theory

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REFERENCE FRAME
Any moving platform or a moving coordinate system which is used by an observer to measure a physical
quantity or to observe an event is called reference frame.
The measured value of a physical quantity depends on the reference frame of the observer, e.g. the
measured value of the speed of a train would be different for the two observers, one on the ground and
other inside a moving car. Thus it is essential to quote the reference frame of the observer along with the
measured value of a physical quantity to make the measurement meaningful.
Out of the infinite number of available reference frames that frame is considered to be most suitable
from where the given motion appears to be simplest.
Reference frames are characterized in the following two categories
Inertial Reference Frames
Reference Frames. These are the reference frames moving with constant velocity with respect to each other.
The observers in different inertial reference frames may obtain different numerical values for measured
physical quantities but laws of physics remain same for all observers in inertial reference frames.
Non-Inertial Reference Frames
would not hold if mass is variable (as in case of falling raindrops, rockets and particles moving with
relativistic speeds).
-Inertial
Reference Frames. The reference frames moving with uniform acceleration with respect to each other are
non-inertial.
LUMINIFEROUS ETHER
In practice, motion is always described in a relative frame of reference. Newton insisted that there must be
a fundamental reference frame which is in absolute rest and with respect to which all the motions must be
measured.
Moreover, Maxwell proved light to be an electromagnetic wave. Since waves require a material medium
for their propagation, it was supposed that there must also be a suitable medium to carry these
electromagnetic waves which travelled even through empty space between stars and Earth. This medium
was called as LUMINIFEROUS ETHER. As light waves are transverse waves and transverse waves require
shearing forces which can occur in solids only, ETHER must be a rigid solid pervading all space, empty or
otherwise.
MICHELSON - MORLEY EXPERIMENT
Object
The objective of the Michelson-Morley experiment was to determine velocity of earth around sun with
respect to stationary ether.
Apparatus
interference by division of amplitude.

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Theory
A beam of light from a monochromatic source S falls on a half silvered plate P placed at an angle of 45o
with respect to incident beam. The incident beam is partially reflected towards plane mirror M1 and partially
transmitted towards mirror M2 by glass plate P. After reflections at
plane mirror M1 and M2, the reflected and transmitted rays retrace
their path and enter into the telescope T. When seen through
telescope T, fringes are observed due to interference between rays
coming after reflections at mirrors M1 and M2.
In Michelson Morley experiment the interferometer is mounted on
earth such that its X - arm is parallel to the direction of motion of earth
around sun. Let the velocity of earth around sun with respect to
stationary ether is . Since apparatus is mounted on earth, it is also
moving with velocity with respect to ether.
In the experiment the distance of mirror M1 and
that of mirror M2 from plate P are kept equal i.e. in
figure
(say) (1)
With respect to the stationary ether, the relative
velocity of light travelling along PA will be ,
where is the velocity of light in free space. Similarly
the relative velocity of light travelling along AP will be
. Hence the time taken by light to travel from
P to A and back to P will be given by
Since ,
therefore (2)
Further, when seen at absolute rest, the apparatus appears to be moving with velocity along X-direction. In
this case, the ray travelling along Y - direction will strike mirror M1
If light takes time
and .
.
This gives
i.e. (3)
Hence, the total time taken by the reflected ray in coming into the telescope after reflection at mirror M1
(4)

