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speed in straight lines with respect to each other. An observer in a reference frame moving
with constant speed in a straight line with respect to the reference frame in which the
object is at rest would conclude that the natural state or motion of the object is one of
constant speed in a straight line, and not one of rest. All inertial observers, in an infinite
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number of frames of reference, would come to the same conclusion. We see, therefore,
that Aristotle's conjecture is not consistent with this fundamental Principle.
4.2 Newton s laws of motion
During his early twenties, Newton postulated three Laws of Motion that form the
basis of Classical Dynamics. He used them to solve a wide variety of problems including
the dynamics of the planets. The Laws of Motion, first published in the Principia in 1687,
play a fundamental rôle in Newton s Theory of Gravitation (Chapter 7); they are:
1. In the absence of an applied force, an object will remain at rest or in its present state of
constant speed in a straight line (Galileo's Law of Inertia)
2. In the presence of an applied force, an object will be accelerated in the direction of the
applied force and the product of its mass multiplied by its acceleration is equal to the
force.
and,
3. If a body A exerts a force of magnitude |FAB| on a body B, then B exerts a force of
equal magnitude |FBA| on A.. The forces act in opposite directions so that
FAB =  FBA .
In law number 2, the acceleration lasts only while the applied force lasts. The applied
force need not, however, be constant in time  the law is true at all times during the
motion. Law number 3 applies to  contact interactions. If the bodies are separated, and
the interaction takes a finite time to propagate between the bodies, the law must be
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modified to include the properties of the  field  between the bodies. This important
point is discussed in Chapter 7.
4.3 Systems of many interacting particles: conservation of linear and angular
momentum
Studies of the dynamics of two or more interacting particles form the basis of a key
part of Physics. We shall deduce two fundamental principles from the Laws of Motion; they
are:
1) The Conservation of Linear Momentum which states that, if there is a direction in
which the sum of the components of the external forces acting on a system is zero, then
the linear momentum of the system in that direction is constant.
and,
2) The Conservation of Angular Momentum which states that, if the sum of the moments
of the external forces about any fixed axis (or origin) is zero, then the angular momentum
about that axis (or origin) is constant.
The new terms that appear in these statements will be defined later.
The first of these principles will be deduced by considering the dynamics of two
interacting particles of masses ml and m2 wiith instantaneous coordinates [xl, y1 ] and [x2,
y2], respectively. In Chapter 12, these principles will be deduced by considering the
invariance of the Laws of Motion under translations and rotations of the coordinate
systems.
Let the external forces acting on the particles be F1 and F2 , and let the mutual
interactions be F21´ and F12´. The system is as shown
N E W T O N I A N D Y N A M I C S 69
y
F1 F2
m2
F12´
m1 F21´
O
x
Resolving the forces into their x- and y-components gives
y
Fy2
Fy1
Fx12´ Fx2
Fy21´ m2
Fx1 Fy12´
m1 Fx21´
O
x
a) The equations of motion
The equations of motion for each particle are
1) Resolving in the x-direction
Fx1 + Fx21´ = m1 (d2x1/dt2) (4.1)
and
Fx2  Fx12´ = m2(d2x2/dt2). (4.2)
Adding these equations gives
Fx1 + Fx2 + (Fx21´  Fx12´) = m1(d2x1/dt2) + m2(d2x2/dt2). (4.3)
2) Resolving in the y-direction gives a similar equation, namely
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Fy1 + Fy2 + (Fy12´  Fy12´) = m1(d2y1/dt2) + m2(d2y2/dt2). (4.4)
b) The rôle of Newton s 3rd Law
For instantaneous mutual interactions, Newton s 3rd Law gives |F21´| = |F12´|
so that the x- and y-components of the internal forces are themselves equal and opposite,
therefore the total equations of motion are
Fx1 + Fx2 = m1(d2x1/dt2) + m2(d2x2/dt2), (4.5)
and
Fy1 + Fy2 = m1(d2y1/dt2) + m2(d2y2/dt2). (4.6)
c) The conservation of linear momentum
If the sum of the external forces acting on the masses in the x-direction is zero, then
Fx1 + Fx2 = 0 , (4.7)
in which case,
0 = m1(d2x1/dt2) + m2(d2x2/dt2)
or
0 = (d/dt)(m1vx1) + (d/dt)(m2vx2),
which, on integration gives
constant = m1vx1 + m2vx2 . (4.8)
The product (mass × velocity) is the linear momentum. We therefore see that if there is
no resultant external force in the x-direction, the linear momentum of the two particles in
the x-direction is conserved. The above argument can be generalized so that we can state:
the linear momentum of the two particles is constant in any direction in which there is no
resultant external force.
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4.3.1 Interaction of n-particles
The analysis given in 4.3 can be carried out for an arbitrary number of particles, n,
with masses m1, m2, ...mn and with instantaneous coordinates [x1, y1], [x2, y2] ..[xn, yn]. The
mutual interactions cancel in pairs so that the equations of motion of the n-particles are, in
the x-direction
" " " " " "
Fx1 + Fx2 + ... Fxn = m1x1 + m2x2 + ... mnxn = sum of the x-components of (4.9)
the external forces acting on the masses,
and, in the y-direction
" " " " " "
Fy1 + Fy2 + ... Fyn = m1y1 + m1y2 + ...mnyn = sum of the y-components of (4.10)
the external forces acting on the masses.
In this case, we see that if the sum of the components of the external forces acting
on the system in a particular direction is zero, then the linear momentum of the system in
that direction is constant. If, for example, the direction is the x-axis then
m1vx1 + m2vx2 + ... mnvxn = constant. (4.11)
4.3.2 Rotation of two interacting particles about a fixed point
We begin the discussion of the second fundamental conservation law by cosidering
the motion of two interacting particles that move under the influence of external forces F1
and F2, and mutual interactions (internal forces) F21´ and F12´. We are interested in the
motion of the two masses about a fixed point O that is chosen to be the origin of Cartesian
coordinates. The system is illustrated in the following figure [ Pobierz całość w formacie PDF ]