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PHYS 1211 - Energy and
Environmental Physics
Michael Ashley
Lecture 2
Mechanical Energy
This Lecture
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A Bit of History
Energy and Work
Units of Energy
Power
Kinetic and Potential Energy
1600
1700
1800
1900
2000
Isaac Newton developed his
system of mechanics entirely
without the concept of Energy.
Newton’s description was in
terms of forces and momentum
and the effects forces had on
the motion of objects.
1600
1700
1800
1900
2000
Newton’s contemporary (and rival) Gottfried
Leibniz introduced the idea of what he called
“vis viva” (living force).
“vis viva” = mv2 and was conserved in some
interactions between particles.
It is what we now know as kinetic energy
(although we now use 1/2 mv2)
The term “vis viva” continued to be used for
mechanical energy up to the 1850s.
But Leibniz’s work was largely ignored
because it was thought to be incompatible
with Newton’s law of conservation of
momentum.
1600
1700
1800
1900
2000
It was clear that “vis viva” was not always
conserved.
At the end of the 18th century several
scientists (such as Antoine Lavoisier and
Pierre-Simon Laplace) begin to suspect that
the lost energy appears as heat.
In 1798 Count Rumford (Benjamin
Thompson) studied the frictional heat
produced in boring the barrel of a cannon.
The mass of the cannon does not change as
it is heated.
An indefinite amount of heat can be
generated by friction.
Heat could not be a substance — as argued
by the then widely accepted “caloric” theory
of heat.
Heat must be “a form of motion”
1600
1700
1800
In 1807 Thomas Young first used the term
“energy” in its modern scientific context.
In the 1840’s James Prescott Joule carries out a
series of experiments, showing the equivalence
of mechanical energy and heat.
When a certain amount of work is done (e.g. by
a falling weight), a corresponding amount of
heat is produced.
But Joule’s ideas still met resistance from
supporters of the “caloric” theory.
1900
2000
1600
1700
1800
1900
2000
In the 1850s scientists including Heinrich
Helmhotz, William Thompson (Lord Kelvin)
and Rudolf Clausius formulate the laws of
thermodynamics.
They show that energy exists in forms such as
mechanical energy, heat, light, electricity and
can be converted between these forms but can
never be created or destroyed (the law of
conservation of energy).
In just a couple of decades from about 18401860 “energy” develops from a largely
unknown term to one of the fundamental
concepts of Physics.
William Thompson (Lord Kelvin)
1600
1700
1800
1900
2000
By 1930 scientists are so convinced of the law
of conservation of energy, that when it
appeared not to be conserved — in the process
of beta decay — Wolfgang Pauli proposes a
new particle “the neutrino” to explain where
the energy has gone.
26 years later the existence of the neutrino was
experimentally confirmed — an experiment
that won Frederick Reines the Nobel Prize in
physics.
Bubble chamber tracks showing the
first detection of a neutrino.
The invisible neutrino hits a proton and
produces three tracks (a mu meson, a
pi meson and the proton).
What is Energy?
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•
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Energy is defined as the “capacity to do work”.
So what is work?
In Physics work is defined as the product of a
force times the distance through which the force
acts.
W=Fd
The Joule
• The S.I. unit of work (and energy) is the Joule
(J).
– Named after James Prescott Joule (1818-1889), the
British physicist who studied the relationship
between work and heat.
• One Joule is the work done when a force of one
Newton moves through a distance of one
metre.
Energy and Work
• Energy and Work are both measured in Joules
so what is the difference between them?
– Work is the action involved when a force acts on a
system.
– Energy is a property of the system.
– We “do work on” a system and the energy of the
system increases.
Energy and Work
• Work is only being done for the time for which
the force acts.
• But the Energy gained by the system continues
to exist — at least until something else happens
to change the state of the system again.
People Doing Work
Rock climbing —
lifting the climbers
weight.
Lifting weights — work is done on the weight.
Ball games such as cricket
work is done on the ball.
Machines doing work
A crane lifting a load
A rocket launch
A car accelerating
Example
Hossein Rezazdeh (“the world’s strongest man”) holds the
world record in the super-heavyweight weightlifting class
and won gold medals at the Sydney and Athens Olympics.
