Transcript Lecture_1 - Department of Mathematics

USC2001 Energy
Lecture 1 Energy and Work
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
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Tel (65) 6516-2749
1
SUMMARY OF THREE DYNAMICS LECTURES
Lecture 1. Energy and Work : work as lifting, levers
and pulleys, gravitational force, springs, statics.
Lecture 2. Kinetic Energy in Motion : Newton’s 2nd
Law, falling bodies, work-kinetic energy theorem,
oscillators, collisions, momentumm Newton’s 3rd Law
Lecture 3. Thermodynamics of Heat : thermometers,
mechanical derivation of ideal gas law, work and heat,
thermodynamic processes, entropy, 1st and 2nd Laws
Related Focus Topics : mechanical engines, steam
and internal combustion engines, refrigeration and
energy conversion, biomechanics.
2
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
3
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
4
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?
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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?
6
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
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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 ?
8
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
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WORK TO COMPRESS A SPRING
The figure below shows a spring being compressed.
k = spring constant
L
Uncompressed
Compressed
Question What is the compression work integral?
Answer Since Hook’s Law gives F ( x )  kx
L
L
W   F ( x) dx   kxdx  kL
0
0
1
2
2
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NEWTON’S 1st LAW
If no force acts on a body, then the body’s velocity
cannot change; that is, the body cannot accelerate.
Note: force is a vector quantity
– it has both magnitude and direction!
What happens if two or more people
pull on an object? This question leads
to the following more precise statement
If no net force acts on a body, then the body’s velocity
cannot change; that is, the body cannot accelerate.
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STATICS
Why is this object static (not moving) ?


mg
Hint: What are the forces acting on this object?
What is the net force acting on this object?
12
VECTOR ALGEBRA FOR STATICS
The tension forces are

Fl 
 a cos θ 
 a sin θ 
The gravity force is



Fg 

Fr 
 0 
 mg
b cos 
 b sin  
13
TUTORIAL 1
1. Design a pulley that a strong person who weights
70 kg can use to lift a 700 kg object.
2. Compute the work required to lift an object with
mass 7 kg from the Earth’s surface to ‘outer space’.
3. Compute the work required to compress gas
with volume V and pressure P to volume V/2.
4. Compute the work required to stretch a spring
with stiffness k by distance L.
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TUTORIAL 1
5. A 100 kg woman stands with her legs making 45
degree angles with respect to the vertical direction.
What is the compressive force in her knees ?
6. How is biomechanics important for orthopaedics ?
7. What is Pascal’s law for fluid statics ?
8. Compute the mass of the object on the side of the
block below that has length 4m so that the system
is in equilibrium (there is no movement). ?
? kg
2 kg
1m
4m
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