MAHARASHTRA STATE BOARD
CLASS XI AND XII
CHAPTER 5
OSCILLATIONS
CONTENT
Introduction
Periodic and oscillatory
motions
Simple harmonic motion
Simple harmonic motion
and uniform circular
motion
Velocity and acceleration
in simple harmonic motion
Force law for simple
harmonic motion
Energy in simple harmonic
motion
Some systems executing
simple harmonic motion
Damped simple harmonic
motion
Forced oscillations and
resonance
This Unit is rely on introduction to Simple Harmonic Motion. the contents was prepared using the Curriculum of NTA level 4 at Mineral Resources Institute- Dodoma.
This Unit is rely on introduction to Simple Harmonic Motion. the contents was prepared using the Curriculum of NTA level 4 at Mineral Resources Institute- Dodoma.
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Gravitational field and potential, escape velocity, universal gravitational l...lovizabasharat
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Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples
Esta es una presentacion que hice con motivo de los requisitos que exige la maestria en fisica en la Unviersidad de Bishops, en Quebec, Canada. Durante mi presentacion, hicieron incapie en un error de subindices durante el desarrollo de las ecuaciones de las ondas gravitacionales. Lamentablemente no recuerdo en que diapositiva me marcaron el error, asi que es un desafio para cualquiera que encuentre mi presentacion interesante para ser utilizada en algun proyecto. Gracias.
MAHARASHTRA STATE BOARD
CLASS XI
PHYSICS
CHAPTER 1
UNITS AND MEASUREMENT
Introduction
The international system of
units
Measurement of length
Measurement of mass
Measurement of time
Accuracy, precision of
instruments and errors in
measurement
Significant figures
Dimensions of physical
quantities
Dimensional formulae and
dimensional equations
Dimensional analysis and its
applications
Chapter 2 - Mechanical Properties of Fluids.pptxPooja M
MARASHTRA STATE BOARD
CLASS XII
PHYSICS
MECHANICAL PROPERTIES OF FLUIDS
CONTENT
Density and pressure.
Buoyant force and Archimedes' principle.
Fluid dynamics.
Viscosity.
Surface tension.
Harmonic motion for college Harmonic motion for college Harmonic motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college Harmonic motion for college Harmonic motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college Harmonic motion for college Harmonic motion for college Harmonic motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college Harmonic motion for college Harmonic motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college Harmonic motion for college Harmonic motion for college Harmonic motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college Harmonic motion for college Harmonic motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs motion for college hajsjjsdjsdddjssksmsjsjjssnnsnsnnsnsnsnsnsnsnsnsnskkcjslfjkskdblandldndnsnsjjsjsjjsjsjjsjs
Gravitational field and potential, escape velocity, universal gravitational l...lovizabasharat
What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples
Esta es una presentacion que hice con motivo de los requisitos que exige la maestria en fisica en la Unviersidad de Bishops, en Quebec, Canada. Durante mi presentacion, hicieron incapie en un error de subindices durante el desarrollo de las ecuaciones de las ondas gravitacionales. Lamentablemente no recuerdo en que diapositiva me marcaron el error, asi que es un desafio para cualquiera que encuentre mi presentacion interesante para ser utilizada en algun proyecto. Gracias.
MAHARASHTRA STATE BOARD
CLASS XI
PHYSICS
CHAPTER 1
UNITS AND MEASUREMENT
Introduction
The international system of
units
Measurement of length
Measurement of mass
Measurement of time
Accuracy, precision of
instruments and errors in
measurement
Significant figures
Dimensions of physical
quantities
Dimensional formulae and
dimensional equations
Dimensional analysis and its
applications
Chapter 2 - Mechanical Properties of Fluids.pptxPooja M
MARASHTRA STATE BOARD
CLASS XII
PHYSICS
MECHANICAL PROPERTIES OF FLUIDS
CONTENT
Density and pressure.
Buoyant force and Archimedes' principle.
