1. Oscillatory motion is a periodic motion that repeats itself after a definite time interval called the period. Linear simple harmonic motion (S.H.M.) is the linear periodic motion of a body where the restoring force is directly proportional to the displacement from the mean position.
2. The differential equation of S.H.M is the second derivative of displacement (x) plus the angular frequency (ω) squared times x equals zero. The expressions for displacement, velocity, and acceleration of a particle in S.H.M involve sinusoidal and cosine functions of angular frequency times time.
3. The amplitude is the maximum displacement, the period is the time for one oscillation, and the frequency is
this is class 12 Maharashtra board physics subject content. this is complete content with notes with easily explaination.
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this is class 12 Maharashtra board physics subject content. this is complete content with notes with easily explaination.
for buying or neet attractive ppt in any subject contact me 8879919898. go to my site akchem.tk
blog akchem.blogspot.com
Gravitation has been the most common phenomenon in our lives but somewhere down the line we don't know musch about it. So here is a presentation whic will help you out to know what it is !! I'll be makin it available for download once i submit it in school :P :P ! Coz last one of the brats showed the same presentation that i uploade and unfortunatele his roll number fell before mine ! I was damned..:D :D :P
This ppt is as per class 12 Maharashtra State Board's new syllabus w.e.f. 2020. Images are taken from Google public sources and Maharashtra state board textbook of physics. Gif(videos) from Giphy.com. Only intention behind uploading these ppts is to help state board's class 12 students understand physics concepts.
This ppt is as per class 12 Maharashtra State Board's new syllabus w.e.f. 2020. Images are taken from Google public sources and Maharashtra state board textbook of physics. Gif(videos) from Giphy.com. Only intention behind uploading these ppts is to help state board's class 12 students understand physics concepts.
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
Gravitation has been the most common phenomenon in our lives but somewhere down the line we don't know musch about it. So here is a presentation whic will help you out to know what it is !! I'll be makin it available for download once i submit it in school :P :P ! Coz last one of the brats showed the same presentation that i uploade and unfortunatele his roll number fell before mine ! I was damned..:D :D :P
This ppt is as per class 12 Maharashtra State Board's new syllabus w.e.f. 2020. Images are taken from Google public sources and Maharashtra state board textbook of physics. Gif(videos) from Giphy.com. Only intention behind uploading these ppts is to help state board's class 12 students understand physics concepts.
This ppt is as per class 12 Maharashtra State Board's new syllabus w.e.f. 2020. Images are taken from Google public sources and Maharashtra state board textbook of physics. Gif(videos) from Giphy.com. Only intention behind uploading these ppts is to help state board's class 12 students understand physics concepts.
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.
Schrodinger wave equation and its application
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Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
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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 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.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
2. 2
Introduction
Oscillatory motion is a periodic motion.
Explanation of Periodic Motion
Any motion which repeats itself after a definite interval of time is called periodic motion. The
time taken for one such set of movements is called its period or periodic time. Circular
motion is periodic but it is not oscillatory.
The smallest interval of time after which the to and fro motion is repeated is called its period
(T) and the number of oscillations completed per unit time is called the frequency (n) of the
periodic motion.
Linear Simple Harmonic Motion (S.H.M.)
The block will begin its to and fro motion on either side of its equilibrium position. This motion
is linear simple harmonic motion.
If x is the displacement, the restoring force f is, f = - k x
where, k is a constant that depends upon the elastic properties of the spring. It is called the
force constant. The acceleration (a) is, a =
𝑓
𝑚
= − (
𝑘
𝑚
)𝑥
Linear S.H.M. is defined as the linear periodic motion of a body, in which force is always
directed towards the mean position and its magnitude is proportional to the displacement
from the mean position.
Differential Equation of S.H.M.
f = - k x
According to Newton’s second law of motion,
f = ma
∴ 𝑚𝑎 = −𝑘 𝑥
The velocity of the particle is, v =
𝑑𝑥
𝑑𝑡
and its acceleration, a =
𝑑𝑣
𝑑𝑡
=
𝑑2𝑥
𝑑𝑡2
𝑚
𝑑2𝑥
𝑑𝑡2 = - k x
∴
𝑑2𝑥
𝑑𝑡2 +
𝑘
𝑚
𝑥 = 0
3. 3
Substituting
𝑘
𝑚
= 𝑤2
, where ω is the angular frequency,
𝑑2𝑥
𝑑𝑡2 + 𝑤2
𝑥 = 0
Acceleration (a), Velocity (v) and Displacement (x) of S.H.M.
