Environmental Engineering Reference
In-Depth Information
pair of particles and rewrite this sum as
F
ij
=
(
F
ij
+
F
ji
).
(2.22)
i
j
i
j<i
The Third Law requires that each term
(
F
ij
+
F
ji
)
is null, therefore the sum is null
and
N
F
(e)
i
F
=
F
i
=
,
(2.23)
i
=
1
i
i.e. the net force acting on any body is the sum of all the forces of external origin.
This allows us to determine a version of the Second Law that is valid for an
extended body. Applying the Second Law to each particle
N
N
F
(e)
i
=
=
=
F
F
i
m
i
a
i
(2.24)
i
=
1
i
=
1
i
now
N
N
d
2
d
t
2
r
i
m
i
a
i
=
m
i
i
=
1
i
=
1
N
d
2
d
t
2
=
m
i
r
i
(2.25)
i
=
1
since the mass of each particle is independent of time. Furthermore, if we divide
by the total mass
M
=
i
m
i
, we obtain
1
M
m
i
r
i
.
N
d
2
d
t
2
F
M
=
(2.26)
i
=
1
The term on the right hand side is the second time derivative of the weighted-mean
position of all the particles in the body, where the “weights” are the masses of the
particles. This special position is known as the centre of mass of the body, it is
located at
N
1
M
R
=
m
i
r
i
.
(2.27)
i
=
1
We are thus led to a version of the Second Law valid for any extended body or
collection of particles:
M
d
2
R
d
t
2
F
=
=
M
A
,
(2.28)