Fluid dynamics.
Viscosity.
Surface tension.
MAHARASHTRA STATE BOARD
CLASS XI AND XII
CHAPTER 4
THERMODYNAMICS
CONTENT
Introduction
Thermal equilibrium
Zeroth law of
Thermodynamics
Heat, internal energy and
work
First law of
thermodynamics
Specific heat capacity
Thermodynamic state
variables and equation of
state
Thermodynamic processes
Heat engines
Refrigerators and heat
pumps
Second law of
thermodynamics
Reversible and irreversible
processes
Carnot engine
MAHARASHTRA STATE BOARD
CLASS XI and XII
CHAPTER 6
SUPERPOSITION OF WAVES
CONTENT:
Introduction
Transverse and
longitudinal waves
Displacement relation in a
progressive wave
The speed of a travelling
wave
The principle of
superposition of waves
Reflection of waves
Beats
Doppler effect
MAHARASHTRA STATE BOARD
CLASS XI AND XII
PHYSICS
CHAPTER 7
WAVE OPTICS
CONTENT:
Huygen's principle.
Huygen's principles & proof of laws of reflection/refraction.
Condition for construction & destruction of coherent waves.
Young's double slit experiment.
Modified Young's double slit experiment.
Intensity of light in Y.D.S.E.
Diffraction due to single slit.
Polarisation & doppler effect.
MAHARASHTRA STATE BOARD
CLASS XI AND XII
PHYSICS
CHAPTER 8
ELECTROSTATICS
Introduction.
Coulomb's law
Calculating the value of an electric field
Superposition principle
Electric potential
Deriving electric field from potential
Capacitance
Principle of the capacitor
Dielectrics
Polarization, and electric dipole moment
Applications of capacitors.
MAHARASHTRA STATE BOARD
CLASS XI AND XII
CHAPTER 9
CURRENT ELECTRICTY
CONTENT
Electric Cell and its Internal resistance
Potential difference and emf of a cell
Combination of cells in series and in parallel
Kirchhoff's laws and their applications
Wheatstone bridge
Metre bridge
Potentiometer – principle and its applications
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
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Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
1. Std : 12th Year : 2022-23
Subject : PHYSICS
Chapter 5: OSCILLATIONS
CLASSXII
MAHARASHTRA STATE BOARD
2. Can you recall?
1. What do you mean by linear motion and angular
motion?
2. Can you give some practical examples of oscillations in
our daily life?
3. What do you know about restoring force?
4. All musical instruments make use of oscillations, can
you identify, where?
5. Why does a ball floating on water bobs up and down,
if pushed down and released?
3. MOTION
“Motion is the phenomenon in which an object changes its
position over time.”
It is described by
• displacement, distance
• velocity, speed
• acceleration
• time
Types of
motion
Rotational
motion
Oscillatory
motion
Linear
motion
Reciprocating
4. “A force acting opposite to displacement
to bring the system back to equilibrium
i.e. at rest position.”
RESTORING FORCE
Periodic motion
“Any motion which repeats itself after a
definite interval of time is called periodic
motion.”
5. OSCILLATION
“Oscillation is defined as the process of repeating vibrations of
any quantity or measure about its equilibrium value in time.”
Or
“Oscillation refers to any periodic motion at a distance about the
equilibrium position and repeat itself and over for a period of
time.”
Oscillation is periodic motion
Displacement, acceleration and velocity for oscillatory motion
can be defined by Harmonic function.
• Sine
• Cosine
6. Linear simple harmonic motion (s.h.m.)
When we pull block right side from mean
position the spring will pull object toward itself
i.e. force produced by spring is opposite.
f ∝ −𝒙
f = - k x
f = m a
∴ 𝒂 =
𝒇
𝒎
Linear S.H.M. is defined as the linear periodic motion of a body,
in which force (or acceleration) is always directed towards the
mean position and its magnitude is proportional to the
displacement from the mean position.
7.