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From Eqs. (2) and (4), it is clear that .
The time difference
(5)
The path difference corresponding to time difference
(6)
This gives us the path difference between the two rays moving along X and Y arms. Now, if the apparatus is
rotated by about a vertical axis then X and Y arms interchange their positions and the path difference
becomes . Hence after a rotation of , the total path difference introduced between the two
beams is . Further, if corresponding to this path difference fringes shift in the field of view then
(7)
From the observed fringe shift , one can calculate velocity of earth with respect to stationary ether.
Result:
When the apparatus was rotated, practically no fringe shift was observed. The experiment was repeated at
difference places on earth and in different seasons but fringe shift could never be detected. Hence
Michelson - Morley experiment has a negative result.
Explanation of the result:
For negative result of the Michelson - Morley experiment there may be either of the following possibilities
1. There is nothing like ether. i.e. it is not possible for anything to come to absolute rest.
2. Earth drags with it the ether in its immediate neighborhood so that there is no relative motion
between earth and ether.
3. The arm of the interferometer parallel to direction of motion suffers a contraction, so that the path
difference between the interfering beams is always zero (length contraction: suggested by Lorentz
and Fitzgerald).
4. Einstein suggested that the velocity of light is absolute and it is independent of relative motion
between source and observer.
The two basic postulates of theory of relativity are
1. Principle of Equivalence
the laws of physics remain same in all inertial
reference frames
Explanation: This postulate expresses the absence of a universal frame of reference. If the laws of
physics had different forms for two observers in relative motion then from these differences it
would have been possible to determine that which of them is actually moving. But this distinction
does not exist in nature.
2. Absoluteness of velocity of light
The velocity of light is absolute and it is independent of the direction of motion as well as the relative
motion between source and observer.

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According to this postulate (where is velocity of light
and is the velocity of observer)
Explanation: Suppose a flash of light is emitted from the origin
when
other. If it is possible to see the wavefront spreading out in space
then, in view of the first postulate, each of the observers in the
two reference frames will claim to be at the center of the
wavefront noticing that the other one is shifting from the center,
i.e. for both the observers the velocity of light will be .
GALILEAN TRANSFORMATIONS
-
axis with constant velocity with respect to
another reference frame S. Suppose two
observe an event. The observer in frame S
notes that the event occurs at position
and at time , while the observer in
and at time .
If time is measured from the instant when
origins of the two frames coincide, the
along -axis is and hence
Since there is no relative motion along y and z directions, therefore
Further, from our everyday experience we can assume that
Hence the transformation equations from reference frame to frame are
(8a)
(8b)
(8c)
(8d)
These are known as Galilean transformations.
Galilean transformations for velocity and acceleration
Differentiating Eq. (8a), (8b) and (8c) with respect to time, we have

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and
i.e. (9a)
(9b)
and (9c)
where and
Equations (9) give Galilean transformations for velocity.
Further, differentiating Eq. (9) with respect to time, the components and of
(10a)
(10b)
and (10c)
Galilean transformations violate the postulates of special relativity
using Galilean transformations, the equations take very different form and they do not remain identical in
the two reference frames. This is contrary to first postulate of special relativity. Moreover, if we measure the
speed of light along -axis as in reference frame S then according to Eq. (9a) the speed of light in system
special theory of relativity.
LORENTZ TRANSFORMATIONS
Lorentz transformations are relativistic
analog of Galilean transformations. These
are fundamental equations of special
relativity and are derived on the basis of
its two postulates.
along -axis with a constant velocity
with respect to another reference frame
S. Let a light flash is emitted from origin
frame S notes that light reaches a point P with coordinates at distance from origin O in time with
velocity , then
or (11)
Since from 2nd
postulate of special relativity, the velocity of light is for both the observers, therefore
at distance
in time . This gives

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or (12)
Further, from 1st
postulate of theory of relativity the two reference frames are equivalent, hence from Eqs.
(11) and (12) we have
Since there is no relative motion along and directions, we have and , therefore above
equation gives
(13)
Further, let the transformation equations for and are of the form
(14)
and (15)
where and are constants.
Substituting for and from Eq. (14) and (15) in Eq. (13) and equating coefficients of , and on
both sides we get three equations in constants and . Solving these equations one obtains
and
Hence from Eq. (14) and (15) the transformation equations for and are given by
and
Thus, Lorentz transformation equations are given by
(16a)
(16b)
(16c)
and (16d)
The inverse Lorentz transformations can be obtained by interchanging coordinates and
and replacing by as below
(17a)
(17b)
(17c)
(17d)

