1
PHY41M1 CLASSICAL MECHANICS
M Chirwa, Chemical & Physical Sciences Department, Faculty of Natural Sciences, Walter
Sisulu University, Nelson Mandela Drive, P/Bag X1, Mthatha 5117, South Africa
[email protected]
,
[email protected]
(12 March 2017 – 29 June 2017)
1.
Mechanics of a System of Particles
1.1.
Equations of motion
In a system of particles, for the
i
th
particle with mass
i
m
and velocity
t
i
i
i
d
d
r
r
v
r
&
r
r
=
≡
at
position
i
r
r
, the equation of motion is
ij
i
i
i
t
F
F
p
p
r
r
r
&
r
+
=
=
d
d
(1.1)
where
i
p
&
r
is the time rate of change of the momentum
i
i
i
m
r
p
&
r
r
=
, and
i
F
r
is the external force on
the
i
th
particle, while
ij
F
r
is the internal force on the
i
th
particle due to another (
j
th
) particle in the
same system. Note that
0
F
r
r
=
ii
as a particle cannot exert a force onto itself. Furthermore, to
satisfy Newton’s third law of motion, the equality of action-reaction forces, we must have the
relation:
0
F
F
r
r
r
=
+
ji
ij
(1.2)
Then for the entire system of particles, the equation of motion is found by summing over all the
particles in the system
(
)
F
F
F
F
F
F
F
r
p
P
r
r
r
r
r
r
r
r
&
r
&
r
∑
∑
∑
∑
∑
∑
∑
=
≡
+
+
≡
+
=
=
=
j
i
i
i
ji
ij
i
i
j
i
ij
i
i
i
i
i
i
i
m
t
,
,
2
2
2
1
d
d
(1.3)
The second equivalence is after applying equation (1.2).
1.2.
Moments as composite physical quantities
1.2.1.
Moment of a scalar quantity
The moment of a scalar physical quantity
b
i
of a single particle located at position
i
r
r
is
defined as the vector
i
i
b
r
r
Then for a system of such particles characterized by the overall physical quantity
∑
=
i
i
b
B
(1.4)

2
the corresponding overall moment of
B
is
∑
=
i
i
i
b
B
r
R
r
r
(1.5)
where the position vector
R
r
is known as the
centre
of the quantity
B
of the system. For a
system where the scalar quantity
B
is continuously distributed, the above summations become
integrals:
∫
=
b
B
d
(1.6)
and
∫
=
b
B
d
r
R
r
r
(1.7)
If the elemental quantity is the mass
=
v
m
d
d
d
d
ρ
σ
λ
a
l
(1.8)
with line
λ
, surface
σ
and volume
ρ
densities, then the total mass is
∫
=
m
M
d
(1.9)
while the overall
mass moment
is
∫
=
m
M
d
r
R
r
r
(1.10)
where vector
R
r
is the position vector of the
c
entre
o
f
m
ass (
com
) relative to the origin.
Fig.1.1:
A system’s
i
th
object has mass
i
m
and the positions
i
r
r
and
i
r
′
r
relative to an origin
O
and the system’s
com
respectively.
O
com
i
m
i
r
r
R
r
i
r
r
′