Download Mechanica Work an al Energy Energy, d Power

Mechanica
al Energy
Energy,
Work and Power
D. Gordon E. Robertson, PhD, FCSB
Biomechanics Laboratory
Laboratory,
School of Human Kinetics,,
University of Ottawa, Ottaw
wa, Canada
Biomechanics Laab, U. of Ottawa
1
Ene
ergy
• Ability to do work
• Measured in jouless (J)
• One joule is the wo
ork done when a one
newton force movees an object through
one metre
• 1 Calorie = 1000 ca
als = 44.186
186 kJ
• Can take many forrms
Biomechanics Laab, U. of Ottawa
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Forms off Energy
• Mass (E = mc2)
• Solar or Light (solar panels, photovoltaic
battery)
f
magnetic induction)
• Electricity (electron flux,
• Chemical (fossil fuelss, ATP, food)
• Thermal or Heat
• Mechanical energy
Biomechanics Laab, U. of Ottawa
3
Types of Mech
hanical Energy
• Translational Kinetiic = ½ m v2
– v2 = vx2 + vy2 ((+ vz2)
– this is usually the larrgest type in biomechanics
• Rotational Kinetic = ½ I ω2
– this is usually the sm
mallest type in biomechanics
• Gravitational Potenttial = m g y
• Elastic Potential = ½ k (x12 – x22)
– Assumed to be zero for
f rigid bodies
Biomechanics Laab, U. of Ottawa
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Laws of Therrmodynamics
• Zeroth
Z th llaw
– When two quantities are in therma
al balance to a third they are in
thermal balance with each other. I.e., they have the same
temperature.
temperature
• First Law (Law of Conservation of Energy)
– Energy is conserved (remains con
nstant) within a “closed system.”
– Energy
E
cannott be
b created
t d or dest
d troyed.
t
d
• Second Law (Law of Entropy)
– When energy is transformed from
m one form to another there is always
a loss of usable energy.
py of the universe.
– All processes increase the entrop
• Third Law
– Absolute zero (absence of all atom
mic motion) cannot be achieved.
Biomechanics Laab, U. of Ottawa
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Law of Cons
servation of
Mechanic
cal Energy
• If the resultant forrce acting on a body is
a conservative forcce then the body’s total
mechanical energyy will be conserved.
• Resultant force willl be conservative if all
external forces aree conservative.
• A force is conserva
ative if it does no work
around a closed pa
ath (motion cycle).
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Conservative Forces
• Gravitational forcees
g
gravity
y
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Conservative Forces
• Gravitational forcees
• Normal force of a frictionless
f
surface
frictionless
surface
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Conservative Forces
• Gravitational forcees
• Normal force of a frictionless
f
surface
• Elastic collisions
elastic collision
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Conservative Forces
•
•
•
•
Gravitational forcees
Normal force of a frictionless
f
surface
Elastic collisions
P d l
Pendulum
pendulum
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Conservative Forces
•
•
•
•
•
Gravitational forcees
Normal force of a frictionless
f
surface
Elastic collisions
P d l
Pendulum
Ideal spring
ideal spring
p g
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Conservative Forces
•
•
•
•
•
•
Gravitational forcees
Normal force of a frictionless
f
surface
Elastic collisions
P d l
Pendulum
force
load
Ideal spring
lever
Lever system
fulcrum
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Conservative Forces
Simple machines:
• Pulleys
• Block & tackle
• Gears
G
• Cams
• Winch
•…
Biomechanics Laab, U. of Ottawa
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Examp
ples of
Nonconservative Forces
•
•
•
•
•
•
Dry friction
Air (fluid) resistan
nce
Viscous forces
Pl ti collisions
Plastic
lli i
Real pendulums
Real springs
Biomechanics Laab, U. of Ottawa
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Direct Errgometry
Treadmill Ergometry
• External work =
m g t v sin θ
• where, m = mass,
g = 9.81,
9 81 t = time,
ti
v = treadmill velocity,
and θ = treadmill’s
angle of incline
Biomechanics Laab, U. of Ottawa
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Direct Errgometry
Cycle Ergometry
• External work =
6nLg
• where, n = number of
pedal revolutions,,
p
L = load in kiloponds
and g = 9.81
• Note,
Note each pedal ccycle
clee
is 6 metres motion of
flywheel
Biomechanics Laab, U. of Ottawa
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Direct Errgometry
Gjessing Rowing
Ergometry
• External work =
nLg
• where,
h
n = number