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For non-relativistic speeds and , therefore Lorentz transformations as given by Eq. (16)
reduce to
and
which are Galilean transformations.
LENGTH CONTRACTION
According to theory of relativity, space is not absolute. The observed length of an object maybe different
when measured from different reference frames. The length of a moving rod is smaller than that of same rod
at rest.
moving with respect to each other with a
constant relative velocity in the positive -
direction. Suppose a rod AB of length is
placed parallel to -
observe the -coordinate of the ends of the
rod as and , then
(18)
and (19)
where L
the observer in system S observes that the rod is moving along x-axis with velocity v . From Lorentz
transformations we have
and
substituting these values in Eq. (18) we get
This gives length of a moving rod as
(20)
Discussion
1. From Eq. (20) it is clear that . Thus length of a rod contracts during motion parallel to its
length.
2. At low velocities when Eq. (20) gives , which is consistent with our daily experience.
3. For , Eq. (20) gives , i.e. whatever be the length of the rod it will reduce to a point when it
moves with velocity of light.

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4. The contraction occurs only in that dimension which is parallel to the
velocity of object. Thus a square becomes rectangle when moving and
sphere becomes ellipsoid.
5. The contraction is reciprocal i.e. if two identical rods are at rest one in S
shorter than the rod of his own system.
TIME DILATION
According to theory of relativity, time is not absolute. The time interval between two events as measured
from different reference frames may be different.
with respect to each other with a constant
relative velocity along -axis. Suppose a gun is
fixed at position P having coordinates in
sy and
(21)
along -axis with respect to system S. let the
time interval between the shots as measured by
an observer in system S is
(22)
From inverse Lorentz transformation we have
and
Substituting these values in Eq. (22) we have
i.e. (23)
Discussion
1. From Eq. (23) it is clear that . Thus the time interval between two events taking place at a
called time dilation.
2. At low velocities , hence , showing Galilean absoluteness of time.
3. At , .
4. Twin Paradox: if one of the two identical twin brothers goes on a long space journey in a rocket
moving with a velocity comparable to velocity of light leaving his brother on earth then the clock in
the moving rocket will appear to go slow according to time dilation formula. As a result when he
returns to earth he will find himself younger than his brother who stayed on earth.

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MESON DECAY: Experimental evidence of Time Dilation and Length Contraction
�-Mesons are unstable subatomic particles which are also found in cosmic ray showers. They have an
average life time of seconds and travel with a velocity of ( being velocity of light). �-
Mesons are produced at high altitudes at a height of about 10 km due to interaction with cosmic photons
and they are projected towards the ground. If length contraction and time dilation phenomena are not real
then the distance covered by �-Mesons in their life time will be
However, �-Mesons can be observed at sea level even though classically they can travel only 600 m distance
in their life time. Further, if length contraction and time dilation phenomena are real then 10 km distance as
seen by �-Mesons will be given by
Similarly with respect to mesons the lifetime of �-Mesons will be given by
and distance travelled by �-Mesons in this time interval
LAW OF ADDITION OF VELOCITIES
The law of addition of velocities as used in Galilean physics is , where is the relative velocity
between two objects moving with velocity and respectively. However, This law holds good only at low
velocities. According to theory of relativity the general formula for velocity addition is
where is velocity of light.
Proof:
moving in space along -axis. Suppose velocity of
. The velocity of object O
and . Let at an instant the position of the object O as measured from two systems are and then
(24)
and (25)
where and
Further, from Lorentz transformations
and
Differentiating above equations we have
(26)

