Download Mechanics 1: Newton`s Laws

Mechanics 1: Newton’s Laws
We now switch our attention from kinematics to dynamics. Let’s recall Newton’s Laws of Motion, which
we will take as axioms.
Newton’s Axioms.
1. Every particle persists in a state of rest or of uniform motion in a straight line (i.e., with constant
velocity) unless acted upon by a force.
2. If F is the force acting on a particle of mass m which as a consequence is moving with velocity v, then
F=
d
dp
(mv) =
,
dt
dt
(1)
where p = mv is called the momentum. If m is independent of time t (1) becomes:
F=m
dv
= ma,
dt
(2)
where a is the acceleration of the particle.
3. If particle 1 acts on particle 2 with a force F12 in a direction along the line joining the two particles,
while particle 2 acts on particle 1 with a force F21 , then F21 = −F12 . In other words, to every action
there is an equal and opposite reaction.
Example. A particle of mass m moves in the x − y plane in such a way that its position vector is given by:
r = a cos ωti + b sin ωtj
(3)
where a, b, and ω are positive constants, with a > b.
1. Show that the particle moves in an ellipse.
From the expression for the position vector we have:
x = a cos ωt,
y = b sin ωt,
which are just the parametric equations for an ellipse with semi-major axis of length a and semi-minor
axis of length b. Since
y2
x2
+ 2 = cos2 ωt + sin2 ωt = 1,
2
a
b
the equation for the ellipse can also be taken as:
x2
y2
+
= 1.
a2
b2
2. Show that the force acting on the particle is always acting towards the origin.
We will assume that the mass m is constant. Then
=
d2
d2 r
dv
= m 2 = m 2 (a cos ωti + b sin ωtj) ,
dt
dt
dt
2
m −ω a cos ωti − ω 2 b sin ωtj ,
=
−mω 2 (a cos ωti + b sin ωtj) = −mω 2 r,
F =
m
from which it follows immediately that the force is directed to the origin.
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Definitions of Force and Mass. Strictly speaking, force and mass are undefined quantities in Newton’s
axioms. Intuitively, I expect most of you to have a good idea of what they are. Mass is a measure of the
“quantity of matter” in an object. Force is a measure of the “push or pull” on an object. The question of “what
is force” and “what is mass” are deep and fundamental. They are questions of current interest in elementary
particle physics and quantum field theory (so beyond the scope of this course). It is possible to “define” force
and mass through Newton’s axioms. Some mechanics books take this approach, but fundamentally, it is not
satisfactory (you might think “why not”?).
Units and Dimensions. It is appropriate at this point to say something about dimensions since we will
seeon be encountering a number of new quantities that are measured with respect to specific types of units.
We saw earlier, that the basic dimensions that we will encounter in this course are length, mass, and
time. There will be two systems of units that we will use to describe each: the centimeter-gram-second (cgs)
system and the meter-kilogram-second (mks) system. Units for various quantities are given in Table 1.
Physical Quantity
Length
Mass
Time
Velocity
Acceleration
Dimensions
L
M
T
LT−1
LT−2
CGS System
cm
gm
sec
cm/sec
cm/sec2
gm cm/sec2
Force
MLT
−2
= dyne
gm cm/sec
MKS System
m
kg
sec
m/sec
m/sec2
kg m/sec2
= newton
kg m/sec
Momentum, Impulse
MLT−1
= dyne sec
ML2 T−2
gm cm2 /sec2
kg m2 /sec2
Energy, Work
= dyne cm = erg
= nt m = joule
gm cm2 /sec3
kg m2 /sec3
Power
Volume
Density
Angle
Angular Velocity
Angular Acceleration
MLT−3
L3
ML−3
none
T−1
T−2
= dyne cm/sec = erg/sec
cm3
gm/cm3
radian (rad)
rad/sec
rad/sec2
= joule/sec = watt
m3
kg/m3
rad
rad/sec
rad/sec2
ML2 T−2
ML2 T−1
ML2
gm cm2 /sec2
Torque
Angular Momentum
Moment of Inertia
= dyne cm
gm cm2 /sec
gm cm2
kg m2 /sec2
= nt m
kg m2 /sec
kg m2
Pressure
−1
gm/(cm sec2 )
2
kg/(m sec2 )
2
ML
T
−2
= dyne/cm
= nt sec
= nt/m
Table 1: Units and Dimensions
In terms of units, we can give a definition of force. A dyne is the force that will give a 1 gm mass an
acceleration of 1 cm/sec2 . A newton is the force that will give a 1 kg mass an acceleration of 1 m/sec2 .
Inertial Frames of Reference and Absolute Motion. It needs to be stated that in the course of
reasoning from experience that led to Newton’s axioms it was always assumed that all measurements or
observations were made with respect to a coordinate system or frame of reference which was fixed in space,
i.e., absolutely at rest. This is the assumption that space or motion is absolute.
We show that to observers in two different coordinate systems a particle appears to have the same force
acting on it if and only if the coordinate systems are moving at constant velocity with respect to each other.
This is sometimes called the classical principle of relativity.
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z'
P
y'
z
r'
r
j'
k'
O'
i'
R = r - r'
k
i
y
j
O
x'
x
Figure 1:
We consider two observers, O and O’, each located at the origin of a different coordinate system, denoted
x − y − z and x′ − y ′ − z ′ , respectively. Each observer observes the motion of a particle P in space. The
position vector of P in the x − y − z coordinates system is denoted my r and the position vector of P in the
x′ − y ′ − z ′ coordinate system is denoted by r′ . The vector R = r − r′ locates the origin of the x′ − y ′ − z ′
coordinate system with respect to the x − y − z coordinate system, se Fig. 1.
Relative to observers O and O’ the forces acting on P according to Newton’s laws are given, respectively,
by:
F=m
d2 r
,
dt2
F′ = m
d2 r′
.
dt2
(4)
The difference in observed forces is:
F − F′ = m
d2 R
d2
′
(r
−
r
)
=
m
,
dt2
dt2
(5)
and this will be zero if and only if:
dR
d2 R
= 0, or
= constant,
(6)
dt2
dt
i.e., the coordinate systems are moving at constant velocity relative to each other. Such coordinate systems
are called inertial coordinate systems.
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