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Work, Energy and Power Mechanical Energy: (units in joules, J, equal to 1 newton metre, similar to units of moments of force) Of a point mass: E = mgy + ½mv 2 = mgy + ½mv x2 + ½mv 2y Of a rigid body: E = mgy + ½mv 2 + ½ Iω 2 Potential energy (gravitational): PE = m g y Kinetic energy: KEt = ½ mv2 KEr = ½ Iω2 Work, Energy and Power Work: (units in joules, J) On a rigid body or point mass is equal to the change in mechanical energy of the rigid body or point mass. A positive value means the force did work on the body increasing its mechanical energy. A negative value means the body did work on some other body or the environment. work = ∆E = Efinal – Einitial Work of a force equals the magnitude of the force times the displacement in the direction of the force. work = F s (F parallel to s) work = F s cos θ (F not parallel to s) Note, θ must be angle between F and s. Work of a moment of force is the product of moment of force times the angular displacement of the body it rotates. work = M θ Work of a force vector is the scalar product of force and displacement. Without displacement no work is done. work = F . s = Fx sx + Fy sy Work, Energy and Power Power: (units of watts, W, equal to 1 joule per second) Power to or from a point mass or rigid body is the rate of work done to or by the body divided by the duration (time) of the work. A negative value indicates the rate at which the body losses energy. A positive value indicates the rate at which the body receives energy. power = P = work / time Instantaneous power of a force is defined as: Pforce = F . v = Fx vx + Fy vy = F v cos θ Note, θ must be angle between F and v. Instantaneous power of a moment of force is defined as: Pmoment = M ω = ( r F sin θ ) ω Note θ must be angle between F and r. Work of Force on a Particle Work = ∫ F ⋅ s = path integral of dot product of force and displacement Work = ∫ F ds = area under force versus displacement If F is constant: Work = F × s Work Done during Treadmill Locomotion • 65 kg person runs at 5.50 m/s on a treadmill with an incline of 10.00 degrees vy = v sin 10E= 5.50 (0.1736) = 0.995 m/s y = vy t = 0.995 (60) = 57.3 m Work = mgy = 65 (9.81) 57.3 = 36 539 J = 36.5 kJ Work of a Moment of Force Work = M θ • given a constant flexor moment of force of 250 N.m acting during 45.0 degrees of flexion Work = M θ = 250 (45.0 × π/180) = 196.3 J Conservation of Mechanical Energy - conservation of mechanical energy occurs when the resultant force acting on a body is a conservative force - a body conserves energy if its total mechanical energy stays constant Conservative forces: - a conservative force is a force that does no work - it is always position dependent - examples of conservative forces: (1) gravity (depends on height) (2) normal force of a frictionless surface (depends on starting height) (3) simple pendulum (depends on starting height) (4) compound pendulum with frictionless joints (5) ideal spring (depends on displacement of spring) - examples of nonconservative forces: (1) frictional forces - dependent on path, normal force and surfaces - negative work only (2) viscous forces - dependent on velocity, surface area etc. (3) muscle forces - dependent on excitation, strength, velocity of contraction, length and type of prestretch - positive (concentric) or negative (eccentric) work (4) viscoelastic forces (muscles, ligaments, tendons)