object is
constant.) What happens to the energy transferred to the system as work is done by the applied
force? Energy is not being transferred to either non-mechanical energy or kinetic energy, during
this constant velocity motion.
Figure 5.5
Spring opposing the motion
Consider a situation where the spring is stretched with the object on the right. Suppose now
that the applied force is reduced and kept slightly less than the force exerted by the spring. The
object will move to the left towards the position where the spring is unstretched. During this
process the force in the spring is doing work on the pusher (the object exerting the applied force),
which is not part of the system. If the spring is ideal, the magnitude of the work done by the
spring equals the work done by the applied force in extending the spring. The energy released by
the system comes from the potential energy (PE) stored when the spring is stretched. Potential
energy should be thought of as a property of the whole system and not of individual parts of the
system.
54
FE5: Energy
The force exerted by the spring is called conservative since the energy transferred to the
system can be recovered directly as mechanical energy. There is always a conservative force
associated with potential energy.
Frictional forces and drag forces are called non-conservative since mechanical energy
cannot be recovered directly by removing the applied force which produced the motion.
5-5
CONSERVATION OF MECHANICAL ENERGY
Terminology
An internal force is one exerted by one part of a system on another part of the system. An
external force is one exerted by something outside the system on some part of the system. An
isolated system is one on which no external forces are doing work, i.e one to which no
mechanical energy is being added.
The total mechanical energy of a system is the sum of its kinetic energy and its potential
energy.
Principle
If there are no non-conservative forces acting within an isolated system, the total mechanical
energy of the system is conserved (i.e. it remains constant as time progresses).
It does not matter what we take as the zero of the potential energy because in any problem
we are interested only in changes in energy :
change in
change in
change in
total mechanical energy = kinetic energy + potential energy .
5-6
CALCULATING POTENTIAL ENERGY
How can the potential energy be found from the associated internal conservative force?
Example
In the spring example above - case (iii) - we usually choose PE = 0 when the spring has its natural
length, neither extended nor compressed. Then the potential energy at extension x is equal to the
work done by the force exerted by the spring in returning the object from x to 0.
PE(x)
Force exerted by
spring at position x
O
x
O
x
55
FE5: Energy
(a)
Figure 5.6
(b)
Force and potential energy for an ideal spring
There is no component of F perpendicular to the motion, so in this instance Fx is the full
force acting. The PE at x indicated in figure 5.6b is therefore represented by the shaded area in
figure 5.6a.
General procedure
The general procedure for finding the potential energies of the various states of a system is as
follows.
(i) Assign zero PE to one state of the system.
(ii) The PE of any other state of the system equals the work done by the internal conservative
forces when the system moves from that state to the zero PE state.
5-7
WHY IS POTENTIAL ENERGY A USEFUL CONCEPT?
In general the state of a system at any instant is specified by giving the positions and the velocities
of all of the parts of the system. The PE of a state does not, however, depend on the velocities of
the particles or on how the system came to be in that state. It depends only on the positions of the
particles in the system (i.e. on their configuration). Therefore, the change in PE when the system
moves from one state to another will depend only on the configurations of the particles in the two
states and not on how the system moves between the states.
Since the gravitational force is conservative and the electrostatic and the two nuclear forces
are all conservative on atomic and nuclear scales, complicated processes can be studied
comparatively easily using PE diagrams and the principle of conservation of mechanical energy.
POST-LECTURE
5-8
ENERGY TRANSFERRED AS WORK - QUESTIONS
The following two questions are designed to help you revise the definitions given in the lecture.
An object of mass m slides down an incline at angle to the horizontal as shown.
Q5.1

Figure 5.7 Object sliding down an incline
What are the forces on the object?
i)
On figure 5.7 draw in all the forces acting on the object.
ii) Which of these forces are doing work on the object ?
iii) Which of the forces doing work on the object are conservative ? Which are non-conservative ?
For the rest of this question assume that the non-conservative forces present are negligible.
iv) What work is done by the gravitational force on the object when the latter slides a distance d down the
incline ?
What is the increase in kinetic energy of the object during this process?
v)
56
FE5: Energy
F

Figure 5.8
Object dragged up an incline
Draw in the forces.
