enginnering physics

1. Physics­I: Newtonian Mechanics &
Prelude to Special Theory of
Relativity*
Rajneesh Atre
rajneesh.atre@juetguna.in
* This discussion is developed broadly along the line of Introduction to Special Relativity,
By Robert Resnick

2. Outline of the Discussion
●
Some Remarks on Newtonian Mechanics
●
Electromagnetism & Maxwell’s Equations
●
Search for Absolute Frame of Reference
●
Basic Philosophy of Michelson­Morley
Experiment
●
Description of the Experiment
●
Chronology of Michelson­Morley
Experiment

3. Some Remarks on Newtonian Mechanics
●
Newton's laws of motion and the equations of motion of a particle would be exactly
the same in all inertial systems. This property is called Galilean Invariance.
●
In mechanics, the conservation principles­such as those for energy, linear momentum,
and angular momentum­all can be shown to be consequences of Newton's laws, it
follows that the laws of mechanics are the same in all inertial frames.
●
Electrodynamics is not included because the interaction between moving electric
charges (that is, between charges and magnetic fields) involves forces whose directions
are not along the line connecting the charges; notice too, that these forces depend not
only on the positions of the charges but also on their velocities.

4. Some Remarks on Newtonian Mechanics
●
An important consequence is that no mechanical experiments carried out entirely in one
inertial frame can tell the observer what the motion of that frame is with respect to any
other inertial frame.
● There is no way at all of determining the absolute velocity of an inertial reference frame
from our mechanical experiments. No inertial frame is preferred over any other.
● There is no physically definable absolute rest frame. All inertial frames are equivalent as
far as mechanics is concerned.
.

5. Some Remarks on Newtonian Mechanics
●
Invariants: We have noticed, transformation laws, in general will change
many quantities but will leave some others unchanged. These unchanged
quantities are called invariants of the transformation.
●
For example under Galilean transformations acceleration is an invariant and­
more important­so are Newton's laws of motion. A statement of what the
invariant quantities are is called a Relativity Principle; it says that for such
quantities the reference frames are equivalent to one another, no one having
an absolute or privileged status relative to the others.

6. Some Remarks on Newtonian Mechanics
Principle of Relativity as expressed by Newton "The motions of
bodies included in a given space are the same amongst themselves,
whether that space is at rest or moves uniformly forward in a
straight line."

7. Electromagnetism & Maxwell’s Equations
●
The Gauss’s Law No Name
Faraday’s Law Ampere’s Law with correction

8. Electromagnetism & Maxwell’s Equations
●
We can deduce from Maxwell's equations that light is electromagnetic wave
and we can identify
as the speed of light in free space (vacuum) with a numerical value

9. Electromagnetism & Maxwell’s Equations
●
According to Galilean transformation, such a velocity cannot be the
same for observers in different inertial frames. It appears that
electromagnetic effects will probably not be the same for different
inertial observers.
●
Also we note that Maxwell’s equations do not preserve their forms
under Galilean transformation (Ref. Ex.7, Tutorial­1).

10. Electromagnetism & Maxwell’s Equations
●
If we accept both the Galilean transformations and Maxwell's
equations as basically correct, then it automatically follows that there
exists a unique privileged frame of reference (the "ether" frame) in
which Maxwell's equations are valid and in which light is propagated
at a speed c.
●
The fact that Galilean relativity principle does apply to the Newtonian
laws but not to Maxwell’s equations of Electromagnetism requires us
to choose the correct consequences from amongst the following
possibilities.

11. The Possibilities
●
A relativity principle exists for mechanics but not for electrodynamics; in
electrodynamics there is a preferred frame of reference i.e., the Ether frame of
reference. Should this alternative be correct the Galilean transformations would
apply and we would be able to locate the ether frame experimentally.
●
A relativity principle exists both for mechanics and for electrodynamics, but the
laws of electrodynamics as given by Maxwell are not correct. If this alternative
were correct, we ought to be able to perform experiments that show deviations
from Maxwell's electrodynamics and reformulate the electromagnetic laws. The
Galilean transformations would apply here also.

12. ●
A relativity principle exists both for mechanics and for electrodynamics, but
the laws of mechanics as given by Newton are not correct. If this alternative is
the correct one, we should be able to perform experiments which show
deviations from Newtonian mechanics and reformulate the mechanical laws. In
this case, the correct transformation laws would not be the Galilean ones (for
they are inconsistent with the invariance of Maxwell's equations) but some
other ones which are consistent with classical electromagnetism and the new
mechanics.
The above possibility led Einstein to Formulate His Special Theory of Relativity

13. Attempts to Locate the Absolute Frame­The Michelson­Morley
Experiment
●
Inspired by the first possibility mentioned earlier, scientists in late 19th Century
tried to carry out experiments in which one can measure the speed of light in a
variety of inertial systems, noting whether the measured speed is different in
different systems, and if so, noting especially whether there is evidence for a
single unique system­the "Ether" (also written as “Æther”) frame­in which the
speed of light is c, the value predicted from electromagnetic theory.
●
During 1881­1887. A.A. Michelson & E.W. Morley carried out an
interferometric experiment to detect the presence of Æther.

