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Chapter 1   The Meaning of Newton's Laws

If the laws of nature are to be tested by experiment all quantities are to be measurable.

1.1   Newton I

Isolated bodies maintain constant velocity, therefore velocity is relative to some reference frame and a time scale, a frame in which Newton's I holds (inertial frame). Assumptions: However if Newton's Laws are valid in a given frame, they will be valid in any frame moving with constant relative velocity.
v' = v - u
where:
v'
velocity seen from frame moving with velocity u with respect to first frame
v
velocity of particle in one inertia frame
Principle of Mechanics states that mechanical experiments cannot determine who is 'at rest', accelerations however can be detected.

For example: Rubber bob on accelerating train
Observer on Platform (inertial frame):
Observer on Train (non-inertial frame)
horizontal force=T sin q
via Newton II T sin q = ma
Requires Balancing Force:
F+Tsin q =0
F=-ma
where 'a' is the acceleration of the frame

F is a fictious force, inertial force resulting from using Newton's Laws in a non-inertial frame. The force does not originate from any physical source.

Inertial Frame is a frame in which a body obey's Newton I when no force acts. However how do we know there is no force? When a body moves with constant velocity. This is self referencing its-self (tautology), so how do we know what is an Inertial Frame? We resort to successive approximations to achieve an inertial frame.
  1. Deep Space - coarset 'clumpiness' of universe is 150 million light years (~ 1025m)
  2. Frame of 'fixed stars' in our galaxy - useful for analysising planetary motion
  3. Earth - difficult to test Newton I as gravity is present in a standard ground based lab frame. Other complications include the centrifeugal, Cariolis force (see later section), inertia forces caused by the earths rotation
Centrifugal term: mwE2RE
acceleration (centrifugal)=wE2RE
where:
wE
angular velocity of earth
RE
radius of earth
force ~ 0.007g
Can we do better by 'getting rid' of gravity? Consider the 'lift cabin':
  1. Observer on ground see's all objects, the lift cabin and the man inside, accelerate downwards with acceleration g=-gk
  2. In the accelerating lift cabin frame the net force is given by mg+F where mg is the bodies weight and F is the inertial force -ma , a being the acceleration of the lift. When a=g the net force is:
    mg+(-ma)=0
    So the net force on any object observed in this frame is zero. It is in this artificially created frame we can test Newton's I Law.
According to Einstein the free-fall situation is in fact identical to the 'free-float' space ship.

1.2   Newton II

Newton II in an inertial frame
F=ma
acceleration a=
d2r
dt2
=r
Measurable mass? ie compare one mass with a standard mass. In order to do this subject test mas and standard mass to the same force and observe the acceleration. Using the idea's of Newton III through a medium such as a spring. The subject mass and test mass to equal and opposite forces.
compare accelerations:
a1=
F1 on 2
m1
; a2=
F2 on 1
m2
m2
m1
=
F2 on 1/a2
F1 on 2/a1
=
a1
a2
since F2 on 1=F1 on 2 (and we only care about magnitude)
m2=(
a1
a2
)m1

1.2.1   Inertial vs Gravitational Mass

Newton's law of gravity, force F on one body (eg test object of mass mg due to a second body (eg earth, MG is
F =
GMEmg
r2
where 'r' is the distance connecting the centres of the two masses. By Newton II: F=mIa we make the distinction between two sorts of mass
  1. gravitational mG the 'charge' appearing in Newton's law of gravitation
  2. inertial mass mI is the ratio of force to accelerate in Newton II
Tower of Pisa Experiment - Galileo:
Applying Newton's laws to the two bodies:
GMGmG1
r2
=mI1a1
GMGmG2
r2
=mI2a2
a1
a2
=
mG1/mI1
mG2/mI2
=
mG1mI2
mG2mI1
this if the ratio mG/mI is the same for all bodies, then all bodies will fall with the same acceleration. ie near the earth's surface a1=a2=g If not then the bodies will seperate as they fall. Modern experiments have tested the equality between gravitational and inertial mass to better than one part in 1012
a1
a2
=1.000000000000
experimentally mI/mG=constant however by a choice of units mI=mG This experiment suggests that gravity is an inertial force - this is the basis of Einstein's gereral relativity.
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