8. A complete oscillation is when the object goes from one extreme to
other and back to the initial position.
The conditions required for simple harmonic motion are:
1. Oscillation of the particle is about a fixed point.
2. The net force or acceleration is always directed towards the fixed
point.
3. The particle comes back to the fixed point due to restoring force.
Harmonic oscillation is that oscillation which can be expressed in terms
of a single harmonic function, such as x = a sin wt or x = a cos wt
Non-harmonic oscillation is that oscillation which cannot be expressed
in terms of single harmonic function.
It may be a combination of two or more harmonic oscillations such as x
= a sin ωt + b sin 2ωt , etc.
9. Differential Equation of S.H.M.
Consider, f → Restoring force, x → Displacement done by the block.
f = - k x ……….(i)
According to newtons second law of motion, f = m a
∴ m a = - k x ……….(ii)
Also, velocity → Rate of change of displacement
∴ v =
𝒅𝒙
𝒅𝒕
Acceleration → Rate of change of velocity
∴ a =
𝒅𝒗
𝒅𝒕
=
𝒅
𝒅𝒙
𝒅𝒕
𝒅𝒕
a =
𝒅𝟐𝒙
𝒅𝒕𝟐
m x
𝒅𝟐𝒙
𝒅𝒕𝟐 = - k x
∴ m
𝒅𝟐𝒙
𝒅𝒕𝟐 + k x = 0
i.e.
𝒅𝟐𝒙
𝒅𝒕𝟐 + 𝒘𝟐 x = 0, Where,
𝒌
𝒎
= 𝝎𝟐
10. Example
A body of mass 0.2 kg performs linear S.H.M. It experiences a
restoring force of 0.2 N when its displacement from the mean
position is 4 cm.
Determine (i) force constant (ii) period of S.H.M. and (iii) acceleration
of the body when its displacement from the mean position is 1 cm.
Solution:
(i) Force constant, k = f / x = (0.2)/ 0.04 = 5 N/m
(ii) Period T = 2𝝅
𝒎
𝒌
= 2𝝅
𝟎.𝟐
𝟓
= 0.4 𝝅 s
(iii)Acceleration
a = - 𝝎𝟐
x = −
𝒌
𝒎
𝒙 = −
𝟓
𝟎.𝟐
× 0.04 = - 1 m 𝒔−𝟐
11. TERMS FOR S.H.M.
𝒅𝟐𝒙
𝒅𝒕𝟐 + 𝒘𝟐
x = 0
𝒅𝟐𝒙
𝒅𝒕𝟐 = - 𝒘𝟐 x
a = − 𝒘𝟐 x
For velocity,
𝒅𝟐𝒙
𝒅𝒕𝟐 + 𝒘𝟐
x = 0
𝒅𝟐𝒙
𝒅𝒕𝟐 = - 𝒘𝟐 x
𝒅𝒗
𝒅𝒕
= - 𝒘𝟐 x
∴
𝒅𝒗
𝒅𝒙
.