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and (27)
Dividing Eq. (26) by Eq. (27)
or
i.e. (28)
should
be replaced by and so we get
i.e. (29)
Eq. (28) gives us relative velocity between two bodies moving with speeds and in same direction and Eq.
(29) gives the same for opposite direction.
Discussion:
1. At low velocities and hence
and from Eqs. (28) and (29) we have which is old classical formula.
2. When one of the velocities is , say then
Thus the relative velocity between two objects one of which is moving with velocity of light is equal
to velocity of light itself. This is in agreement with 2nd
postulate of special relativity.
3. From Eq. (29), the relative velocity between two photons moving towards each other may be
obtained by substituting as
RELATIVITY OF SIMULTANEITY
Two events occurring at same time are called simultaneous events and this phenomenon is called
simultaneity. It can be shown by theory of relativity that two events occurring at same time in a physical
system may appear to be occurring at different times from other systems.
Proof
along -axis. Suppose an observer
in system S observes two events E1 and E2 occurring at positions and and at time and respectively.
and and at time
and respectively.
From time transformation equation of Lorentz

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(30)
and (31)
Subtracting Eq. (30) from Eq. (31) we get the time interval between the two events as observed from system
and (32)
Discussion:
1. If these two events are simultaneous in system S i.e. then
(33)
It is clear that . Further, for
we have from Eq. (33
will appear to be occurring first.
2. Space time equivalence
From Eq. (33) we have
This equation gives a relation between time interval and space interval . It shows relativity of
simultaneity as well as the equivalence of time and space. It indicates that space and time are inter-
convertible and hence they are equivalent.
3. From Eq. (33), one has when . Hence two simultaneous events in system S are
occurring at same place.
RELATIVITY OF MASS
According to theory of relativity mass of a
body increases with its velocity i.e. mass of a
moving body is greater than its mass when it
is at rest.
Proof:
with respect to each other with a constant
relative velocity in the positive -direction.
Suppose there are two identical balls A and B of mass hing each other at equal speeds
and . Let the balls collide and after collision they coalesce ( ) to form a combined mass. In
Momentum of body A + Momentum of body B = Momentum of combined mass.

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Further, let us consider the same collision from system S. Let the velocities of the balls A and B as observed
from system S are and and there masses are and , then from law of addition of velocities
(34)
and (35)
observed from system S will be .
Now, applying law of conservation of momentum in system S, we have
(36)
Substituting for and from Eq. (34) and (35), we get
which gives
Simplifying we get
i.e. (37)
Further, using Eq. (34) we can evaluate
i.e.
or (38)
Similarly we can obtain
(39)
Dividing Eq. (39) by Eq. (38) we get
(40)
From Eq. (37) and Eq. (40), we have

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or (41)
Since both sides of above equation are independent of each other, therefore above result can be true only if
each side of the above equation is a constant (say ). Hence,
or, in general, for a body of mass moving with velocity we have
From above equation, if then . Hence represents mass of the body when the body is at
rest. Thus mass of a moving body is given by
(42)
Discussion:
1. When , we have . This shows Galilean absoluteness of mass at low velocities.
2. At , Eq. (42) gives i.e. an object having a finite rest mass and moving with velocity of light
will have infinite mass.
3. From Eq. (42)
which indicates that if then i.e. the particles having zero rest mass (like photons, neutrino
etc) always move with velocity of light.
MASS ENERGY RELATION
According to theory of relativity, mass and energy are equivalent. A large amount of energy concentrated
- energy equivalence relation is
given by
where is the energy equivalent to mass and is velocity of light.
Proof:
We know that the force acting on a particle is defined as the rate of change of linear momentum i.e.
Since according to theory of relativity mass varies with velocity of the particle therefore
(43)
Let the force displaces the body of mass through a distance then increase in kinetic energy of
the body
Substituting for from Eq. (43)
i.e. (44)
Further, from mass transformation formula