Suppose that a force F with magnitude F greater than mg sin is used to pull the object a distance d up
the incline as shown in figure 5.8.
What work is done by F ?
What work is done by the gravitational force?
What is the increase in the kinetic energy of the object during this process?
Q5.2
An object of mass m swings at the end of a taut, straight unstretchable rope whose other end is fixed.
Figure 5.9 A swinging object
Draw in the forces. Which forces do work?
i)
On figure 5.9 draw in all the forces acting on the object.
ii) Which of these forces are doing work on the object?
iii) Which of the forces doing work on the object are conservative? Which are non-conservative?
Assume that the non-conservative forces present are negligible. Suppose that the object is released
from rest from some raised position with the rope taut. The work done on the object by the gravitational
force when the object moves from the point of release to the lowest point of its swing equals the kinetic
energy of the object at the lowest point. The object continues its swing until it momentarily comes to rest
at some highest point on the other side. At this highest point the kinetic energy of the object is zero.
Therefore the work done by the gravitational force when the object moves from the point of release to this
highest point is zero. (Here the object moves along a curved path so it is difficult to calculate the work
done by the gravitational force. This problem will be considered from a different viewpoint in Q5.4
below.)
5-9
GRAVITATIONAL PE NEAR THE EARTH'S SURFACE
Consider a system consisting of an object of mass m and the Earth. For our purposes the Earth
may be considered immovable so that all forces acting on it do no work. In any small region near
the Earth's surface it is possible to choose any horizontal level and say that the system has zero PE
when the object is on that level. If the object is at height h above this horizontal level the
gravitational potential energy of this system is equal to mgh. If the object is a vertical distance
h below this horizontal level, the PE is equal to -mgh . (A proof of this result will be outlined in
Q5.10.)
|U| =
|mgh| ... (5.4)
57
FE5: Energy
PE = mgh
h
PE = 0
h
PE = -mgh
Figure 5.10 Gravitational potential energy
Note that these simple expressions hold only if h is small compared to the distance from the
centre of the Earth. If this condition is not satisfied, the change in the gravitational force with
height must be taken into account.
5-10
QUESTIONS
Conservation of mechanical energy
Q5.3
Consider the object sliding down the incline described in Q5.1. Assume that the non-conservative forces
acting on the object are negligible.
The system consisting of the object and the Earth is isolated according to the definition given in the
lecture. (The incline can be treated as part of the Earth.) The gravitational force on the object is now
internal to the system.
i)
Can you explain why this system is isolated?
ii) Use the idea of conservation of mechanical energy and the above expressions for the gravitational
potential energy to find the increase in kinetic energy of the object when it slides a distance d down the
incline. (Do you get the same answer as that obtained using the method of Q5.1?)
Q5.4
Consider the swinging object described in Q5.2. Assume that the non-conservative forces acting on the
object are negligible.
The system consisting of the object and the Earth (the string is not part of the system) is isolated. The
gravitational force is now internal to the system.
i)
Can you explain why this system is isolated ?
ii) Suppose that the object is released from rest from height h above the lowest point of its swing. Find the
maximum kinetic energy of the object and the maximum height to which it rises on the other side.
iii) How would your answers to part (ii) differ if non-conservative forces were not negligible? (Note that the
system would then no longer be isolated.)
Energy dissipation
Q5.5
Use energy considerations to explain why an object thrown vertically into the air takes longer to fall than it
does to rise if air resistance is taken into account (c.f. Q4.4 in FE4). Hint: consider the speed of the object
at a given height on the upward and downward portions of its motion.
Energy balances
Q5.6
Flow of sap in trees
A typical tall tree manages to raise water through a height of 20 m from roots to leaves at a rate of 700 kg
per day. (Approximately 90% of this water is evaporated from the leaves, the remainder is used in the
photosynthesis process.) What energy must be supplied in a day to raise this water?
It is impossible to raise water to a height of more than about 10.3 metres using a vacuum pump which
utilises atmospheric pressure. How do trees manage to accomplish this? (See Scientific American, March
1963, pp 132-142.)
Q5.7
58
FE5: Energy
v
Figure 5.11 At what rate does the possum work?
