
In mechanics a force does work on a particle only when the particle undergoes a displacement in the direction of the force.

If a force F acts on a particle and the particle experiences a differential displacement dr , the work done is dU

F ds cos
 where ds is the magnitude of dr and
θ
is the angle between F and dr .

The equation may also be written as d U

 r
Note that if 0 o
≤
θ
< 90 o
, the work is positive, but if 90 o
<
θ
≤ 180 o
, the work is negative.

Work is a scalar quantity. The unit of work is the joule (J) in SI system and ft.Ib
in FPS system.

If the force on the particle varies with position, F = F ( s ), the work is determined by integration as
U
1

2
  r
2 r
1
F
 d r
  s
1 s
2
F cos
 ds (1)

The integral in equation (1) can be interpreted as the area under the F cos
θ
versus s curve (see
Fig. 14-2).

If the force has a constant magnitude and acts at a constant angle from its straight line path, integration of equation (1) yields
U
1

2

F
C cos
  s
2
 s
1

(2)

If a particle is displaced in the vertical direction, the work of the weight is
U
1-2
= ± W Δ y (3)
Work of a Spring Force

Consider a linear elastic spring displaced a distance s from its unstretched position. o The magnitude of force developed in the spring is F s
= ks , where k is the spring stiffness. o When the spring is compressed from position s
1
to position s
2
, the work done on the spring is positive and given as

U
1

2


 s
1 s
2
F s
1
2 k s
2
2 ds


1
2 k s
1
2
 s
1 s
2 ks ds o If a particle is attached to the spring, the force exerted on the particle is opposite to that exerted on the spring. Consequently, the force will do negative work on the particle when the particle is moving so as to further elongate (or compress) the spring. Hence,
U
1

2 


1
2 k s
2
1
2 k s
1
2




Consider a particle of mass m which moves from point 1 to point 2 under the action of a system of external forces, as shown in Fig. 14-7. o The equation of motion in the tangential direction is Σ F t
= ma t
. o Applying the kinematic equation a t
= v dv / ds and integrating both sides yields
 s
1 s
2
F t ds

1
2 mv
2
2 
1
2 mv
1
2
(5) o Noting that Σ
F t
= Σ F cos
θ
and using the definition of work, equation (5) may be written as

U
 1 
Sum of forces the acting work on by the all particle

1 mv
2
2
  final the
K .
E .
particle of

1 mv
2
1
  initial the
K .
E particle
.
of
(6) o Equation (6) is often symbolized in the form o Equation (7) states that the particle’s initial kinetic energy plus the work done by all forces acting on the particle as it moves from its initial to its final position is equal to the particle’s final kinetic energy
. o Equation (7) expresses the principle of work and energy. It provides a convenient substitution for Σ
F t
= ma t
when solving kinetic problems which involve force, velocity, and displacement (see pages 164-165 for details).


The principle of work and energy can be extended to include a system of n particles isolated within an enclosed region of space as shown in Fig. 14-8.

For a system of particles, the work done by external and internal forces must be taken into account. The principle of work and energy for the system can be written as
 Equation (8) states that the system’s initial kinetic energy ( ΣT
1
) plus the work done by all the external and internal forces acting on the particles of the system (Σ U
1-2
) is equal to the system’s final kinetic energy
(Σ
T
2
).
 Note that although the internal forces on adjacent particles occur in equal but opposite collinear pairs, the total work done by each of these forces will, in general, not cancel out since the paths over which corresponding particles travel will be different. [See more details on page 166.]
Power

Power is defined as the amount of work performed per unit of time. Hence,
P
 dU
(9) dt

Power can also be defined in terms of the velocity at a point and the force which acts at that point, as follows:
P
 dU dt

F
 d r dt

F
 d r dt
or
P

F
 v (10)

Power is a scalar quantity. The unit of power is the watt (W) in SI system and ft.Ib/s in FPS system. Also, the horsepower (hp) is used, where
1 hp = 746 W = 550 ft.Ib/s
Efficiency

The mechanical efficiency of a machine is defined as the ratio of the output of useful power produced by the machine to the input power supplied to the machine. Hence,

  power power output
(11) input

If energy input and output occur during the same time interval, the efficiency can also be expressed as ratio of energy output to input. Hence,
  energy output
(12) energy input

Due to frictional forces in machines, the energy output is always less than energy input.
Consequently, the efficiency of a machine is always less than 1 .
Conservative Force
 When the work done by a force in moving a particle from one point to another is independent of the path followed by the particle, then this force is called a conservative force .

The weight of a particle and the force of an elastic spring are two examples of conservative forces often encountered in mechanics.

In contrast, frictional forces are nonconservative, because the work done by frictional forces depends on the path.
Potential Energy

Energy may be defined as the capacity for doing work.

When energy comes from the motion of the particle, it is referred to as kinetic energy .
 When energy comes from the position of the particle, measured from a fixed datum or reference plane, it is called potential energy .

Thus, potential energy is a measure of the amount of work a conservative force will do when it moves from a given position to the datum.
 If a particle is located above an arbitrarily selected datum, the particle’s weight has a positive gravitational potential energy , V g
. Likewise, if the particle is located below the datum, V g
is negative.
 In general, the gravitational potential energy of a particle of weight W located a distance y above a datum is
V g
= W y (13)


When an elastic spring is elongated or compressed a distance s from its unstretched position, the elastic potential energy V e
due to the spring’s configuration can be expressed as
V e
 
1
2 ks
2
(14)
Note that V e
is always positive since, in its deformed position, the force of the spring has the capacity for always doing positive work on the particle when the spring is returned to its unstretched position.
Potential Function

In the general case, if a particle is subjected to both gravitational and elastic forces, the particle’s potential energy can be expressed as a potential function, which is
V = V g
+ V e
(15)

The work done by a conservative force in moving a particle from a location 1 to another location 2 in space is the difference in the potential function of the particle, i.e.,
U
1-2
= V
1
– V
2
(16)
(See the illustration on page 192.)

Recall the principle of work and energy
T
1
+ Σ U
1-2
= T
2
(17)

When a particle is acted upon by a system of both conservative and nonconservative forces, the portion of the work done by the conservative forces can be written in terms of the difference in their potential energies, using equation 16. As a result, the principle of work and energy can be written as
T
1
+ V
1
+ (Σ U
1-2
) noncons
= T
2
+ V
2
(18)

If only conservative forces are applied to the body, equation 18 reduces to

Equation 19 is referred to as the conservation of mechanical energy (or simply the conservation of energy). It states that during the motion the sum of the particle’s kinetic and potential energies remains constant. [See the illustration in Fig. 14-20.]


For a system of particles, the conservation of energy principle is expressed as

It should be noted that only problems involving conservative forces (weights and springs) may be solved by using conservation of energy theorem.