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Squaring both sides and simplifying we get
or
Differentiating above equation we get
or (45)
from Eq. (44) and Eq. (45)
(46)
which shows that a change in kinetic energy of the particle is directly related to a change in mass. Moreover,
when body is at rest we have , and kinetic energy and when body is moving its mass
is and kinetic energy is . Integrating Eq. (46) within these limits we get
or (47)
This is relativistic formula for kinetic energy. When the body is at rest the energy stored in the body is
which is called rest mass energy. The total energy of the body is the sum of rest mass energy and kinetic
energy. Hence
or (48)
Discussion:
1. Eq. (48) indicates that mass and energy are equivalent and they are inter-convertible.
2. The formula for kinetic energy reduces to the classical formula for as follows
or
Expanding right hand side by binomial theorem and neglecting higher terms as , we get
3. Eq. (48) forms the basis of all nuclear reactions like fission and fusion.
RELATIVISTIC MOMENTUM
is given by
where .
Therefore (49)

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The linear momentum and from Eq. (49) we have
i.e.
or
Substituting in the second term on left hand side
i.e.
or
or (50)
Above equation gives relation between total energy, rest mass energy and linear momentum. It shows that
for a mass at rest when then . Further a particle with no rest mass i.e. can still have
linear momentum given by
Since therefore From Eq. (50)
or
i.e.
or
This gives relativistic formula for linear momentum in terms of kinetic energy of the particle. The first term in
right hand side is identical with the classical formula for momentum ( ). The second term may
be called relativistic correction term which reduces to zero at low velocities.
EXERCISE
Short Answer Questions
1. Is earth inertial frame of reference? Explain.
2. What do you understand by inertial and non-inertial frames? (UPTU 14 CO).
3. What are inertial and non-inertial reference frames? (UPTU 14)
4. What do you understand by time dilation? (UPTU 13)
5. What are mass-less particles? (UPTU 13, 12)
6. What is length contraction? (UPTU: 12 II)
7. What do you understand by variant and invariant under Galilean transformations? (UPTU 11)
8. Show that Galilean transformations violate the postulates of special relativity.
9. Show that a mass-less particle has energy and momentum and moves with the speed of light.
10.
11. What was the objective and outcome of Michelson-Morley experiment?
12. How the negative result of MM Expt interpreted? (UPTU 15)

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13. Establish the relation , where symbols have their usual meaning.
14. Find relativistic relation between energy and momentum. (UPTU 15)
Long Answer Questions
1. Show that the relativistic invariance of the law of conservation of momentum leads to the concept of
variation of mass with velocity. (UPTU 15, 10)
2. Derive the Galilean transformation equations and show that its acceleration components are invariant.
(UPTU 15)
3. What do you mean by proper length? Derive the expression for relativistic length. (UPTU 15)
4. of special theory of relativity. Explain why Galilean relativity failed to explain
actual results of Michelson-Morley experiment. [(UPTU 13, 12, 10)]
5. State the fundamental postulates of the special theory of relativity. Deduce the Lorentz transformation
equations. (UPTU 09)
6.
transformations reduce to Galilean transformations. (UPTU 14)
7. Show that the relativistic invariance of the law of conservation of momentum leads to the concept of
variation of mass with velocity. (UPTU 10)
8. Show that the relativistic invariance of the law of conservation of momentum leads to the concept of
variation of mass with velocity and equivalence of mass and energy. (UPTU 13, 12)
9. Explain Michelson Morley Experiment and its outcome. (UPTU 12 II)
10. Discuss the objective and outcome of Michelson Morley experiment. (UPTU 12)
11. What was the objective of Michelson Morley experiment? Describe the experiment. How is the
negative result of the experiment interpreted? (UPTU 14 CO, 08)
12. What do you understand by time dilation? How the time dilation is experimentally verified? (UPTU 12)
13. Explain why a moving clock appears to go slow to a stationary observer. (UPTU 07)
14. What is Time dilation effect? Show that time dilation is a real effect.
15. Show that no signal can travel faster than the velocity of light. (UPTU 10)
16.
postulate. (UPTU 14, 14 CO 09)
17. Show from Lorentz transformation that two simultaneous events ( ) at different positions
(
08)
18. Show that for small velocities the relativistic kinetic energy of a body reduces to the classical kinetic
energy, which is less than the rest energy. (UPTU 07)
19. Show that the mass-less particles can exist only if they move with the speed of light and their energy E
and momentum p must have the relation E = pc. (UPTU 07)
20.
Give experimental evidence which verifies Einstein mass energy equivalence relation.
21. ion and give its experimental evidence.
Numerical Problems
1. Calculate the length of one meter rod moving parallel to its length when its mass is 1.5 times its rest
mass. (UPTU 13)
2. What is the length of a meter stick moving parallel to its length when its mass is 3/2 times its rest
mass?