𝒅𝒙
𝒅𝒕
= - 𝒘𝟐
x
∴
𝒅𝒗
𝒅𝒙
. v = - 𝒘𝟐
x
Integrating both side,
𝒗 𝒅𝒗 = − 𝝎𝟐 𝒙 𝒅𝒙
∴
𝒗𝟐
𝟐
= −
𝝎𝟐𝒙𝟐
𝟐
+ 𝒄 …………..(i)
12. Now, if object is at extreme position
x = A, v = 0
∴ C =
𝝎𝟐𝑨
𝟐
From equation (i)
∴ 𝒗𝟐
= − 𝝎𝟐
𝒙𝟐
+ 𝝎𝟐
𝑨𝟐
v = ± 𝝎 𝑨𝟐 − 𝒙𝟐
For displacement,
We know that, v =
𝒅𝒙
𝒅𝒕
v = 𝝎 𝑨𝟐 − 𝒙𝟐
∴
𝒅𝒙
𝒅𝒕
= 𝝎 𝑨𝟐 − 𝒙𝟐
∴ 𝒙 = 𝑨 𝒔𝒊𝒏 (𝝎𝒕 + 𝝓)
13. Case (i) If the particle starts S.H.M. from the mean position, x = 0 at t = 0
𝒙 = 𝑨 𝒔𝒊𝒏 𝝎𝒕 + 𝝓
0= 𝑨 𝒔𝒊𝒏 (𝝎 × 𝟎 + 𝝓)
𝝓 = 𝟎 𝒐𝒓 𝝅
i.e. x = ±𝑨 𝒔𝒊𝒏 𝒘𝒕
Case (ii) If the particle starts S.H.M. from the extreme position, x = ± A at t = 0
𝒙 = 𝑨 𝒔𝒊𝒏 (𝝎𝒕 + 𝝓)
A= 𝑨 𝒔𝒊𝒏 (𝝎 × 𝟎 + 𝝓)
1 = 𝒔𝒊𝒏 𝝓
𝝓 =
𝝅
𝟐
or
𝟑𝝅
𝟐
𝒙 = ± 𝑨 𝒄𝒐𝒔 𝝎𝒕
14. Expressions of displacement (x),velocity(v) and acceleration(a) at timet
𝒙 = 𝑨 𝒔𝒊𝒏 𝝎𝒕 + 𝝓
∴ 𝒗 =
𝒅𝒙
𝒅𝒕
=
𝒅 [𝑨 𝐬𝐢𝐧 𝝎𝒕 + 𝝓 ]
𝒅𝒕
v = 𝑨 𝒄𝒐𝒔 (𝝎𝒕 + 𝝓).(w+0)
v = Aw cos (𝝎𝒕 + 𝝓)
∴ 𝒂 =
𝒅𝒗
𝒅𝒕
=
𝒅 [𝑨𝒘 𝐜𝐨𝐬 𝝎𝒕 + 𝝓 ]
𝒅𝒕
a = −𝑨𝒘𝟐
𝐬𝐢𝐧 𝝎𝒕 + 𝝓
Extreme values of displacement (x), velocity(v) and acceleration(a):
1) Displacement: 𝒙 = 𝑨 𝒔𝒊𝒏 (𝝎𝒕 + 𝝓)
At mean position, (𝝎𝒕 + 𝝓) = 0 or 𝝅
∴ 𝒙𝒎𝒊𝒏 = 𝟎
At extreme position, (𝝎𝒕 + 𝝓) =
𝝅
𝟐
𝒐𝒓
𝟑𝝅
𝟐
∴ 𝒙 = ±𝑨 𝒔𝒊𝒏
𝝅
𝟐
∴ 𝒙𝒎𝒂𝒙 = ±𝑨
15. 2) Velocity : v = ± 𝝎 𝑨𝟐 − 𝒙𝟐
At mean position, 𝒙 = 0
∴ 𝒗𝒎𝒂𝒙 = ±𝑨𝝎
At extreme position, 𝒙 = ±𝑨
∴ 𝒙𝒎𝒊𝒏 = 𝟎
3) Acceleration: a = 𝝎𝟐
𝒙
At mean position, 𝒙 = 0
∴ 𝒂𝒎𝒊𝒏 = 𝟎
At extreme position, 𝒙 = ±𝑨
∴ 𝒂𝒎𝒂𝒙 = ∓𝝎𝟐𝑨
Amplitude
The maximum displacement of a particle performing S.H.M. from its mean position is
called the amplitude of S.H.M.
𝒙 = 𝑨 𝒔𝒊𝒏 (𝝎𝒕 + 𝝓)
For maximum displacement 𝒔𝒊𝒏 (𝝎𝒕 + 𝝓) = ±1
i.e. x = ±𝐀
16.
17. Period of S.H.M.
The time taken by the particle performing S.H.M. to complete one oscillation is
called the period of S.H.M.
