Module-1-2.pptx

Module -1
Newton’s Applications &
Special Theory of Relativity

INTRODUCTION
Periodic motion – Motion which repeats itself after a regular interval
of time is called periodic motion.
Oscillation (Vibration)- The motion in which a particle moves in a to
and fro motion about a mean position over the same path is called
oscillation or vibration.
Restoring force – The force which
brings oscillator back to its
original position is called
restoring force.
Damping force – The opposing
force which reduces the
oscillation of the oscillator in a
system is called damping force.

SIMPLE HARMONIC
OSCILLATIONS
Statement – The motion in which acceleration of the body is
directly proportional to its displacement from its mean point
is called simple harmonic oscillation
In S.H.O.,
RESTORING FORCE ∝ DISPLACEMENT
𝐹 ∝ 𝑥
𝐹 = −
𝑘
𝑥 ------(1)
Where F = restoring force
x = displacement of oscillator
k = force constant or spring constant (unit: N/m)
Negative sign indicates that the direction of the restoring force
and displacement of the oscillator are opposite to each-other.

Time period (T): Time taken to complete one oscillation
𝑇=
𝜔𝑜 𝑘
= = 2
𝜋
2
𝜋 2
𝜋 𝑚
𝑘
𝑚
Frequency (f): Number of oscillations per unit time.
1 1 𝑘
𝑓=
𝑇
=
2
𝜋 𝑚
SIMPLE HARMONIC OSCILLATIONS

SIMPLE HARMONIC
OSCILLATIONS
Displacement, Velocity and
Acceleration of the S.H.O.
Displacement of S.H.O. at ɵ= 0
is
Velocity of of S.H.O. at ɵ= 0 is
𝑑
𝑡
V = 𝑑
𝑥
=
Acceleration of S.H.O. at ɵ= 0
is
𝑑
𝑡
2
2
a = 𝑑 𝑥

SIMPLE HARMONIC
OSCILLATIONS
Displacement, Velocity and Acceleration of theS.H.O.
Displacement

SIMPLE HARMONIC OSCILLATIONS
From the above figure it is clear that
 Amplitude of velocity is ω0 times the displacement.
 Phase difference between displacement and
velocity graph is 2 𝜋 .
 Amplitude of acceleration is ω0 times the velocity.
 Phase difference between acceleration and velocity
graph is 2𝜋.

Examples of S.H.O.
SIMPLE PENDULUM
A point mass suspended from a rigid
support with the help of massless,
flexible and inelastic string. When the
bob of the simple pendulum is displaced
through a small angle from its mean
position, it will execute SHM.
Here, angular frequency 𝜔=
𝑔
𝑙
𝜔
Time Period, 𝑇=2
𝜋
= 2
𝜋
𝑙
𝑔
Frequency, 𝑓= =
𝜔 1 𝑔
2
𝜋 2
𝜋 𝑙

Examples of S.H.O.
Oscillation of the loaded vertical spring
When the mass m is displaced from the mean position
and released, it starts executing S.H.M.
Here, angular frequency 𝜔=
𝑘
𝑚
𝜔
Time Period, 𝑇=2
𝜋
= 2
𝜋
𝑚
𝑘
Frequency, 𝑓= =
𝜔 1 𝑘
2
𝜋 2
𝜋 𝑚
If l is the extension of spring due to the load, then the
time period of oscillation of the spring is given by
𝑇= 2
𝜋
𝑙
𝑔

Problems
1. The differential equation of motion of freely
2
oscillating body is given by 2𝑑 𝑥
+1
8
𝜋
2
𝑥=0.
𝑑
𝑡
2
Calculate the natural frequency of the body.
2. The total energy of a simple harmonic oscillator is
0.8 erg. What is its kinetic energy when it is
midway between the mean position and an extreme
position?
3. The displacement of a one dimensional simple
harmonic oscillator of mass 5gram is y(t) =
2cos(0.6t + θ) where y & t are in cm & second
respectively. Find the maximum kinetic energy of
the oscillator.