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3. Calculate the length and orientation of a rod of length 5 m in a frame of reference moving with a
velocity of 0.6c in the direction making an angle 30 degrees with the rod. (UPTU 15, 10) Hint:
[4.272 m, 35.8o]
4. Calculate percentage contraction of a rod moving with a velocity of 0.8c in a direction inclined at 60
deg to its own length. (UPTU 14 CO, 08) Hint: [8.3%]
5. How fast would a rocket have to go relative to an observer for its length to be contracted to 99% of
its length? (UPTU 07-08) Hint: [0.141 c]
6. A rod 1 m long is moving along its length with a velocity 0.6c. Calculate its length as it appears to an
observer (a) on the earth (b) moving with the rod itself.
7. A train, whose length is 150 meter when at rest, has to pass through a tunnel of length 125 m. The
train is moving with uniform speed of 2.4 �108
m/s towards the tunnel. Find the length of the train
and that of the tunnel as observed by an observer (i) at the tunnel (ii) at the train.
8. A clock in a spaceship emits signals at intervals of 1 second as observed by an astronaut in the space
ship. If the spaceship travels with a speed of m/s. What is the interval between successive
signals as seen by an observer at the control center on the ground?
9. A particle with a proper lifetime of 1 �s moves through the laboratory at m/s, (a) what is
its lifetime as measured by the observer in the laboratory? (b) What will be the distance traversed by
it before disintegrating?
10. A man leaves the earth in a rocket that makes a round trip to the nearest star which is 4 light years
away at a velocity of 0.8c. How much younger will he be on his return than his twin brother who
preferred to stay behind?
11. Compute the +
mesons traveling with velocity 0.8c if their proper mean life time is
s. What will be the distance traveled in one mean lifetime with a velocity 0.8c? What will
be this distance if the relativistic effect is not considerable?
12. A clock measures the proper time. With what velocity it should move relative to an observer so that
it appears to go slow by 30 s in 24 hrs.Hint: , [ ]
13. A wrist watch keeping correct time on the earth is worn by the pilot of a spaceship. How much will it
appear to go slow per day with respect to an observer on the earth when spaceship leaves the earth
with velocity of 107
m/sec.
14. At what speed should a clock be moved so that it may appear to lose 1 minute in each hour? (UPTU
09) Hint: , [ ]
15. The proper life of mesons is sec. If a beam of these mesons of velocity 0.8c is
produced, calculate the distance the beam can travel before the flux of the meson beam is reduced
to 1/e2
times the initial flux.
16. An experimenter observes a radioactive atom moving with a velocity of . The atom then emits
a particle which has a velocity of relative to the atom in the direction of its motion. What is
the velocity of the particle as observed by the experimenter? Hint: ,
[ ]
17. An electron is moving with a speed of in a direction opposite to that of a moving photon.
Calculate the relative velocity of the photon with respect to the electron. Hint:
, [ ]
18. A particle has a velocity m/s in a coordinate system moving with velocity 0.8c
relative to laboratory along +ve direction of x-axis. Find in laboratory frame.

