Download Work, Energy and Power KEr = ½ Iω2

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)