Displacement of the particle at time t,
𝒙 = 𝑨 𝒔𝒊𝒏 𝝎𝒕 + 𝝓
After some time,
𝒙 = 𝑨 𝒔𝒊𝒏 [𝝎 𝒕 +
𝟐𝝅
𝝎
+ 𝝓]
𝒙 = 𝑨 𝒔𝒊𝒏 𝝎𝒕 + 𝟐𝝅 + 𝝓
𝒙 = 𝑨 𝒔𝒊𝒏 𝝎𝒕 + 𝝓
Where
𝟐𝝅
𝝎
= 𝑻
k = m 𝝎𝟐
∴ 𝝎𝟐 =
𝒌
𝒎
18. ∴ 𝝎𝟐 =
𝑭𝒐𝒓𝒄𝒆 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕
𝒎𝒂𝒔𝒔
∴ 𝝎𝟐
=
𝒂
𝒙
Now, T =
𝟐𝝅
𝒂𝒄𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕
T = 2 𝝅 X
𝒎
𝒌
Frequencyof S.H.M.
The number of oscillations performed by a particle performing S.H.M. per unit time is called the
frequency of S.H.M.
n =
𝟏
𝑻
=
𝝎
𝟐𝝅
=
𝟏
𝟐𝝅
𝒌
𝒎
PHASEIN S.H.M.
Phase in S.H.M. is basically the state of oscillation.
Requirements to know the state of oscillation
- Position of particle (displacement)
- Direction of velocity
- Oscillation number
19.
20.
21.
22. PHASEIN S.H.M.
Phase in S.H.M. is basically the state of oscillation.
Requirements to know the state of oscillation
- Position of particle (displacement)
- Direction of velocity
- Oscillation number
Commonly, 𝜽 = 𝝎𝒕 + 𝝓
Expressions of displacement (x), velocity (v) and acceleration(a) at time t
𝒙 = 𝑨 𝒔𝒊𝒏 (𝝎𝒕 + 𝝓)
v =
𝒅𝒙
𝒅𝒕
= 𝑨𝝎 𝒄𝒐𝒔 (𝝎𝒕 + 𝝓)
a =
𝒅𝒗
𝒅𝒕
= 𝑨𝝎𝟐
𝒔𝒊𝒏 (𝝎𝒕 + 𝝓)
24. 𝑪𝑶𝑴𝑷𝑶𝑺𝑰𝑻𝑰𝑶𝑵 𝑶𝑭 𝑻𝑾𝑶 𝑺. 𝑯. 𝑴
Consider, two S.H.M having same period and along same path.
𝒙𝟏 𝒂𝒏𝒅 𝒙𝟐 are displacements of both S.H.M.
Composition of two S.H.M.
𝒙 = 𝒙𝟏 + 𝒙𝟐
x = 𝑨𝟏 𝒔𝒊𝒏 𝝎𝒕 + 𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝎𝒕 + 𝝓𝟐
x = 𝑨𝟏 𝒔𝒊𝒏 𝝎𝒕. 𝒄𝒐𝒔𝝓𝟏 + 𝒄𝒐𝒔 𝝎𝒕. 𝒔𝒊𝒏𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝎𝒕. 𝒄𝒐𝒔𝝓𝟐 + 𝒄𝒐𝒔 𝝎𝒕. 𝒔𝒊𝒏𝝓𝟐