DAMPED HARMONICOSCILLATIONS
• Damped oscillation – The oscillation which takes place in
the presence of dissipative force are known as damped
oscillation
• Here amplitude of oscillation decreases w.r.t. time
• Damping force always acts in a opposite direction to that of
motion and is velocity dependence.
• For small velocity the damping force is directly
proportional to the velocity
• Mathematically
𝐹𝑑 ∝ 𝑣
𝐹𝑑 = −𝑏𝑣 (1)

Where 𝐹𝑑 = dampingforce
b = damping force constant
v = velocity of oscillator
Negative sign indicates that the direction of damping force and
velocity of oscillator are opposite to each other .
In D.H.O. two types of forces are acting such as restoring force and
damping force.
Restoring force can be written as
𝐹𝑟 = −𝑘𝑥
So the net force acting on the body is
𝐹𝑛𝑒𝑡 = 𝐹𝑟+ 𝐹𝑑
-----------------------------------(2)
𝑑 𝑡
= −𝑏 𝑑𝑥
−𝑘𝑥
(3)

𝑑𝑡2
2
= ma = m𝑑 𝑥
----------
From Newton’s 2nd law of motion Fnet
-----(4)
After solving the equation (3)and (4) we can write
2
m𝑑 𝑥
= −𝑏 𝑑𝑥
−𝑘𝑥
𝑑𝑡2 𝑑𝑡
2
𝑑 𝑥 𝑑
𝑥
𝑑𝑡2 𝑑𝑡 0
2
+ 2𝛽 + 𝜔 𝑥 = 0 --------------------------------
Or,
(5)
0
Where b = 2𝑚𝛽 and 𝜔2 = 𝑘
𝑚
Eq(5) is a homogeneous, 2nd order differential equation.
The general solution of eq(5) for 𝛽 ≠ 𝜔𝑜 is
𝑥 = 𝑒−𝛽𝑡
1
𝐴 𝑒 2
0 0
𝛽2−𝜔2 𝑡 − 𝛽2−𝜔2 𝑡
+ 𝐴 𝑒 ------------------(6)

𝐴1 𝑎𝑛𝑑 𝐴2 are constants depend on the initial position
and velocity of the oscillator.
Depending on the values of 𝛽 and 𝜔𝑜, three types of motion
are possible.
• Such as
4. 𝛽2 =0, SHM
0
1. Under damped (𝜔2 > 𝛽2)
0
2. Over damped (𝜔2 < 𝛽2)
0
3. Critical damped (𝜔2 = 𝛽2 )

Case-1: Underdamped
−(𝜔𝑜
2 − 𝛽2) =𝑖𝜔
Condition: 𝛽2 < 𝜔𝑜
2
So, 𝛽2−𝜔𝑜
2 = −𝑣𝑒 , Hence 𝛽2−𝜔𝑜
2 =
Where 𝜔 = (𝜔𝑜
2 − 𝛽2)
Hence the solution becomes
𝑥 𝑡 = 𝑒−𝛽𝑡(𝐴1𝑒𝑖𝜔𝑡+A2𝑒−𝑖𝜔𝑡)
= 𝑒−𝛽𝑡[(𝐴1 cos𝜔𝑡 + 𝑖𝐴1 sin𝜔𝑡) + (𝐴2 cos𝜔𝑡 − 𝑖𝐴2 sin𝜔𝑡)]
= 𝑒−𝛽𝑡[(𝐴1 +𝐴2)cos 𝜔𝑡 + 𝑖(𝐴1 −𝐴2)sin 𝜔𝑡]
= 𝑒−𝛽𝑡[(𝐴 sin𝜑)cos 𝜔𝑡 + (𝐴 cos𝜑)sin 𝜔𝑡]
𝒙 𝒕 = 𝑨𝒆−𝜷𝒕 𝐬𝐢𝐧𝝎𝒕 + 𝝋
Or, 𝒙 𝒕 = 𝑨𝒆−𝜷𝒕 𝐬𝐢𝐧 ( 𝜔𝑜
2 − 𝛽2)𝒕 + 𝝋 ----------[7]

Case-1: Underdamped
Where 𝐴1 + 𝐴2 = 𝐴 sin𝜑 and i(𝐴1 − 𝐴2) = 𝐴 cos𝜑
Equation (7) represents damped harmonic oscillation with
amplitude 𝑨𝒆−𝜷𝒕 which decreases exponentially with time and
the time period of vibration is 𝑻 =
𝟐
𝝅
(𝝎𝒐
𝟐−𝜷𝟐
)
which is greater
than that in the absence of damping.
Example: Motion of Simple pendulum in air medium.

