55Relativity and Universe

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RELATIVITY AND UNIVERSE
▢ Theory of Relativity:
❖ Formulated By: Albert Einstein
❖ Classical Newtonian View: Space, time and mass are absolute and independent of relative motion of body, observer and frames of reference
❖ Special Relativity View:
  • •
    Space is relative
  • •
    Time is relative
  • •
    Mass is relative
  • •
    Speed of light is invariant
  • •
    Valid strongly when velocity is comparable to speed of light
▢ Frame of Reference:
❖ Definition: System of coordinate axes used to define position of a particle or event in 2D or 3D

Table 1: Types of Frame of Reference

Frame
Meaning
Example
Inertial frame
Unaccelerated frame where Newton's laws and fundamental laws of mechanics hold good
Train moving with constant velocity
Non-inertial frame
Accelerated frame where Newton's laws of motion do not hold exactly
Train moving with variable velocity
❖ Important Point: Reference frame attached to earth is taken as inertial frame by definition
▢ Postulates of Special Theory of Relativity:

Table 1: Einstein's Postulates

Postulate
Statement
First postulate
Laws of physics have same identical form in all inertial frames of reference
Second postulate
Velocity of light in free space has same value in all inertial frames
Light velocity
Invariant under transformation from one inertial frame to another
Time
Not absolute; time is relative
▢ Michelson-Morley Experiment:
❖ Importance: Led to development of theory of relativity
❖ Conclusion: Velocity of light is same in all inertial frames
▢ Length Contraction:
❖ Also Called: Lorentz-Fitzgerald length contraction
❖ Condition: Observed when body moves along its length with speed \(v\)

Table 1: Length Contraction

Quantity
Formula / Meaning
Observed length
\(L=L_0\sqrt{1-\frac{v^2}{c^2}}\)
Proper length
\(L_0\)
Observed length in moving condition
\(L
Contraction factor
\(\sqrt{1-\frac{v^2}{c^2}}\)
Direction of contraction
Only along direction of motion
❖ Important Point: Moving rod appears shorter than its actual length
▢ Time Dilation:
❖ Definition: Moving clock appears to run slow

Table 1: Time Dilation

Quantity
Formula / Meaning
Observed time interval
\(t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}\)
Proper time
\(t_0\)
Observed time
\(t>t_0\)
Moving clock
Runs slow
❖ Important Point: Observed time interval between two events is greater due to relative motion
▢ Relativistic Mass:
❖ Definition: Mass of body depends on its velocity

Table 1: Relativistic Mass

Quantity
Formula / Meaning
Relativistic mass
\(m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\)
Rest mass
\(m_0\)
Moving mass
\(m>m_0\)
At rest
\(v=0\Rightarrow m=m_0\)
At speed of light
\(v=c\Rightarrow m\to\infty\)
❖ Important Point: Mass of body in moving frame observed from stationary frame is greater than rest mass
▢ Mass-Energy Equivalence:

Table 1: Mass-Energy Formulae

Quantity
Formula
Rest energy
\(E_0=m_0c^2\)
Total energy
\(E=mc^2\)
Kinetic energy
\(K=mc^2-m_0c^2\)
Kinetic energy
\(K=(m-m_0)c^2\)
Mass defect energy
\(E=\Delta mc^2\)
❖ Important Points:
  • •
    When mass disappears, equivalent energy is produced
  • •
    Electron-positron annihilation produces energy
  • •
    Electron + positron annihilation energy = \(1.02MeV\)
  • •
    Rest mass of photon is zero
▢ Relativistic Velocity Addition:

Table 1: Velocity Addition

Condition
Formula
Same direction
\(u=\frac{u'+v}{1+\frac{u'v}{c^2}}\)
Opposite direction
\(u=\frac{u'-v}{1-\frac{u'v}{c^2}}\)
Limit
Resultant velocity never exceeds \(c\)
▢ Important Relativity Results:

Table 1: Relativity Read and Digest

Fact
Answer
Newtonian mechanics
Valid only at velocities much smaller than light
Speed of light
Same in all inertial frames
Moving body length
Appears shorter along direction of motion
Moving clock
Runs slow
Shape of moving circular body
Appears elliptical
Minor axis of moving circular body
Along direction of motion
Maximum possible speed
Speed of light
Charge of particle
Does not change with velocity
Photon rest mass
Zero
▢ Facts About Universe:

Table 1: Astronomy Facts

Fact
Answer
Temperature of planets
Usually estimated using Stefan's law
Spectrum of sun
Continuous spectrum
Luminosity of stars
Closely related to colour
Mercury
Has no atmosphere
Comets
Small rock-like masses surrounded by vapours / large masses revolving in highly elliptical orbits
Halley's comet period
76 years
Asteroids
Small rocky pieces revolving around sun between Mars and Jupiter
Temperature of sun
Found by Wien's displacement law
Outer space to astronaut
Appears black
Heliocentric theory
Given by Copernicus
Milky Way
Spiral galaxy
Mass of Milky Way
\(3\times10^{41}kg\)
Venus
Retrograde planet
Major atmospheric gas of Venus and Mars
\(CO_2\)
Temperature of sun measured by
Radiation pyrometer
▢ Stars and Universe:

Table 1: Stars and Cosmology

Fact
Answer
Death of star when original mass is 2M to 5M
Neutron star
Death of star when original mass is greater than 5M
Black hole
Most common stars like sun
Dwarfs
Most plausible origin of universe
Big Bang theory
Constellation
Group of bright and faint stars
Expansion of galaxies
Supported by red shift
Planetary motion
Example of conservation of angular momentum
Mars
Has longest day
▢ Hubble's Law:
❖ Statement: Speed of recession of galaxy is directly proportional to its distance

Table 1: Hubble's Law

Quantity
Formula / Meaning
Proportionality
\(v\propto r\)
Equation
\(v=Hr\)
\(H\)
Hubble's constant
Use
Evidence for expansion of universe
▢ Solar System Data:

Table 1: Important Data

Quantity
Value / Meaning
Mean distance from Sun to Earth
\(1AU\)
1 astronomical unit
\(1.49\times10^{11}m\)
Albedo
Reflecting power of heavenly body
Total solar eclipse spectrum
Line emission spectrum
▢ Objective Answer Key:

Table 1: Relativity and Universe MCQ Answers

Q
Ans
1
a
2
b
3
c
4
c
5
b
6
d
7
d
8
c
9
c
10
a
11
b
▢ High-Yield Recall:

Table 1: Relativity and Universe One-Liners

Fact
Answer
Relativity formulated by
Albert Einstein
Inertial frame
Unaccelerated frame
Non-inertial frame
Accelerated frame
Speed of light
Invariant in all inertial frames
Length contraction
\(L=L_0\sqrt{1-\frac{v^2}{c^2}}\)
Time dilation
\(t=\frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}\)
Relativistic mass
\(m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}\)
Rest energy
\(E_0=m_0c^2\)
Total energy
\(E=mc^2\)
Kinetic energy
\(K=(m-m_0)c^2\)
Photon rest mass
0
Electron-positron annihilation energy
\(1.02MeV\)
Velocity addition
\(u=\frac{u'+v}{1+\frac{u'v}{c^2}}\)
Michelson-Morley experiment
Led to theory of relativity
Planets temperature
Stefan's law
Sun temperature
Wien's displacement law / radiation pyrometer
Comets
Highly elliptical orbits
Halley's comet period
76 years
Asteroids
Between Mars and Jupiter
Big Bang
Origin of universe
Red shift
Expansion of galaxies
Hubble's law
\(v=Hr\)
1 AU
\(1.49\times10^{11}m\)
Milky Way
Spiral galaxy
Venus
Retrograde planet
Outer space
Appears black
Heliocentric theory
Copernicus
Albedo
Reflecting power
Q1.
According to pulsation theory, the universe relaxes and contracts at the time interval of
📅BP 2012
Q2.
Two photons approach each other. Their relative velocity will be
Q3.
A charged particle has charge q when at rest. When it acquires a velocity c/3, its charge becomes
Q4.
The apparent length of a meter stick measured by an observer at rest when the stick moves along its length with velocity 86.6% of velocity of light will be
Q5.
If the rest mass of a particle is m0, then its mass when it moves with velocity 0.8c is
Q6.
At what velocity is the kinetic energy of a particle equal to the rest mass energy?
Q7.
When a particle and its antiparticle are annihilated, energy E is released. What is mass of each particle?
📅MOE
Q8.
A spaceship moving away from Earth with velocity 0.6c fires a rocket away from Earth whose velocity relative to spaceship is 0.6c. The velocity of rocket as observed from Earth will be
Q9.
At what speed should a clock be moved so that it appears to lose 2 minutes in each hour?
Q10.
The kinetic energy of a particle is double its rest mass energy. Then the dynamic mass of the particle in terms of rest mass is
Q11.
A proton is accelerated to a kinetic energy of 1.60 × 10^-9 J in a modern synchrotron. If rest mass of proton is 1.67 × 10^-27 kg, the increase in mass of proton is