x = 𝑨𝟏𝒔𝒊𝒏 𝝎𝒕. 𝒄𝒐𝒔𝝓𝟏 + 𝑨𝟏𝒄𝒐𝒔 𝝎𝒕. 𝒔𝒊𝒏𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝎𝒕. 𝒄𝒐𝒔𝝓𝟐 + 𝑨𝟐 𝒄𝒐𝒔 𝝎𝒕. 𝒔𝒊𝒏𝝓𝟐
x = (𝑨𝟏𝒄𝒐𝒔𝝓𝟏 + 𝑨𝟐 𝒄𝒐𝒔𝝓𝟐) 𝒔𝒊𝒏 𝝎𝒕 + (𝑨𝟏𝒔𝒊𝒏 𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝓𝟐) 𝐜𝐨𝐬 𝝎𝒕 ……………….(i)
R cos 𝜹 = 𝑨𝟏𝒄𝒐𝒔𝝓𝟏 + 𝑨𝟐 𝒄𝒐𝒔𝝓𝟐 ………….(ii)
R sin 𝜹 = 𝑨𝟏𝒔𝒊𝒏 𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝓𝟐 ………….(iii)
x = R cos 𝜹 . sin 𝝎𝒕 + 𝑹 𝒔𝒊𝒏 𝜹 . 𝒄𝒐𝒔 𝝎𝒕
x = R [cos 𝜹 . sin 𝝎𝒕 + 𝒔𝒊𝒏 𝜹 . 𝒄𝒐𝒔 𝝎𝒕]
x = R sin (𝝎𝒕 + 𝜹)
25. RESULTANT AMPLITUDE
R = (𝑹 𝒔𝒊𝒏 𝜹)𝟐+(𝑹 𝒄𝒐𝒔 𝜹)𝟐
From equation (ii) and (iii)
𝑹𝟐
= 𝑨𝟏
𝟐
+ 𝑨𝟐
𝟐
+ 𝟐𝑨𝟏𝑨𝟐𝒄𝒐𝒔(𝝓𝟏 − 𝝓𝟐)
R = 𝑨𝟏
𝟐
+ 𝑨𝟐
𝟐
+ 𝟐𝑨𝟏𝑨𝟐𝒄𝒐𝒔(𝝓𝟏 − 𝝓𝟐)
SPECIAL CASES
(i) If the two S.H.M are in phase,
(𝝓𝟏 − 𝝓𝟐) = 𝟎𝟎, ∴ 𝒄𝒐𝒔 (𝝓𝟏 − 𝝓𝟐) = 1
∴ R = 𝑨𝟏
𝟐
+ 𝑨𝟐
𝟐
+ 𝟐𝑨𝟏𝑨𝟐 = ±(𝑨𝟏 + 𝑨𝟐)
If, 𝑨𝟏 = 𝑨𝟐 = 𝑨, 𝒘𝒆 𝒈𝒆𝒕 𝑹 = 𝟐𝑨
(ii) If the two S.H.M.s are 𝟗𝟎𝟎
out of phase,
(𝝓𝟏 − 𝝓𝟐) = 𝟗𝟎𝟎
, ∴ 𝒄𝒐𝒔 (𝝓𝟏 − 𝝓𝟐) = 0
∴ R = 𝑨𝟏
𝟐
+ 𝑨𝟐
𝟐
If, 𝑨𝟏 = 𝑨𝟐 = 𝑨, 𝒘𝒆 𝒈𝒆𝒕 𝑹 = 𝟐𝑨
26. SPECIAL CASES
(iii) If the two S.H.M.s are 𝟏𝟖𝟎𝟎 out of phase,
(𝝓𝟏 − 𝝓𝟐) = 𝟏𝟖𝟎𝟎, ∴ 𝒄𝒐𝒔 (𝝓𝟏 − 𝝓𝟐) = -1
∴ R = 𝑨𝟏
𝟐
+ 𝑨𝟐
𝟐
+ 𝟐𝑨𝟏𝑨𝟐
∴ 𝑹 = I 𝑨𝟏 − 𝑨𝟐I
If, 𝑨𝟏 = 𝑨𝟐 = 𝑨, 𝒘𝒆 𝒈𝒆𝒕 𝑹 = 𝟎
Initial Phase (𝜹)
Dividing equation (ii) and (iii)
𝑹 𝒔𝒊𝒏 𝜹
𝑹 𝒄𝒐𝒔 𝜹
=
𝑨𝟏𝒔𝒊𝒏 𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝓𝟐
𝑨𝟏𝒄𝒐𝒔𝝓𝟏 + 𝑨𝟐 𝒄𝒐𝒔𝝓𝟐
∴ 𝒕𝒂𝒏 𝜹 =
𝑨𝟏𝒔𝒊𝒏 𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝓𝟐
𝑨𝟏𝒄𝒐𝒔𝝓𝟏 + 𝑨𝟐 𝒄𝒐𝒔𝝓𝟐
∴ 𝜹 = 𝒕𝒂𝒏−𝟏
𝑨𝟏𝒔𝒊𝒏 𝝓𝟏 + 𝑨𝟐 𝒔𝒊𝒏 𝝓𝟐
𝑨𝟏𝒄𝒐𝒔𝝓𝟏 + 𝑨𝟐 𝒄𝒐𝒔𝝓𝟐
27. Energy of a Particle
Fig.: Energy in an S.H.M.