The metabolism of a possum of mass m is such that the possum is capable of doing work at a maximum
rate of P. When the possum climbs a tall vertical tree at constant speed, it must do work at a rate equal to
the rate of increase of the potential energy of the possum-Earth system. (We can neglect any nonconservative forces for the purposes of this discussion.)
i)
If the possum is moving at speed v, at what rate is it doing work?
ii) If m = 3.0 kg and P = 60 W what is the shortest possible time in which the possum can climb a tree 20 m
high ? (These ideas will be used in chapter FE8 on Scale)
Machines
Machines usually are devices for doing work using a small force travelling a large distance. For
example on an inclined plane like a winding road, only a force large enough to overcome the
component of gravity parallel to the plane (and any non-conservative forces present) is required.
However, one must travel a longer distance to reach any given height.
Q5.8
When using a lever like a crowbar, one applies a force at one end to raise a load at the other (figure 5.12).
a
b
Applied
force
Load
Figure 5.12 Lifting with a lever
Compare forces and works.
i)
By considering torques about the pivot, find the ratio of the load, mg, to the minimum applied force F
required to raise the load, in terms of the ratio of the lengths a and b of the lever on either side of the pivot
point. You may neglect the weight of the lever.
ii) If the load is raised a small height h, the work done by the minimum applied force must equal the increase,
mgh, in the PE of the load-Earth system. How far does the applied force have to move its end of the
lever? (Ignore the fact that the ends of the lever move in slightly curved paths.)
Q5.9
Draw a schematic diagram of the main bones in the arm and the biceps muscle used to raise the forearm.
Discuss qualitatively how this ‘machine’ in the human body differs from the machine used in Q5.8. Hint:
consider the work done by the biceps muscle in raising a load held in the hand.
Gravitational force and potential energy
In questions 5.3 to 5.9 you used the result that the gravitational potential energy of an object-Earth
system depends only on the height of the object. Question 5.10 outlines a proof of this result.
Q5.10
59
FE5: Energy
B
A
E
D
C
F
Figure 5.13 Work done by gravity
Show that the work done by gravity does not depend on the path taken.
In figure 5.13 A is on the same horizontal level as B which is vertically above C , a distance h below.
i)
Find the work done by the gravitational force on an object of mass m when the object moves from A to B
as shown.
ii) Find the work done when the object moves from B to C and hence the total work done when the object
moves from A to C along the path A  B  C.
iii) The path A  D E F C is made up of straight vertical and horizontal segments as shown. Verify
that the total work done by the gravitational force on the object when it moves from A to C along this path
is equal to the result of (ii).
A
C
Figure 5.14
Straight-line approximation to a curved path
Any curved path from A to C can be approximated as closely as desired by a series of vertical and
horizontal segments - see figure 5.14. The work done along the curved path is then approximately equal to
the total work done along the straight line segments, i.e. the result of (ii) .
Thus we can unambiguously define the difference in PE between the states of the object-Earth system
represented by A and C to be the work done by the gravitational force when the object moves from A to C
along any path, i.e. PE is equal to mgh in every case.
Other conservative forces and potential energy
Q5.11
Figure 5.15 Object striking a spring buffer
What happens to its kinetic energy?
Figure 5.15 shows a spring-buffer arrangement to prevent a moving object from striking a wall. The force
F exerted by the spring-buffer on the object is plotted in figure 5.16 as a function of the distance x of the
object from the wall.
60
FE5: Energy
F / kilonewton
8
4
G
E
D
C
B
A
2
3
0
0
Figure 5.16
1
x / metre
Force exerted by the buffer spring on the block
Choose the PE of the system to be zero when the moving object is 3 metres from the wall, i.e. at the
point labelled A .
i)
Find the potential energies at points B, C, D, E and G by calculating the work done by F when the object
moves from those points to A . (Hint: Use areas under the graph.)
ii) Sketch the potential energy diagram for the system.
iii) Use the diagram to find the minimum kinetic energy of the object for it to strike the wall.
iv) How would the answers to (i) - (iii) be altered if the PE of the system had been chosen to be zero at x = 0?
61
FE5: Energy
Example: Atoms in a diatomic molecule
The force between atoms is a complicated one due to individual electrical forces exerted by the
nucleus and the electrons of one atom on those of the other atom. There is an infinite repulsive
force at zero separation and the force decreases and becomes attractive at larger separations
(figure 5.17).
