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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 2 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 4 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 5 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 6 Examp ples of Conservative Forces • Gravitational forcees g gravity y Biomechanics Laab, U. of Ottawa 7 Examp ples of Conservative Forces • Gravitational forcees • Normal force of a frictionless f surface frictionless surface Biomechanics Laab, U. of Ottawa 8 Examp ples of Conservative Forces • Gravitational forcees • Normal force of a frictionless f surface • Elastic collisions elastic collision Biomechanics Laab, U. of Ottawa 9 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 10 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 11 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 12 Examp ples of Conservative Forces Simple machines: • Pulleys • Block & tackle • Gears G • Cams • Winch •… Biomechanics Laab, U. of Ottawa 13 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 14 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 15 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 16 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 17 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 18 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 19 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 20 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 21 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 22 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 23 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 24 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 26 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 27 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 28 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