When particle performing S.H.M. then it passes both kinetic and potential energy.
Velocity of particle performing S.H.M
𝑣 = 𝜔 𝐴2 − 𝑥2 = A𝜔 cos 𝜔𝑡 + 𝜙
Kinetic Energy: 𝐸𝐾 =
1
2
𝑚𝑣2
=
1
2
𝑚[𝜔2
𝐴2
− 𝑥2
]
𝐸𝐾 =
1
2
𝑘 𝐴2
− 𝑥2
………(i)
Displacement x
𝐸𝐾 =
1
2
𝑚𝑣2
=
1
2
𝑚 [A𝜔 cos 𝜔𝑡 + 𝜙 ]2
𝐸𝐾 =
1
2
𝑘𝐴2𝑐𝑜𝑠2 𝜔𝑡 + 𝜙 ………….(ii)
External work done (dw)
dW = f (-dx)
dW = -kx (-dx)
dW = kx dx
28. Total work done on the particle,
W = 0
𝑥
𝑑𝑊 = 0
𝑥
𝑘𝑥
W =
1
2
𝑘 𝑥2
𝐸𝑃 =
1
2
𝑘 𝑥2 =
1
2
𝑚 𝜔2𝑥2
𝐸𝑃 =
1
2
𝑚 𝐴2𝜔2𝑐𝑜𝑠2(𝜔𝑡 + 𝜙) ………….(iii)
Total energy = 𝐸𝑘 + 𝐸𝑝
E=
1
2
𝑚𝜔2 𝐴2 − 𝑥2 +
1
2
𝑚 𝜔2𝑥2
E=
1
2
𝑚𝜔2 𝐴2 − 𝑥2 + 𝑥2
E=
1
2
𝑚𝜔2𝐴2
E=
1
2
𝑘𝐴2
For frequency, 𝜔 = 2 𝜋 𝑛
E=
1
2
𝑚𝜔2𝐴2 =
1
2
𝑚 (2𝜋𝑛)2𝐴2
E =
2 𝜋2 𝑚 𝐴2
𝑇2
29. Simple Pendulum
An ideal simple pendulum is a heavy particle suspended by a massless,
inextensible, flexible string from a rigid support.
A practical simple pendulum is a small heavy (dense) sphere (called bob)
suspended by a light and inextensible string from a rigid support.
In the displaced position (extreme position),
two forces are acting on the bob.
(i) Force T' due to tension in the string,
directed along the string, towards the
support and
(ii) Weight mg, in the vertically downward
direction.
30. Simple Pendulum
At the extreme positions, there should not be any net force along the
string. The component of mg can only balance the force due to tension.
Thus, weight mg is resolved into two components;
(i) The component mg cos θ along the string, which is balanced by the
tension T ' and
(ii) The component mg sin θ perpendicular to the
string is the restoring force acting on mass m
tending to return it to the equilibrium position.