His record for the “clean and jerk” is 263.5 kg.
How much work is he doing in such a lift?
The gravitational force is mg = 263.5 * 9.81
= 2585 Newtons.
Assume the distance the weight is lifted is 2m.
The work done is force x distance
= 2 x 2585
= 5170 Joules.
= 5.17 kJ
How much is a Joule?
“AA” alkaline battery
10,800 J
10.8 kJ
Tim Tam biscuit
400,000 J
400 kJ
A litre of unleaded petrol 34,800,000 J 34.8 MJ
Average daily household
electricity usage
72,000,000 J 72 MJ
Other Work (Force x distance) Energy
Units
• Erg
– The unit on the old cgs system. An erg is the work
done when a force of one dyne moves through one
cm. 1 erg = 10–7 J
• Foot pound (ft-lb)
– Imperial unit of work. A force of one pound force
moving through one foot. 1 ft-lb = 1.356 J
“Heat” Energy Units
• 1 calorie is the energy needed to heat 1 gram
of water through 1 degree C. 1 cal = 4.184 J
• 1 kcal (or food Calorie) = 1000 calories = 4.184
kJ.
• 1 Btu (British Thermal Unit) is the energy
needed to heat 1 lb of water through 1 degree
F. 1 Btu = 1055 J
– Despite the name now mostly used in the USA.
Other Units
• 1 kilowatt-hour (kWh) is the energy
corresponding to 1 kW of power used for 1
hour (since 1 kW = 1000 J/sec and 1 hour =
3600 sec, 1 kWh = 3.6 x 106 J).
• Tonne of oil equivalent (toe) is often used in
statistics of national and global energy usage.
– 1 toe = 41868 MJ = 4.1868 x 1010 J (according to IEA
convention).
Power
• Power is defined as the rate of doing work or
converting energy.
• The SI unit of power is the Watt (W)
– Named after Scottish inventor James Watt who made
major improvements to the steam engine.
– One Watt is one Joule per second.
Desktop Computer (iMac 20”)
200 W
Electric Heater
2000 W
Small Car (Honda Jazz)
61 kW
Formula 1 Car
550 kW
Queen Mary 2
86,000 kW
Other Power Units
• A horsepower (hp or HP) is the unit James Watt
actually used to measure power.
– When steam engines were first introduced this was
a useful unit as it indicated how many horses the
engine could replace.
• One hp = 746 W.
• Horsepower is still sometimes used to describe
the power of engines.
Energy Units & Conversion Factors
Prefixes:
Micro
Milli
Kilo
Mega

m
k
M
10-6
10-3
103
106
Giga
Tera
Peta
Exa
G
T
P
E
109
1012
1015
1018
Energy Units
1 Btu = 1055 J = 252 cal [British Thermal Unit]
1 cal = 4.184 J
[calorie]
1kcal = 1000 cal = 1 food Calorie
1kWh
= 3.6 x 106 J
= 3413 Btu [Kilowatt hour]
1 Quad
= 1015 Btu
= 1.055 x 1018 J
1 GJ = 109 J = 948,000 Btu
Power Units
1 W = 1 J/s
1 hp = 2545 Btu/hr
= 3.41 Btu/hr
= 746 W
[Watt]
[Horse Power]
Fuel
1 barrel crude oil = 5.8 x 106 Btu
= 6.12 x 109 J
1 standard ft3 natural gas (SCF) = 1000 Btu = 1.055 x 106 J
1 therm
= 100,000 Btu
1 ton bituminous coal = 25 x 106 Btu
1 ton 238U = 70 x 1012 Btu
1 ton
= 907.2 kg
1 metric ton = 1 tonne = 1000kg
Electrical Energy and Power
• The power of an electrical appliance is the
product of the voltage (V in volts) and the
current (I in Amps) flowing through it:
Power = VI
I = 2 Amps
Mains
Supply
240 V
Power = VI
= 240  2
= 480 Watts
Energy in a Battery
• Energy content of a battery is usually quoted in amperehours (Ah).