Repulsive force
Component of
force exerted
by atom at
0 on atom
at position
x
x
0
Separation
Attractive force
Figure 5.17
Force between atoms
The graph shows the force exerted by one atom on the other. The component of the force is taken in the
direction of the line joining the atoms. Positive values of the component correspond to repulsion;
negative values represent attraction
The zero of PE for the two atoms is conventionally defined to occur when the atoms are a
very large (infinite) distance apart. Can you see why it would not be sensible to define PE = 0
when x = 0?
The PE is then the work done by F(x) when the separation is increased from x to  , i.e. the
shaded area in figure 5.17. Although the curve extends to very large distances () this shaded
area is finite.
PE( x)
0
x
0
Separation
C
A
B
Figure 5.18
Potential energy for the system of two atoms
By convention the PE is taken to be zero when the separation is infinite.
Q5.12
Use the section, Sign convention for graphs, on page 51 to explain why the PE is negative at the point
marked A in figure 5.18 .
At point B , the PE is minimum. To which point on the force-separation curve does this correspond?
How does the force behave on either side of this point ? What happens to a stationary object placed at that
point? At point C, the PE is zero. To which point on the force-separation curve does this correspond?
62
FE5: Energy
5-11
FINDING THE CONSERVATIVE FORCE FROM THE PE CURVE
Questions 5.10, 5.11 and 5.12 were practice in finding the PE curve for a system from the
appropriate conservative force. It is worth noting in passing that since we obtain the PE curve
from the conservative force by integration (areas under curves) we may also find the force by
differentiating (finding the slope) of the PE curve (c.f. velocity and distance).
d
Conservative force component = - dx (potential energy)
= - (slope of PE vs x curve).
The minus sign is needed to give the correct direction for the conservative force.
This result can be used to give a quick check on whether the PE curve obtained from a
conservative force is correct : the force should be negative if the PE increases as x increases,
positive if the PE decreases and zero if the PE is constant. Try this out on the PE curves you have
met so far.
5-12
CONCEPTUAL MODELS FOR POTENTIAL ENERGY
If you are finding PE curves to be rather abstract, you may like to think of them as depicting a
succession of hills and valleys viewed from the side (see figure 5.19).
PE(x)
PE(x2 )
PE( x 1)
0
x1
Figure 5.19
x
2
x
Potential energy hills and valleys
The difference between the potential energies at x1 and x2 is just like the difference between
the gravitational potential energies when the object is at different heights on the hills above x1 and
x2.
Consider an object sliding without friction up and down the ‘hills’. When viewed from
directly above, only the horizontal motion in a straight line can be seen. The object appears to
speed up and slow down so there must be a horizontal force acting on it. This force at any point is
obviously related to the slope of the hill at that point (c.f. previous section). The PE surface seen
in the TV lecture was a generalisation of the PE curve to a case where the object can undergo a
two dimensional motion.
5-13
POWER
Power, the rate at which energy is supplied to, released by, or dissipated within a system is often a
quantity of interest.
The average power during a time interval is the energy transferred during that time interval
divided by the time interval.
Q5.13
Calculate the average power required to raise the water in the tree of Q5.6.
You will meet energy again in all of the other units of this course. The unit Thermal Physics in
particular is concerned mainly with transfers of energy to and from systems.
Summary: Graphs
63
INTERLUDE 5 - EARTH'S ENERGY BALANCE AND FLOW
Note:
12
1 TW = 1  10 W.
Solar radiation
178 TW
Short wavelength
radiation
Long wavelength
radiation
Direct reflection
62 000 TW (35%)
Tidal
energy
Tides, tidal
currents etc.
3 TW
Direct conversion
to heat
76 000 TW (43%)
Evaporation, precipitation,
runoff etc.
40 000 TW (22%)
Storage:
water & ice
Convection,volcanoes,
hot springs
0.3 TW
Winds, waves, convection and curre nts
Photosynthesis
40 TW
Storage:
plants
Decay
Conduction
32 TW
Animals
Terrestrial enegy
Fossil fuels
Nuclear, thermal &
gravitational enrgy