USC2001 Energy
Lecture 1 Energy and Work
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email matwml@nus.edu.sg
http://www.math.nus/~matwml
Tel (65) 6874-2749
1
WHAT IS ENERGY ?
[1] The American Heritage Dictionary of the English
Language, Houghton Mifflin, Boston, 1992.
1 The capacity for work or vigorous activity, strength
2 Exertion of vigor or power
‘a project requiring a great deal of time and energy’
3 Usable heat or power
‘Each year Americans consume a high percentage
of the world’s energy’
4 Physics. The capacity of a physical system to do
work -attributive. energy – conservation, efficiency
2
WORK IS ENERGY
[1] Appendix: PIE http://www.bartleby.com/61/roots/IE577.html
(old form 5.5-7 thousand years ago) Werg – to do
(suffixed form) Werg-o
derivatives handiwork,boulevard,bulwark, energy, erg, ergative,-urgy;
adrenergic,allergy,argon,cholinergic,demiurge, dramaturge,endergonic,
endoergic,energy,ergograph,ergometer, ergonomics,exergonic,exergue,
exoergic,georgic,hypergolic,lethargy,liturgy,metallurgy,surgery,synergids
ynergism,thaumaturge,work
Greek: ergon  energos  energeia  Latin: energia  French:energie
Germanic: werkam  Old High German: werc, Old English: weorc,werc
(zero-grade form) Wig
derivatives wrought, irk, wright
(o-grade form) Worg
derivatives organ, organon (= tool), orgy
3
WEIGHT LIFTING
Physicists define Work  Force Distance
d
or
W  F d
F
m
g
d
F  mg
in energy units called Joules (J)
Newtons (N)
is the mass of an object in Kilograms (kg)
2
is the acceleration of gravity = 9.8 m / s
is the distance that the object is lifted Meters (m)
is force required to lift the object in
Questions What is weight? Can F be exactly constant?
4
ARCHIMEDES
“ Give me a place to stand and I will move the Earth”
Earth
load arm
lightweight
braggart
effort arm
fulcrum
http://www.shu.edu/projects/reals/history/archimed.html
https://www.cs.drexel.edu/~crorres/Archimedes/contents.html
http://wow.osu.edu/experiments/simplemachines/levers.html
Questions The Earth’s mass is 5.98E24 kg, if Archimedes’
is 65 Kg what is the geometry of his lever? What is his
lever principle and what are some tools that employ it?
5
PULLEYS
In the balance shown below, the heavier/lighter mass
may be lifted by lowering the lighter/heavier mass.
1m
2kg
1kg
2m
The objects move in opposite directions by distances
that are inversely proportional to their masses ?
Question What is the golden rule of mechanics?
http://www.hp-gramatke.net/pmm_physics/english/page0200.htm
6
Distance Dependent Forces
R  6.37 x 10 m
Our formula F  mg is only an approximation valid
6
The Earth’s radius is
for objects whose distance r from the Earth’s centre is
very close to R
Isaac Newton’s Universal Law of Gravitation gives
mMG
F
2
r
where M is the Earth’s mass
and G is the gravitational constant
Question Why is G  6.673 x 10
-11
2
2
Nm /kg ?
7
d2
How to Compute Work
F ( s ) ds where F is in the s-direction
W 
d1
and d1  d 2
are the initial and final values of s
This integral is the area between the graphs of
s  d , s  d and y  0, y  F ( s )
1
2
y
d1
d2
s
Question What work is required to lift an object,
having mass m, from the Earth’s surface to height d?
Rd
Answer
mgd
2
2
W 
R
mgR / s ds  1 d / R
8
WORK TO COMPRESS A SPRING
The figure below illustrates a spring being compressed
k = spring constant
xi
xf
Initial Distance
Final Distance
Question What is the compression work integral?
Answer Hook’s Law states that F ( x )  k ( x  xi )
therefore
W 
xf
xi
F ( x) dx  k 
xf
xi
( x  xi ) dx
9
TUTORIAL 1
1. Compute the work to lift an object, having mass
1kg, from the Earth’s surface to ‘infinite’ height.
2. Compute the work to compress a spring by 1m if
the final force equals 5N.
3. How do snakes crush prey with > 6 x their strength?
4. Compute the work to compress a volume V of gas
with pressure P by a small amount dV. Compute the
work to compress it to one tenth of its initial volume at
constant temperature under ideal gas law assumptions.
5. Compute the work that the force of gravity does
to an object having mass m as it slides a distance d
along an inclined plane. How are vectors involved?10