b
of flywheel cycles,
L = workload in
kiloponds and
g = 9.81
Biomechanics Laab, U. of Ottawa
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Biomechanic
cal Methods
Point Mass
M
Method
M th d
– Simplest, least accurate
e, ignores rotational energy
• Mechanical Energy = E = m g y + ½ m v2
• External work = Efinaal – Einitial
Biomechanics Laab, U. of Ottawa
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Biomechanic
cal Methods
Single Rigid Body Method
– Simple, usually planar,
includes rotational energy
Carriage load
• Mechanical Energy
gy =
E= mgy + ½mv2 + ½Iω2
• External Work =
Efinal – Einitial
Biomechanics Laab, U. of Ottawa
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Biomechanic
cal Methods
Multiple Rigid Body
Method
– Difficult
Difficult, usually planar
planar,
more accurate, accuracy
increases with number of
segments
• External Work =
Efinal – Einitial
• E = sum of segmental
total energies (kinetic
plus potential energies)
Biomechanics Laab, U. of Ottawa
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Biomechanic
cal Methods
Inverse Dynamics
Method
–M
Mostt diffi
difficult,
lt usually
ll
planar, requires force
platforms
• E
External
t
l Work
W k=
Σ ( Σ Μj ωj ∆t )
• Sum over all joint
moments and over
duration of movement
Biomechanics Laab, U. of Ottawa
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Biomechanic
cal Methods
Absolute Power Method
– similar to previous method
d
• Total Mechanical Work
k = Σ ( Σ | Μj ωj | ∆t )
• Sum over all joint moments and over
d
duration
ti off movementt
• Notice positive and negaative moment
powers do not cancel (ab
bsolute values)
• Internal Work =
Total mechanical work – External work
Biomechanics Laab, U. of Ottawa
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Physiologic
cal Methods
• Oxygen Uptake
– Difficult, accurate,
expensive invasive
expensive,
• Physiological Work =
c ((VO2)
• Where, c is the energyy
released by
metabolizing O2 and
VO2 is the volume of
O2 consumed
Biomechanics Laab, U. of Ottawa
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Mechanicall Efficiency
Mouthpiece for
collecting expired
gases and
physiological costs
• Measure both
mechanical and
physiological costs
• ME = mechanical
cost divided by
physiological cost
times 100%
Monark ergometer used to
measure mechanical work
done
Biomechanics Laab, U. of Ottawa
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Mechanicall Efficiency
Internal work + External work
ME = ———————
———————— × 100 %
Physiologiical cost
Internal work is measu
ured by adding up the work
done by all the joint moments
m
of force. Most
researchers ignore the internal work done.
Biomechanics Laab, U. of Ottawa
25
Work of a Force
Work of a Force is producct of force (F) and
displacement (s) when F and s are in the same
direction.
Work = F s
(w
when F is parallel to s)
= F s cos φ
((w
when F is not p
parallel to s
an
nd is φ angle between F and s)
= F . s = Fx sx + Fy sy + Fz sz ((dot p
product))
= Ef – Ei
(cchange of energy)
=Pt
(p
power times time)
Biomechanics Laab, U. of Ottawa
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Work of a Mom
ment of Force
Work of a Moment of Force
F
is product of
moment of force (M) and angular displacement
(θ).
(θ)
Work = M θ
= r F (sin
( i φ) θ (φ
φ is
i angle
l between
b
r and
d F)
=Pt
(p
power times time)
= Σ (M ω ∆t) (ttime integral of moment
po
ower)
Biomechanics Laab, U. of Ottawa
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Average
e Power
Power is the rate of doin
ng work.
– measured in watts (W), 1 watt = 1 joule per second (J/s)
Power = work / time
= (Ef – Ei) / time
= (F s) / t = F v
= (M θ) / t = M ω
(work rate)
(change in energy over
time)
(force times velocity)
(moment of force times
angular velocity)
Biomechanics Laab, U. of Ottawa
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Instantaneous P
Power of a Force
or Momen
nt of Force
Power = F v
= F v cos φ
(w
when F is parallel to v)
(w
when F is not parallel to v
an
nd is φ angle between F
an
nd v)
= F . v = Fx vx + Fy vy + Fz vz (dot product)
=Mω
(moment times
(m
i
angular
l
velocity)
Biomechanics Laab, U. of Ottawa
29
Isokinetic Dy
ynamometers
• Controls speed of
motion therefore
lever has constant
angular velocity (ω)
• Measures force
against
i t a lever
l
arm
• Moment = force times
lever arm
• Instantaneous Power
= moment times
angular
l velocity
l it
KinCom 500H
Biomechanics Laab, U. of Ottawa
hydraulically
controlled motion
lever arm
force sensor
30