4.5. PRINCIPLE OF CONSERVATION OF ENERGY 51
where v is the speed of the particle and is given by Equation (2.16) as v D
ds
dt
. e above equation
can be written as
F
t
D m
dv
dt
D m
dv
ds
ds
dt
D m v
dv
ds
:
Substituting this equation into Equation (4.2b) for the work of the particle, one has
U
1!2
D
s
2
Z
s
1
F
t
ds D
v
2
Z
v
1
m v dv D
1
2
m v
2
2
1
2
m v
2
1
or simply
U
1!2
D T
2
T
1
(4.5)
in which the kinetic energy of the particle is denoted by T
i
D
1
2
mv
2
i
; i D 1; 2. is equation
states that the work of the force
*
F is equal to the change in kinetic energy of the particle. is is
known as the principle of work and energy.
Rearranging terms in Equation (4.5), one has
T
1
C U
1!2
D T
2
: (4.6)
is equation states that the sum of the kinetic energy at position P
1
and the work done by the force
*
F during the displacement from P
1
to P
2
is equal to the kinetic energy of the particle at position P
2
.
As both work and kinetic energy are scalar quantities and therefore when several forces
act on the particle the expression U
1!2
represents the total work of the forces acting on the
particle. at is, U
1!2
is obtained as an algebraic sum of the work of the various forces.
4.5 PRINCIPLE OF CONSERVATION OF ENERGY
In the preceding sections if the force
*
F acting on the particle is conservative, meaning if the
work U
1!2
is independent of the path traced by the particle P as it moves from position P
1
to
P
2
, one can write
U
1!2
D V
1
V
2
(4.7)
in which V
i
; i D 1; 2 are the potential energy of the conservative force (for example, the weight
of the particle or the force exerted by a spring) acting on the particle. In this case, the principle
of work and energy defined by Equation (4.5) can be modified to
U
1!2
D V
1
V
2
D T
2
T
1
or
T
1
C V
1
D T
2
C V
2
: (4.8)
is is known as the principle of conservation of energy. It states that the total energy (sum of kinetic
energy and potential energy) of a particle acting on by a conservative force or forces is constant.
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