14. Albert Abraham Michelson
●
Albert A. Michelson (December 19, 1852 – May 9,
1931) was was an American physicist known for his
work on measuring the speed of light and especially
for the Michelson–Morley experiment. In 1907 he
received the Nobel Prize in Physics, becoming the
first American to win the Nobel Prize in a science.
Prize Motivation read "for his optical precision instruments
and the spectroscopic and metrological investigations
carried out with their aid."
More about him: https://en.wikipedia.org/wiki/Albert_A._Michelson

15. Edward Williams Morley
●
Edward W. Morley (January 29, 1838 –
February 24, 1923) was an American scientist
famous for his extremely precise and accurate
measurement of the atomic weight of oxygen
and for the Michelson–Morley experiment.
More about him: https://en.wikipedia.org/wiki/Edward_W._Morley

16. More about Æther concept
●
Let us try to understand Æther a bit further. Whenever we describe a wave
phenomena, e.g., when we say that speed of sound in dry air is at 0°C is 331.3
m/s we have in mind an observer, and a corresponding reference system, fixed
in the air mass through which the sound wave is moving. The speed of sound
for observers moving with respect to this air mass is correctly given by the
usual Galilean velocity transformation.
●
In sharp contrast to above example, when we say that the speed of light in a
vacuum is 2.9979x108 m/s it is not at all clear what reference system is
implied.

17. More about Æther concept
●
A reference system fixed in the medium of propagation of light presents
difficulties because, in contrast to sound, no medium seems to exist.
●
We must bear in mind that, the the thought process of physicists of 19th
Century was dominated by Mechanistic View of Nature and hence it was
inconceivable that light and other electromagnetic waves, in contrast to all
other kinds of waves, could be propagated without a medium.

18. More about Æther concept
●
A reference system fixed in the medium of propagation of light presents
difficulties because, in contrast to sound, no medium seems to exist.
●
We must bear in mind that, the the thought process of physicists of 19th
Century was dominated by Mechanistic View of Nature and hence it was
inconceivable that light and other electromagnetic waves, in contrast to all
other kinds of waves, could be propagated without a medium.

19. More about Æther concept
●
This Conformist approach naturally led to to postulate such a medium, called
the Æther, even though it was necessary to assume unusual properties for it,
such as zero density and perfect transparency, to account for its
undetectability. The Æther was assumed to fill all space and to be the medium
with respect to which the speed c applies.
●
It followed then that an observer moving through the ether with velocity v
would measure a velocity c' for a light beam, where c' = c + v. It was this
result that the Michelson­Morley experiment was designed.

20. Basic Philosophy of Michelson­Morley Experiment
●
If we believe in Æther then the spinning and rotating earth should be moving
through it. An observer on earth would sense an "ether wind," whose velocity is
v (say earth's orbital speed about the sun about 30 km/s or 30,000 m/s)
relative to the earth then v/c 10
≈ −4. The optical instruments available, which
were accurate up to first order in v/c, were not able to detect the absolute
motion of the earth through Æther.
●
For an unambiguous test of Æther hypothesis Optical instruments that can
measure second order effects i.e., one that can measure (v/c)2 were required.
Fortunately Interferometer designed and built by Albert Michelson possessed
remarkable accuracy to capture second order effects!!

21. A Simplified Arrangement of Michelson­Morley Experiment
Beam 1
Beam 2
v
l1
l2

22. The Michelson­Morley’s Experimental Set­up

23. Description of Michelson­Morley Experiment
●
The interferometer is fixed on the earth. If we imagine the "Æther" to be fixed
with respect to the sun, then the earth (and interferometer) moves through the
ether at a speed of 30 km/sec, in different directions, as shown in the
following Fig. Also the spinning effect are neglected in this experiments.

24. Description of Michelson­Morley Experiment
●
A parallel beam of light emerges from the Monochromatic Coherent Source of
light S and is directed towards mirrors M1 and M2 with the help of Beam
Splitter G. After reflection from both the mirrors, light waves are made to
interfere and the patterns is recorded by Detector D (it can be a
telescope/photographic film).
●
In the field of view one detects a bright/dark fringe depending on the path
difference (or phase difference) of the beams.

25. Analysis of Michelson­Morley Experiment
●
We now calculate the phase difference between the beams 1 and 2.
●
To do this let us calculate time taken by light beam 1 to reach from beam
splitter to mirror 1 and back

26. Analysis of Michelson­Morley Experiment
● The path of beam 2 from G to M2 and back is a cross­stream through Æther as
depicted in the following Fig., enabling the beam to return to the advancing G.
We can easily note here,
Which simplifies to

28. Analysis of Michelson­Morley Experiment
●
The difference in transit times is
●
Now if the entire set up is rotated by 90 degree, i.e., the role of arms 1 & 2 gets
interchanged. If the corresponding times are now designated by primes, the same
analysis as above gives the transit­timc difference as

29. Analysis of Michelson­Morley Experiment
●
The difference in transit times after 90 degree rotation is
●
Hence the rotation changes the difference by:

30. Analysis of Michelson­Morley Experiment
●
Using the binomial expansion and dropping terms higher than the second­
order, we find:

31. Analysis of Michelson­Morley Experiment
●
The difference in transit times in initial and rotated configurations should
cause a shift in the fringe pattern, since it changes the phase relationship
between Beams 1 and 2.
●
If the optical path difference between the beams changes by one wavelength,
for example, there will be a shift of one fringe across the cross­hairs of the
viewing telescope.

32. Analysis of Michelson­Morley Experiment
●
If N
Δ represents the number of of fringes moving past the cross­hairs as the
pattern shifts. Then, if light of wavelength λ is used, so that the period of one
vibration is
●

33. Analysis of Michelson­Morley Experiment
●
In the original experiment carried out by Michelson & Morley they had taken

34. Analysis of Michelson­Morley Experiment
●
With this data the predicted fringe shift can be calculated as,
●
A shift of four­tenths a fringe was predicted!!
Experimentally no fringe shift was detected, the Interferometer was sensitive
enough to detect a shift as small as 1/100 of a fringe!!

36. Now it is the right time to enter in a New Era of Thinking