Decrement
• Decrement: The ratio between amplitudes of two successive
maxima.
Let A1, A2,A3 ---- are the amplitudes at time t=t, t+T, t+2T, ----
respectively where T is time period of damped oscillation.Then
𝐴1 = 𝐴𝑒−𝛽𝑡
𝐴2 = 𝐴𝑒−𝛽(𝑡+𝑇)
𝐴3 = 𝐴𝑒−𝛽(𝑡+2𝑇)
Hence decrement 𝑑 = 𝐴1
= 𝐴2
= 𝑒𝛽𝑇
𝐴2 𝐴3
Hence logarithmic decrement is given by
𝒆
𝝀 = log 𝑒𝛽𝑇 = 𝜷𝑻 =
𝟐 𝝅
𝜷
𝑜
(𝜔 2 − 𝛽2)

Case-2: OverDamped
• Condition: 𝛽2 > 𝜔𝑜
2
• 𝛽2 − 𝜔𝑜
2 = 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
• Let 𝛽2 − 𝜔𝑜
2 = 𝛼, so from eq(6), we have
𝑥 = 𝑒−𝛽𝑡 𝐴1𝑒𝛼𝑡 + 𝐴2𝑒 −𝛼 𝑡 = 𝐴1𝑒− 𝛽−𝛼 𝑡+ 𝐴2𝑒− 𝛽+𝛼 𝑡----------(8)
Since both the powers are
negative, the body once displaced
comes to the equilibrium position
slowly without performing
oscillations

Case-3: CRITICALDAMPING
Condition: 𝛽2 = 𝜔𝑜
2
Solution: 𝑥 𝑡 = 𝐶 + 𝐷𝑡 𝑒−𝛽𝑡
The motion is non oscillatory and the displacement approaches
zero asymptotically.
The rate of decrease of displacement
in this case is much faster than that of
over damped case.
•Example – suspension of spring
of automobile.
X(t)
t

Damped HarmonicOscillations
I
III
t
II
O
I: UNDER DAMPED
II: OVER DAMPED
III:CRITICAL DAMPED
X(t)
A
-A

Problems
1. What is the physical significance of damping coefficient? What is
its unit (2marks)
2. Give the graphical comparison among the following three types of
harmonic motion:
a) Under damped harmonic motion
b) Over damped harmonic motion
c) Critically damped harmonic motion
3. What is logarithmic decrement? Find the ratio of nth amplitude
with 1st amplitude in case of under damped oscillation.(2 mark)
4. The natural angular frequency of a simple harmonic oscillator of
mass 2gm is 0.8rad/sec. It undergoes critically damped motion
when taken to a viscous medium. Find the damping force on the
oscillator when its speed is 0.2cm/sec. (2marks)(Ans: 0.64dyne)

17
Small Oscillations– Energy of Bound System

Small Oscillationsin a Bound System
It is well known that any "well behaved" function f(x) can be expanded in a Taylor's series about a
point x0. Thus

Twisting Scissoring Rocking
Symmetric Stretching Asymmetric Stretching Wagging
The normal modes of vibration are: asymmetric, symmetric, wagging, twisting,
scissoring, and rocking for polyatomic molecules.
Molecular Vibrations
The degrees of vibrational modes for linear molecules can be calculated using the for mula: 3N−5.
The degrees of freedom for nonlinear molecules can be calculated using the formula: 3N−6

How many vibrational modes are there in the linear CO2 molecule ?
How many vibrational modes are there in the tetrahedral CH4 molecule ?
How many vibrational modes are there in the nonlinear C60 molecule ?

Molecular Vibrations
• A molecule has translational and rotational motion as a whole while each atom has it's own motion.
The vibrational modes can be IR or Raman active.
• For a mode to be observed in the IR spectrum, changes must occur in the permanent dipole (i.e. not
diatomic molecules). Diatomic molecules are observed in the Raman spectra but not in the IR
spectra. This is due to the fact that diatomic molecules have one band and no permanent dipole, and
therefore one single vibration. An example of this would be O2, or N2 .
• However, unsymmetric diatomic molecules (i.e. CN ) do absorb in the IR spectra. Polyatomic
molecules undergo more complex vibrations that can be summed or resolved into normal modes of
vibration.
Consider the two atoms connected by a spring of equilibrium length r0 and spring constant k, as shown
below. The equations of motion are

This vibrational motion, characteristic of all molecules, can be identified by the light the molecule
radiates. The vibrational frequencies typically lie in the near infrared (3 X 1013 Hz), and by
measuring the frequency we can find the value of d2U/dr2 at the potential energy minimum.
The harmonic oscillator approximation therefore has a wide
range of applicability, even down to internal motions in nuclei.
It is more natural to write the energies in terms of a variable other than linear displacement. For
instance, the energies of a pendulum are

More generally, the energies may have the form
Since the system is conservative, E is constant. Differentiating the energy equation with
respect to time gives
Hence q undergoes harmonic motion with frequency 𝐴/𝐵.