∴ Restoring force, F = - mg sin θ
As θ is very small (θ < 10°), sin θ ≅ 𝜃𝑐
∴ 𝐹 ≅ −𝑚 𝑔 𝜃
Small angle, 𝜃 =
𝑥
𝐿
∴ 𝐹 = - m g
𝑥
𝐿
31. Simple Pendulum
∴ 𝐹 = - m g
𝑥
𝐿
∴ 𝐹 ∝ −x
∴ 𝑚 𝑎 = −𝑚 𝑔
𝑥
𝐿
∴
𝑎
𝑥
= −
𝑔
𝐿
For time period, T =
2 𝜋
𝜔
T =
2 𝜋
𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
T =
2 𝜋
𝑔
𝐿
= 2 𝜋 ×
𝐿
𝑔
For frequency, n =
1
𝑇
=
1
2 𝜋
×
𝑔
𝐿
32. Second’s Pendulum
A simple pendulum whose period is two seconds is called second’s pendulum.
T = 2 𝝅
𝑳
𝒈
For a second s pendulum, 2 = 2 𝝅
𝑳𝑺
𝒈
Where, 𝑳𝑺 𝒊𝒔 𝒕𝒉𝒆 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒔𝒆𝒄𝒐𝒏𝒅′𝒔 𝒑𝒆𝒏𝒅𝒖𝒍𝒖𝒎, 𝒉𝒂𝒗𝒊𝒏𝒈 𝒑𝒆𝒓𝒊𝒐𝒅 𝑻 = 𝟐 𝒔
𝑳𝑺 =
𝒈
𝝅𝟐
33. Angular S.H.M. and its Differential Equation
Thus, for the angular S.H.M. of a body, the restoring torque acting upon it, for angular
displacement θ, is
𝝉 ∝ − 𝜽 𝒐𝒓 𝝉 = −𝒄 𝜽 ……….(i)
The constant of proportionality c is the restoring torque per unit angular displacement.
𝝉 = 𝑰 𝜶
Where, 𝜶 𝒊𝒔 𝒕𝒉𝒆 𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒂𝒄𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏
𝑰 𝜶 = − 𝒄 𝜽
∴ 𝑰
𝒅𝟐𝜽
𝒅𝒕𝟐
+ 𝒄 𝜽 = 𝟎
𝜶 =
𝒅𝟐𝜽
𝒅𝒕𝟐 = −
𝒄 𝜽
𝑰
Since c and I are constants, the angular acceleration α is directly
proportional to θ and its direction is opposite to that of the
angular displacement.
Hence, this oscillatory motion is called angular S.H.M.
34. Angular S.H.M. is defined as the oscillatory motion of a body in which the
torque for angular acceleration is directly proportional to the angular
displacement and its direction is opposite to that of angular displacement.
The time period T of angular S.H.M. is given by,
T =
𝟐 𝝅
𝝎
T =
𝟐 𝝅
𝑨𝒏𝒈𝒖𝒍𝒂𝒓 𝒂𝒄𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕
Magnet Vibrating in UniformMagnetic Field
If a bar magnet is freely suspended in the plane of a uniform
magnetic field.
Consider, μ be the magnetic dipole moment and
B the magnetic field.
35. The magnitude of this torque is, 𝝉 = 𝝁 𝑩 𝐬𝐢𝐧 𝜽
If θ is small, 𝐬𝐢𝐧 𝜽 ≅ 𝜽
𝝉 = 𝝁 𝑩 𝜽
Here, restoring torque is in anticlockwise
𝝉 = 𝑰 𝜶 = − 𝝁 𝑩 𝜽
Where, I – Moment of inertia
∴ 𝜶 = −
𝝁 𝑩
𝑰
𝜽
∴
𝜶
𝜽
= −
𝝁 𝑩
𝑰
T =
𝟐 𝝅
𝑨𝒏𝒈𝒖𝒍𝒂𝒓 𝒂𝒄𝒄𝒆𝒍𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕
=
𝟐 𝝅
𝜶
𝜽
T =
𝟐 𝝅
𝝁 𝑩
𝑰
𝑻 = 𝟐 𝝅
𝑰
𝝁 𝑩