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Mechanical Energy,Work and Power
D. Gordon E. Robertson, PhD, FCSB
Biomechanics Laboratory,
School of Human Kinetics,
University of Ottawa, Ottawa, Canada1Biomechanics Lab, U. of Ottawa
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Energy
• Ability to do work
• Measured in joules (J)
• One joule is the work done when a one newton force moves an object through one metre
• 1 Calorie = 1000 cals = 4.186 kJ
• Can take many forms
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Forms of Energy
• Mass (E = mc2)
• Solar or Light (solar panels, photovoltaic battery)
• Electricity (electron flux, magnetic induction)
• Chemical (fossil fuels, ATP, food)
• Thermal or Heat
• Mechanical energy
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Types of Mechanical Energy
• Translational Kinetic = ½ m v2
– v2 = vx2 + vy
2 (+ vz2)
– this is usually the largest type in biomechanics
• Rotational Kinetic = ½ I 2
– this is usually the smallest type in biomechanics
• Gravitational Potential = m g y
• Elastic Potential = ½ k (x12 – x2
2)–Assumed to be zero for rigid bodies
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Laws of Thermodynamics
• Zeroth law– When two quantities are in thermal balance to a third they are in
thermal balance with each other. I.e., they have the same temperature.
• First Law (Law of Conservation of Energy)– Energy is conserved (remains constant) within a “closed system.”
– Energy cannot be created or destroyed.
• Second Law (Law of Entropy)– When energy is transformed from one form to another there is always
a loss of usable energy.
– All processes increase the entropy of the universe.
• Third Law– Absolute zero (absence of all atomic motion) cannot be achieved.
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Law of Conservation of Mechanical Energy
• If the resultant force acting on a body is a conservative force then the body’s total mechanical energy will be conserved.
• Resultant force will be conservative if all external forces are conservative.
• A force is conservative if it does no work around a closed path (motion cycle).
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Examples ofConservative Forces
• Gravitational forces
gravity
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Examples ofConservative Forces
• Gravitational forces
• Normal force of a frictionless surface
frictionless surface
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Examples ofConservative Forces
• Gravitational forces
• Normal force of a frictionless surface
• Elastic collisionselastic collision
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Examples ofConservative Forces
• Gravitational forces
• Normal force of a frictionless surface
• Elastic collisions
• Pendulumpendulum
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Examples ofConservative Forces
• Gravitational forces
• Normal force of a frictionless surface
• Elastic collisions
• Pendulum
• Ideal spring ideal spring
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Examples ofConservative Forces
• Gravitational forces
• Normal force of a frictionless surface
• Elastic collisions
• Pendulum
• Ideal spring
• Lever system lever
force
fulcrum
load
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Examples ofConservative Forces
Simple machines:
• Pulleys
• Block & tackle
• Gears
• Cams
• Winch
• …
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Examples ofNonconservative Forces
• Dry friction
• Air (fluid) resistance
• Viscous forces
• Plastic collisions
• Real pendulums
• Real springs
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Direct Ergometry
Treadmill Ergometry
• External work = m g t v sin
• where, m = mass, g = 9.81, t = time, v = treadmill velocity, and = treadmill’s angle of incline
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Direct Ergometry
Cycle Ergometry• External work =
6 n L g• where, n = number of
pedal revolutions, L = load in kiloponds and g = 9.81
• Note, each pedal cycle is 6 metres motion of flywheel
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Direct Ergometry
Gjessing Rowing Ergometry
• External work = n L g
• where, n = number of flywheel cycles, L = workload in kiloponds and g = 9.81
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Biomechanical Methods
Point Mass Method– Simplest, least accurate, ignores rotational energy
• Mechanical Energy = E = m g y + ½ m v2
• External work = Efinal – Einitial
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Biomechanical Methods
Single Rigid Body Method– Simple, usually planar,
includes rotational energy
• Mechanical Energy = E= mgy + ½mv2 + ½I2
• External Work = Efinal – Einitial
Carriage load
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Biomechanical Methods
Multiple Rigid Body Method– Difficult, usually planar,
more accurate, accuracy increases with number of segments
• External Work =
Efinal – Einitial
• E = sum of segmental total energies (kinetic plus potential energies)
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Biomechanical Methods
Inverse Dynamics Method– Most difficult, usually
planar, requires force platforms
• External Work =
jjt )
• Sum over all joint moments and over duration of movement
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Biomechanical Methods
Absolute Power Method– similar to previous method
• Total Mechanical Work = jjt )
• Sum over all joint moments and over duration of movement
• Notice positive and negative moment powers do not cancel (absolute values)
• Internal Work =
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Physiological Methods
• Oxygen Uptake– Difficult, accurate,
expensive, invasive
• Physiological Work = c (VO2)
• Where, c is the energy released by metabolizing O2 and VO2 is the volume of O2 consumed
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Mechanical Efficiency
• Measure both mechanical and physiological costs
• ME = mechanical cost divided by physiological cost times 100%
Monark ergometer used to measure mechanical work done
Mouthpiece for collecting expired gases and physiological costs
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Mechanical Efficiency
Internal work + External work
ME = —————————————— × 100 %
Physiological cost
Internal work is measured by adding up the work done by all the joint moments of force. Most researchers ignore the internal work done.
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Work of a Force
Work of a Force is product of force (F) and displacement (s) when F and s are in the same direction.
Work = F s (when F is parallel to s)= F s cos (when F is not parallel to s
and is angle between F and s)
= F . s = Fx sx + Fy sy + Fz sz (dot product)
= Ef – Ei (change of energy)= P t (power times time)
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Work of a Moment of Force
Work of a Moment of Force is product of moment of force (M) and angular displacement ().
Work = M = r F (sin ) is angle between r and
F)= P t (power times time)= (M t) (time integral of moment
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Average Power
Power is the rate of doing work.–measured in watts (W), 1 watt = 1 joule per second (J/s)
Power = work / time (work rate)
= (Ef – Ei) / time (change in energy over time)
= (F s) / t = F v (force times velocity)
= (M / t = M (moment of force times
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Instantaneous Power of a Force or Moment of Force
Power = F v (when F is parallel to v)
= F v cos (when F is not parallel to v
and is angle between F and v)
= F . v = Fx vx + Fy vy + Fz vz (dot product)
= M (moment times angular velocity)
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Isokinetic Dynamometers
• Controls speed of motion therefore lever has constant angular velocity ()
• Measures force against a lever arm
• Moment = force times lever arm
• Instantaneous Power = moment times angular velocity
KinCom 500H
lever arm
force sensor
hydraulically controlled motion
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