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19. A spaceship moving away from the earth with velocity 0.5 c fires a rocket whose velocity relative to
the space is 0.5 c. Calculate the velocity of the rocket as observed from the earth in following two
cases: (i) away from the earth (ii) towards the earth.
20. In an inertial frame S, a red light and blue light are separated by a distance = 2.45 km. The red light is
velocity u=0.855 c. What is the distance between the two flashes and the time between them as
21. At what speed is a particle moving if the mass is equal to three times its rest mass? Hint:
, [ ]
22. Find the speed that a proton must be given if its mass is to be twice its rest mass of kg.
What energy must be given to the proton to achieve this speed?
23. How fast must an electron move in order to have its mass equal to the rest mass of the proton?
24. What is the length of a meter stick moving parallel to its length when its mass is 3/2 times its rest
mass?
25. At what velocity will the mass of a body be 2.25 times its rest mass?
26. Calculate the velocity of a particle if the kinetic energy of particle is three times the rest mass
energy. (UPTU 12-II) Hint: , mass transformation formula, [ ]
27. The total energy of a moving meson is exactly twice its rest energy. Find the speed of the meson.
(UPTU 12) Hint: , [ ]
28. The mass of a moving electron is 11 times its rest mass. Find its kinetic energy and momentum.
(UPTU 11, 09) Hint: , , [ ]
29. How much does a proton gain in mass when accelerated to a kinetic energy of 500 MeV. (UPTU: 07)
Hint: , [ kg]
30. If 4 kg of a substance is fully converted into energy, how much energy is produced?
31. Calculate the rest energy of an electron in joules and in electron volts.
32. Calculate the kinetic energy of an electron moving with a velocity in the laboratory system.
33. If the Kinetic energy of a body is double its rest mass energy, calculate its velocity. (UPTU 15)
34. Calculate the amount of work to be done to increase the speed of electron from 0.6 c to 0.8c, given
rest energy of electron = 0.5 Mev?
(UPTU 14)
35. Find the speed of 0.1 MeV electrons according to classical and relativistic mechanics.
36. What will be the fringe shift according to the ether theory in the Michelson Morley experiment, if
the effective path length of each path is 7 m and light has 7000 � wavelength? The velocity of earth
is 3x104
m/s. [0.2] Hint:
37. In Michelson Morley experiment the length of the paths of the two beams is 11 m each. The
wavelength of the light used is 6000 �. If the expected fringe shift is 0.4 fringes, calculate the
velocity of the earth relative to ether. Hint: , [31.3 km/s]
38. Calculate the expected fringe shift in a Michelson Morley experiment if the distance of each path is
11 m and the wavelength of light is 5.6x10-7
m. The experimental set up was now rotated through
90o
. The linear velocity of earth may be taken as 30 km/s. (0.196) Hint:
39.
the X-axis is given by (11, 9, 8). Calculate its position with respect to the frame S, if the two frames
were in coincidence only 0.5 second before. (16,9,8) Hint:

19. Dr.SujeetAgarwal
Engineering Physics Vol - I
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40. If at the time
produced at the common origin, show that the speed of propagation of spherical wavefront is the
.
41. Show that the circle
with velocity relative to S.
42. A circular lamina moves with its plane parallel to the -plane of a reference frame S at rest.
Assuming its motion to be along the axis of , calculate the velocity at which its surface area
would appear to be reduced to half to an observer in frame S. (UPTU 13)
43. Obtain the volume of a cube, the proper length of each edge of which is , when it is moving with a
velocity along one of its edge?
44. Show that space time interval is invariant under Lorentz transformation.
45. Prove that magnitude of momentum of a particle of rest mass and the kinetic energy T is given
by
46. Establish the relation , where is the linear momentum, is the total energy of
the particle.
47. If frame S' is moving with velocity with respect to frame S, and the components of velocity in
frame S' are and then prove that for the frame S, .
48.
49. As seen by an inertial observer S an event takes place at at time . Another
event takes place at at time so that for S the two events are simultaneous.
-axis at velocity with respect to the
events are not simultaneous and where .