• Consider a 6 V battery with a capacity of 10 Ah
• It can deliver 60 W for 1 hour (=3600 seconds)
• Energy = 60  3600 J
= 216,000 J = 216 kJ
Electrical Energy will be discussed in more detail later in the
course.
Kinetic Energy
• The Kinetic Energy of an object is the energy it
possesses because of its motion at a velocity v.
KE = 1/2 mv2
• This expression can be derived from Newton’s
2nd Law.
F = ma
Kinetic Energy
• Consider an object accelerating from rest with
constant acceleration a because of a force
applied to it. After time t:
v = at
s = 1/2 at2
F
m
So t = v/a
t=0
s = 1/2 a(v/a)2 = 1/2 v2 /a
v =0
Work = Fs = 1/2 F v2/a and F = ma
= 1/2 mv2
• So the work done by a force accelerating an
object is 1/2 mv2 and this must be the energy
gained by the accelerating object.
s
m
t=t
v = at
Potential Energy
• Potential Energy is the energy of an object by virtue of its
position.
• Gravitational potential energy is energy due to its position
in the gravitational field.
Since the force of gravity on an object is:
F = mg
where g is the gravitational acceleration (9.81 ms–2) then:
PE = Force x Distance = mgh
where h is the height of the object.
Work, PE and KE
• When work is done on an object its potential or
kinetic energy (or both) is changed.
• For example:
– Lifting a weight — the potential energy of the
weight is increased.
– Throwing a ball — the kinetic energy of the ball is
increased.
Law of Conservation of Energy
• If we consider just mechanical energy then the
following relations hold:
E = PE + KE
Total Energy = kinetic + potential energy
W = E = KE + PE Work done on a system
changes its total energy
If no external work is done on a system:
E = 0
KE = –PE
i.e. Total energy cannot change, but energy can change
from kinetic to potential or vice versa.
Elastic Potential Energy
• Another type of potential energy is the energy
stored in a spring or elastic material.
• The force due to a spring is kx. k is the spring
constant and x is the amount the spring is
compressed (Hooke’s Law)
• The PE is then given by:
PE = 1/2 kx2
Potential Energy as a Power Source
We can use potential energy to power machines.
For example Clocks:
Weight driven clocks use the
gravitational potential energy of
a falling weight to drive the clock
mechanism.
Wind-up spring-driven clocks
use elastic potential energy
stored in a spring.
Exchange of Potential and Kinetic
Energy
• Many systems involve exchange of potential and kinetic
energy.
• Simplest example is a falling object (in the absence of air
resistance).
– After falling a vertical distance s
–
PE = –mgs (–ve sign since PE is lost)
– And it gains an equal amount of KE
1/ mv2 = mgs
–
2
–
v = (2gs)
• We could have got the same result using Newton’s 2nd Law
— using energy is an alternative approach to such problems.
Roller Coaster
A roller coaster operates by converting energy
between potential energy and kinetic energy.
Roller Coaster
Initial
Position (max PE)
As energy is lost due to
friction, successive peaks
have to be lower or the car
would not have enough
energy to reach them
Gaining
velocity and
KE
Minimum PE,
maximum KE
and velocity
Escape Velocity
• Escape velocity is the velocity needed for a
spacecraft to completely escape the Earth’s
gravitational field.
– Spacecraft sent to other planets (e.g. Mars) need to
reach escape velocity.
NASA’s Mars
Reconnaissance
Orbiter
Escape Velocity
• Previously we have used the expression mgh for PE,
but this is only correct near the surface of the Earth
where the gravitational acceleration has the fixed
value g (= 9.81 ms–1).
• In general we have to use the full form of Newton’s
law of gravity.
F = GMm/r2
Where M is the mass of the Earth, r is the distance from the
Earth’s centre, and G is the gravitational constant (G = 6.67 
10–11)
• From F = ma we can now see that the acceleration due
to gravity is GM/r2
PE = 0
at r = 
Escape Velocity
• So in the expression for PE (mgh) we need to
replace g with GM/r2 and h with r giving:
PE = –GMm/r
The – sign is needed so that energy increases upwards.
