Post on 26-Dec-2015
CONSERVATION LAWS
PHY1012
F
WORK
Gregor Leighgregor.leigh@uct.ac.za
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2
WORK
Learning outcomes:At the end of this chapter you should be able to…
Extend the law of conservation of energy to include the thermal energy of isolated systems.
Calculate the work done on and by systems and apply the work-kinetic energy theorem to the solution of problems.
Distinguish between conservative and nonconservative forces.
Calculate the rate of energy transfer (power).
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THERMAL ENERGY
An object as a whole has:Kinetic energy, K (due to movement)
Potential energy, U (due to position)
Mechanical energy, Emech
Particles within an object (i.e. atoms or molecules)
have:Kinetic energy (associated with the substance’s temperature)
Potential energy (associated with the substance’s phase)
Thermal energy, Eth
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SYSTEM ENERGY
The sum of a system’s mechanical energy and the thermal energy of its internal particles is called the system energy, Esys.
Esys = Emech + Eth = K + U + Eth
Conversions between energy types within the system
are called energy transformations.
Energy exchanges between the system and its environment are called energy transfers.
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ENERGY TRANSFORMATIONS
Isolated system no energy enters or leaves the system.
Transformations are indicated with arrows: e.g. K Eth .
Conversions between K and U are easily reversible, but we say that Emech is dissipated when it is
transformed into Eth since it is
extremely difficult to transform Eth back into Emech.
Friction is a common cause of the dissipation of mechanical energy.
SYSTEM
K
U
Eth
Esys
Emech + Eth
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Esys = W
ENERGY TRANSFERS
The exchange of energy between a system and its environment by mechanical means (i.e. through the
agency of forces) is called work, W.
Energy can also be transferred by the non-mechanical process of heat. (Thermodynamics is not covered by this course.)
Work is regarded as a system asset:
work done on the system by the environ-ment increases the system’s energy: W > 0.
work done by the system on the environ-ment decreases the system’s energy: W < 0.
SYSTEM
= K + U + Eth
K
U
Eth
ENVIRONMENT
ENVIRONMENT
W < 0
Q < 0
W > 0
Q > 0work heat
work heat
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WORK and KINETIC ENERGY
Consider a body sliding on a frictionless surface, under the action of some (possibly varying) force…
…as it moves from an initial position, si, to a final position, sf …
F
ssi , vis sf , vfs
Fs Fs
sdv dsm
ds dt
Newton II: sdvm
dt
sF ss
dvmv
ds(chain rule)
s s sF ds mv dv
F
s sF ma
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WORK and KINETIC ENERGY
… mvs dvs = Fs ds
f f
i i
v ss s sv s
mv dv F ds
½ mvf2 – ½ mvi
2 f
i
sss
F ds
Work done by moving the object from si to
sf.
f
i
sss
F ds W F
Hence K = W
No work is done if sf = si. I.e. To do work, the
force must cause the body to undergo displacement.
Units: [N m = (kg m/s2) m = kg m2/s2 = joule, J]
Notes:
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WORK DONE ON A SYSTEM
Work-Kinetic energy theorem:When one or more forces act on a particle as it is displaced from an initial position to a final position, the net work done on the particle by these forces causes the particle’s kinetic energy to change by K = Wnet.
f
i
s
ssK W F ds
Fs
s
Force curveThe work, W, done on a system is given by the area under a F-vs-s graph.
(cf. Impulse, J, and F-vs-t graphs.)
K p. I.e. you cannot change one without changing the other, since… 2 2
212 2 2
pmvK mvm m
displacement
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f
i
coss
sW F ds
f
i
coss
sF ds
WORK DONE BY A CONSTANT FORCE
In the special case of a constant force…
F
ssi sf
Fs s
W Energy transfer
0° to < 90°
90°
90° to 180°
F(s)…F(s) cos Esys incr; K (and v) incr.
F(s) cos…–F(s)
0 Esys, K (and v) constant.
Esys decr; K (and v) decr.
f
i
sss
W F ds
f icosF s s cosF s
s
F
and
CONSERVATION LAWS
ˆ ˆ ˆ ˆi i = i i cos0 = 1
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THE DOT PRODUCT
The quantity F(s) cos is the product of the two
vectors, force, , and displacement, , and is more elegantly written as the dot product of the two vectors, .
