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EGR 280
Mechanics
12 – Work and Energy of Particles
Work and Energy
Newton’s Second Law describes the vector relationship between force,
acceleration and time. Work and Energy is a restatement of Newton’s Second
Law, and describes the scalar relationship between force, position and speed.
The work done by a force
Let the change in the position of a particle be displacement = ∆r
The work done by a force in moving a particle from point A to another point B
is defined as
F
UAB =

B
A
F ∙ dr
A
B
Special cases:
1. Work of a constant force in rectilinear motion:
UAB = ∫AB F ∙ dr
= ∫AB F cosθ dx
= F cosθ (xB – xA)
= F cosθ ∆x
B
F
θ
A
2. Work done by a particle’s weight
B
UAB = ∫AB F ∙ dr
= ∫AB -W dy
= -W (yB – yA)
= -W ∆y
y
W
yB
A
yA
3. Work of the force exerted by a linear spring
L0
UAB = ∫AB F ∙ dr
= ∫AB -kx dx
= -½ k (xB2 – xA2)
= ½ k (xA2 – xB2)
F = -kx
xA
xB
Kinetic Energy of a Particle
Consider the motion of a particle in intrinsic coordinates. The component of
any force applied to the particle in the normal direction does no work on the
particle, since it is always perpendicular to the direction of motion. Therefore,
all the work on the particle is done by the tangential force:
Ft = m dv/dt = m(dv/ds)(ds/dt) = mv(dv/ds)
∫AB Ft ds = ∫AB mv dv
UAB = ½ m(vB2 – vA2)
et
en
Define the kinetic energy of the particle, T, as ½ mv2, so that
TA + UAB = TB
UAB = TB - TA
Power
Power is defined as the rate at which work is done:
P = dU/dt
= d/dt(U) = (F ∙ dr)/dt = F ∙ dr/dt
P=F∙v
1 W = 1 J/s = 1 N-m/s
1 hp = 550 ft-lb/s = 746 W
Efficiency is the ratio of the output power to the input power:
η = (power output) / (power input)
Potential Energy
We have seen that the work in moving a body against gravity is
UAB = -W ∆y = -W (yB – yA) = WyA - WyB
The work done on the particle does not depend on the path taken between
points A and B, it only depends on the heights of A and B above a datum at
y=0. This function Wy is called the potential energy of the body with respect
to gravity, or Vg. Thus,
UAB = VgA - VgB
Likewise, if we define the potential energy of a linear spring as
Ve = ½ k x2 , then we can write the work done by the spring as
UAB = VeA - VeB
Conservation of Mechanical Energy
If the work done by all of the forces acting on a particle is independent of the
path taken by the particle, then the force system is called conservative and the
total mechanical energy, T+V, is conserved:
UAB = VA – VB = TB – TA
VA + TA = VB + TB
If friction acts on the particle, then the total mechanical energy is not
conserved and this expression can be written as
VA + TA = VB + TB + energy lost