• At infinite distance (r = ) from the Earth PE = 0.
• At the surface of the Earth (r = R)
PE = –GMm/R
• To launch a spacecraft so it escapes from the
Earth we need to give it a KE (= 1/2mv2) equal to
the PE change from the surface to infinite
distance.
1/ mv2 = GMm/R
2
v2 = 2GM/R
PE = –GMm/R
Earth
Radius R
Mass M
Escape Velocity
• For Earth:
– M = 5.97  1024 kg
– R = 6.37  106 m
• So:
v2 =
2  6.67  10–11  5.97  1024
6.37  106
v = 11,181 ms–1 = 11.2 km s–1
= 40,300 kph.
Soyuz-Fregat rocket
launching the ESA Venus
Express spacecraft
The Pole Vault
• The pole vault illustrates an efficient way of
converting kinetic energy into potential energy.
The Pole Vault
The run up — the pole vaulter must run up
as fast as possible. World record holder
Sergey Bubka has been measured at 22.2
mph = 9.93 ms–1. His kinetic energy is then:
KE = 1/2mv2 = 1/2  80  9.932
KE = 3944 J
(for mass m = 80 kg)
The Pole Vault
As the vaulter plants the pole, he starts to
rise (converting some of his KE to PE), but
also bends the pole. Much of the original
kinetic energy is now stored as elastic
potential energy in the pole.
The Pole Vault
As the pole starts to straighten it
releases its stored elastic potential
energy and converts this to
gravitational potential energy of the
vaulter.
The Pole Vault
The pole is now straight and the vaulter
has reached his maximum height. All the
original KE should now be converted to the
vaulter’s PE.
PE = mgh = original KE = 3944 J
h = 3944/(mg) = 3944/(80  9.81)
h = 5.03 metres.
The Pole Vault
• But Sergey Bubka’s world record was 6.14m
and we have calculated only 5.03m - How is
this possible?
– In fact he starts off with his centre of mass already at a
height of about 1 metre.
– When he crosses the bar his centre of mass would be
only slightly above the bar.
– In addition he can make an extra push off the pole at
the top of the swing.
• However, it would seem like his performance
is near the limit of what is physically
possible.
– Perhaps not surprising that his 1994 world record was
not broken until 2014 (6.16m Renauld Lavillenie).
The 100m Sprint
The 100m event is often
referred to as the Blue
Riband event of the
Olympics and the men who
run it as the “fastest men on
Earth”.
The 100m Sprint
An olympic sprinter does the 100m run in
about 10s (world record is 9.58s). This
means an average speed of about 10 ms–1
However the top speed is more like 12
ms–1 (26.95 mph).
The plot at right shows that in the first
20m the sprinter has accelerated to about
10 ms–1 (22mph).
Using:
v2 = 2as
and
v = at
We can calculate that:
a = 2.5 ms–2
t=4s
i.e. the sprinter takes about 4 seconds to run the first 20m
accelerating at 2.5 ms–2 to 10 ms–1
Then KE = 1/2 mv2 = 4000 J
Power = 4000/4 = 1000 W
( = 1.34 horsepower)
Tour de France
Climb of the Cime de la Bonnete:
Starts at 1152m
Ends at 2802m
A climb of 1650m
The race leaders completed this climb
in 69 minutes.
Speed = 6.4 ms–1 (compared to ~12 ms–
1 on flat stages)
For m = 80kg (cyclist+bike):
PE = mgh = 80  9.81  1650
= 1,295,000 J = 1.29 MJ
Power = 1295000/(69 60)
= 313 W
This probably underestimates total
power as it only accounts for PE gain not air drag etc.
Stage 16 of the 2008 Tour de France
Power Output of Human Body
• So while about 1000W can be achieved over short periods, for
long duration events power outputs are more like 400 W
maximum.
• We will look at the reasons for this difference later — but briefly.
– In endurance events power is limited by the ability of the cardiovascular
system to supply O2 to the muscles.
– Over short periods a different process (“anaerobic respiration”) can be used
to supply energy at higher rates.
Next Lecture
• We will continue our introduction to energy by
looking at thermal energy.