𝑑2𝑥
𝑑𝑡2 + 𝑤2
𝑥 = 0
𝑑2𝑥
𝑑𝑡2 = - 𝑤2
𝑥
a = - 𝑤2
𝑥
This is the expression for acceleration in terms of displacement x.
𝑑2𝑥
𝑑𝑡2 = - 𝑤2
𝑥
∴
𝑑
𝑑𝑡
(
𝑑𝑥
𝑑𝑡
) = - 𝑤2
𝑥
∴
𝑑𝑣
𝑑𝑡
= - 𝑤2
𝑥
∴
𝑑𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑡
= - 𝑤2
𝑥
∴ 𝑣
𝑑𝑣
𝑑𝑥
= - 𝑤2
𝑥
∴ v dv = - 𝑤2
𝑥 𝑑𝑥
Integrating both the sides,
∫ 𝑣 𝑑𝑣 = −𝑤2 ∫ 𝑥 𝑑𝑥
∴
𝑣2
2
= −
𝑤2
𝑥2
2
+ 𝐶
where C is the constant of integration.
Let A be the maximum displacement (amplitude) of the particle in S.H.M.
When the particle is at the extreme position, velocity (v) is zero.
Thus, at x = ±𝐴, 𝑣 = 0
0 = −
𝑤2
𝐴2
2
+ 𝐶
∴ 𝐶 = +
𝑤2
𝐴2
2
Using C in Equation,
Substituting v =
𝑑𝑥
𝑑𝑡
Here φ is the constant of integration.
This is the general expression for the displacement (x) of a particle performing linear S.H.M.
at time t.
4. 4
Case (i) If the particle starts S.H.M. from the mean position, x = 0 at t = 0
φ = 𝑠𝑖𝑛−1 (
𝑥
𝐴
) = 0 𝑜𝑟 𝜋
x = ± 𝐴 si n (𝑤𝑡)
Case (ii) If the particle starts S.H.M. from the extreme position, x = ±𝐴 at t = 0
Expressions of displacement (x), velocity (v) and acceleration (a) at time t
Extreme values of displacement (x), velocity (v) and acceleration (a)
1) Displacement: The general expression for displacement x in S.H.M. is, x = A sin (𝜔𝑡 + 𝜙)
At the mean position, (𝜔𝑡 + 𝜙) = 0 or 𝜋.
Thus, at the mean position, the displacement of the particle performing S.H.M. is minimum
(i.e., zero).
(wt + Φ) =
𝜋
2
0r
3𝜋
2
Thus, at the extreme position the displacement of the particle performing S.H.M. is
maximum.
2) Velocity: According to Equation of the magnitude of velocity of the particle performing
S.H.M. is, v = ± 𝜔 √𝐴2 − 𝑥2.
At the mean position, x = 0. ∴ 𝑉
𝑚𝑎𝑥 = ±𝐴 𝜔.
Thus, the velocity of the particle in S.H.M. is maximum at the mean position.
At the extreme position, x = ± 𝐴. ∴ 𝑉𝑚𝑖𝑛 = 0.
Thus, the velocity of the particle in S.H.M. is minimum at the extreme positions.
3) Acceleration: The magnitude of the acceleration of the particle in S.H.M is 𝑤2
𝑥.
At the mean position x = 0, so that the acceleration is minimum. ∴ 𝑎𝑚𝑖𝑛 = 0.
At the extreme positions x = ± 𝐴,
so that the acceleration is maximum, 𝑎𝑚𝑎𝑥 = ∓ 𝜔2
𝐴.
Amplitude(A), Period(T) and Frequency (n) of S.H.M.
Amplitude of S.H.M.
Fig. S.H.M. of a particle.
The displacement of the particle is,
The particle will have its maximum displacement when
i.e., when x = ± 𝐴. This distance A is called the amplitude of S.H.M.
The maximum displacement of a particle performing S.H.M. from its mean position is called
the amplitude of S.H.M.
5. 5
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 is,
After a time, t = (𝑡 +
2𝜋
𝑤
) the displacement will be
𝜔2
=
𝑘
𝑚
=
𝑓𝑜𝑟𝑐𝑒 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑚𝑎𝑠𝑠
= 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
∴ 𝑇 =
2 𝜋
√𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
Also, 𝑇 = 2 𝜋 √
𝑚
𝑘
Frequency of 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.
In time T, the particle performs one oscillation. Hence in unit time it performs
1
𝑇
oscillations.