F
s
ˆ ˆ ˆ ˆi j i jx y x yF s F F s s
F s
ˆ ˆ ˆ ˆ ˆ ˆi i i j j jx x x y y x y yF s F s F s F s F s
y
x
1
1
i
j
Note first:ˆ ˆ ˆ ˆi j = i jcos90 0 and:
x x y yF s F s F s
I.e. the dot product is the sum of the products of the components.
ˆ ˆ= j j
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THE DOT PRODUCT
is the angle between the two vectors.
Since it is a scalar quantity, the dot product is also known as the scalar product.
Vectors can also be multiplied using a different procedure (the cross product) to produce a vector product (q.v.).
Notes:
cosx x y yA B A B A B AB
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Instead…
If the force varies in a simple way, we can calculate the work geometrically, by plotting and determining the area under a F-vs-s graph.
Otherwise the integral must evaluated mathematically.
WORK DONE BY A VARIABLE FORCE
If the force applied to a system varies during the course of the motion, we cannot take Fs out of the
integral…
f
i
sss
W F ds
Fnet s (N)
s (m)2 4 60
4
8
0
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WORK DONE BY GRAVITY
y
w
s
s
dy = –cos ds
ds
Consider an object sliding down an arbitrarily-shaped frictionless surface as it moves a short distance ds.
grav cosdW w s mg ds
gravdW mg dy
f f
i igrav
s y
s ydW mg dy f img y y
Work done by gravity is thus path-independent.
Gravity is therefore a conservative force.
Notes:
Wgrav = –mgy
n
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CONSERVATIVE FORCES
The work done by a conservative force on a particle moving between two points does not depend on the path.
The net work done by a conservative force on a particle moving around any closed path is zero.
Conservative forces transform mechanical energy losslessly between the two forms, kinetic and potential.
Any conservative force has associated with it its own form of potential energy: the work done by a conservative force in moving a particle from an initial position i to a final position f, denoted Wc(if) , changes
the potential energy of the particle according to: U = –Wc(if)
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Potential energy… Work done by…
WORK DONE BY CONSERVATIVE FORCES and POTENTIAL ENERGY
Force of gravity, :
U = –Wc
Spring force, :
Wgrav = –mgy
Wsp = –½ k(s)2
Gravitational, Ug:
Elastic, Usp:
Usp = ½ k(s)2
Ug = mgy
w
spF
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NONCONSERVATIVE FORCES
The work done by nonconservative forces is path-
dependent. s
s
kf
Wfric = fk(s)cos180° = –kmgs
Whether the block slides directly to point A, or via point B, makes a difference to s and hence to Wfric.
B
A
All kinetic frictional forces and drag forces are nonconservative forces.
E.g. The work done by friction is
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NONCONSERVATIVE FORCES
A nonconservative force has no associated form of potential energy. Instead, the work done by a nonconservative force increases the thermal energy, Eth, of the system – a form of energy which has no
“potential” for being reconverted to mechanical energy.
A nonconservative force is consequently known as a dissipative force. Thus: Eth = –Wdiss.
Wdiss is always negative since the force opposes motion.
Thus Eth is always positive.
Hence dissipative forces always increase the thermal
energy of a system, and never decrease it.
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CONSERVATION OF ENERGY
(work-kinetic energy theorem)
K = Wnet
K = Wc + Wnc
K = –U + Wnc (i.e. K + U = Emech = Wnc)
K = –U + Wdiss + Wext
K = –U – Eth + Wext
K + U + Eth = Wext
Esys = Wext (energy equation of the system)
(choose system carefully to include all dissipative forces)
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LAW OF CONSERVATION OF ENERGY
The total energy Esys = Emech + Eth of an isolated system is
a constant.
The kinetic, potential and thermal energies within the system can be transformed into each other, but their sum cannot change.
Further, the mechanical energy Emech = K + U is
conserved if the system is both isolated and non dissipative.
CONSERVATION LAWS
dWPdt
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POWER
Power is the rate at which energy is transformed or transferred:
sysdEP
dt
Units: [J/s = watt, W]
Power is also the rate at which work is done:
Hence: P = Fv cos
F dsdt
dsFdt
F v
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WORK
Learning outcomes:At the end of this chapter you should be able to…
Extend the law of conservation of energy to include the thermal energy of isolated systems.
Calculate the work done on and by systems and apply the work-kinetic energy theorem to the solution of problems.
Distinguish between conservative and nonconservative forces.
Calculate the rate of energy transfer (power).