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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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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 (18181889), 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 10 6 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 m 10-3
Kilo k
103
Mega
M
10-6 Giga
Tera
T
Peta
P
106 Exa E
G 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 = 3.41 Btu/hr
[Watt]
1 hp= 2545 Btu/hr
= 746 W [Horse Power]
Fuel
1
1
1
1
1
1
1
barrel crude oil = 5.8 x 106 Btu = 6.12 x 109 J
standard ft3 natural gas (SCF) = 1000 Btu = 1.055 x 10 6 J
therm = 100,000 Btu
ton bituminous coal = 25 x 106 Btu
ton 238U
= 70 x 1012 Btu
ton
= 907.2 kg
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 = VII = 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
ampere-hours (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
Sot = 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
– 1/2mv2 = mgs
– 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.