Problems: Energy and Stability & Small Oscillations—The Teeter Toy

 Einstein’s special theory of relativity deals with the physical
law as determined in two references frames moving with
constant velocity relative to each other.
 Event : An event is something that happens at a particular
point in space and at a particular instant of time, independent
of the reference frame. Which we may use to described it.
 A collision between two particles, an explosion of bomb or
star and a sudden flash of light are the examples of event.
 Observer : An observer is a person or equipment meant to
observe and take measurement about the event. The
observer is supposed to have with him scale, clock and
other needful things to observe that event.
Special Theory of Relativity

FRAME OF REFERENCE (INERTIAL AND NON-INERTIAL FRAMES OF REFERENCE):
Frame of Reference: sAny object can be located or any event can be described using a
coordinate system. This coordinate system is called the Frame of Reference.
Inertial Frame: An inertial frame is defined as a reference frame in which the law of
inertia holds true. i.e. Newton’s first law. Such a frame is also called un-accelerated frame.
e.g. a distant star can be selected as slandered inertial frame of reference.
Non-inertial Frame: It is defined a set of coordinates moving with acceleration relative to
some other frame in which the law of inertia does not hold true. It is an accelerated frame.
e.g., applications of brakes to a moving train makes it an accelerated (decelerated) frame.
So it becomes a non-inertial frame.

Satya, Moving Deva, and the spark
In Satya's frame of reference ...
In Deva's frame of reference ...

Satya, moving Deva, and TWO sparks

GALILEAN TRANSFORMATION EQUATIONS

Imagine that a train passes you and after 6 seconds you see a flash of light from the train. The
train is moving at 80 mph. How far away was the train when the light flashed? To make our
calculations easier we will convert miles per hour into meters per second: 80 mph is about 130
km/h, which is about 36 metres per second
For example, imagine a rod accelerating to a very high speed, say 90% of the speed of
light (0.9 c). We can label the ends of the rod (L1 and L2) and then apply the x-co-
ordinate transformation equation to measure its length. The diagram below shows what
Galileo's system predicts

PSOSTULATE EINSTEIN’S SPECIAL THEORY OF RELATIVITY
Einstein proposed the special theory of relativity in
1905. This theory deals with the problems of mechanics
in which one frame moves with constant velocity
relative to the other frame.
The two postulates of the Special Theory of Relativity
are :
1.The laws of physics are the same in all inertial
systems. No preferred inertial system exists.
2.The speed of light(c) in free space has the same value
in all the inertial systems.

Consequences of Lorentz transformations:
Now, let us discuss the consequences of Lorentz transformations
regarding the lengths of the bodies and time intervals between
given intervals.
Length Contraction:
Time Dilation:

Problem: Let consider a rod of length L moving with a velocity which is equal to 0.6c. Then its length
as measured from another frame is given by L0. Here we have v = 0.6 c.
Problem-Twin Paradox: Consider two twins A and B, each 20 years of age. Twin A remains at rest at the
origin O on the earth and twin B takes a round trip in space voyage to a star with velocity v=0.99c relative
to A. The star is 10 light years away from O. What are the ages of the A and B when B finishes his
journey.
Problem: Determine the length and the orientation of a rod of length 10 meters in a frame of reference
which is moving with 0.6c velocity in a direction making 30o angle with the rod.
Problem: if Vo is the rest volume of a cube of side lo, then find the volume viewed from a reference
frame moving with velocity of v in a direction parallel to an edge of cube.
Problem: A meson has a speed 0.8c relative to the ground. Find how fas the meson travels relative to the
ground. If its peed remains constant and the time of its flight, relative to the system, in which it is at rest,
is 2 x 108.
Problem: A beam of particles of half life 2 x 10-6 sect travels in the laboratory with speed 0.96 time the
speed of light. How much distance does the beam travel before the flux falls to ½ times the initial flux.?

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