Hence, frequency n of S.H.M. is,
𝑛 =
1
𝑇
=
𝜔
2 𝜋
=
1
2 𝜋
√
𝑘
𝑚
Reference Circle Method
A rod rotating along a vertical circle in the x-y plane.
Fig: Projection of a rotating rod.
Its angular positions are decided with the reference OX.
It means, at F it is 900
=
𝜋
2
𝑐
, at G it is 1800
= 𝜋𝑐
, and so on.
Let 𝑟
⃗ = OP be the position vector of this particle.
Fig: S.H.M. as projection of a U.C.M.
Projection of displacement: The position vector OM = OP sin 𝜃 = y = r sin (wt + 𝜙)
6. 6
Fig: Projection of velocity.
Projection of velocity: 𝑉
𝑦 = 𝑟 𝜔 cos 𝜃 = 𝑟 𝜔 cos(𝜔𝑡 + 𝜙)
Projection of acceleration: Its projection on the reference diameter will be
The projection of a U.C.M. on any diameter is an S.H.M.
Phase in S.H.M.
Phase in S.H.M. is basically the state of oscillation.
In order to know the state of oscillation in S.H.M.
- displacement (position)
- direction of velocity
- and the oscillation number at that instant of time.
The angular displacement 𝜃 = (𝑤𝑡 + Φ)
Special cases
(i) Phase θ = 0 indicates that the particle is at the mean position. Phase angle 𝜃 =
3600
𝑜𝑟 𝜋𝑐
is the beginning of the second oscillation.
(ii) Phase 𝜃 = 1800
or 𝜋𝑐
indicates that during its first oscillation, the particle is at the mean
position and moving to the negative. Similar state in the second oscillation will have phase
𝜃 = (360 + 180)0
or (2𝜋 + 𝜋)𝑐
.
(iii) Phase 𝜃 = 900
or (
𝜋
2
)
𝑐
indicates that the particle is at the positive extreme position. For
the second oscillation it will be 𝜃 = (360 + 90)0
or (2𝜋 +
𝜋
2
)
𝑐
.
(iv) Phase 𝜃 = 2700
or (
3𝜋
2
)
𝑐
indicates that the particle is at the negative extreme position.
For the second oscillation it will be 𝜃 = (360 + 270)0
or (2𝜋 +
3𝜋
2
)
𝑐
.
Graphical Representation of S.H.M.
(a) Particle executing S.H.M., starting from mean position, towards positive:
As the particle starts from the mean position, towards positive, φ = 0
∴ displacement x = A sin wt, Velocity v = A ω cos wt, Acceleration a = - A 𝑤2
sin wt
Conclusions from the graphs:
• Displacement, velocity and acceleration of S.H.M. are periodic functions of time.
• Displacement time curve and acceleration time curves are sine curves and velocity time
curve is a cosine curve.
• There is phase difference of
𝜋
2
radian between displacement and velocity.
• There is phase difference of
𝜋
2
radian between velocity and acceleration.
7. 7
• There is phase difference of π radian between displacement and acceleration.
• Shapes of all the curves get repeated after 2π radian or after a time T.
Fig.: (a) Variation of displacement with time, (b) Variation of velocity with time,
(c) Variation of acceleration with time.
(b) Particle performing S.H.M., starting from the positive extreme position.
As the particle starts from the positive extreme position, 𝜙 =
𝜋
2
∴ 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡, 𝑥 = 𝐴 sin (𝜔𝑡 +
𝜋
2
) = 𝐴 cos 𝜔𝑡
Velocity, v =
𝑑𝑥
𝑑𝑡
=
𝑑(𝐴 cos𝜔𝑡)
𝑑𝑡
= −𝐴 𝜔 sin(𝜔𝑡)
Acceleration, a =
𝑑𝑣
𝑑𝑡
=
𝑑 (−𝐴 ω sin (𝜔𝑡))
𝑑𝑡
= −𝐴 𝜔2
cos (𝜔𝑡)
Fig.: (a) Variation of displacement with time, (b) Variation of velocity with time,
(c) Variation of acceleration with time.
Composition of two S.H.M.s having same period and along the same path
8. 8
Equations of displacement of the two S.H.M.s along same straight line (x-axis) are
The resultant displacement (x) at any instant (t) is, 𝑥 = 𝑥1 + 𝑥2
𝑥 = 𝐴1si n(𝜔𝑡 + Φ1) + 𝐴2si n(𝜔𝑡 + Φ2)
𝑥 = 𝐴1si n 𝜔𝑡 𝑐𝑜𝑠 Φ1 + 𝐴1cos ωt 𝑠𝑖𝑛 Φ1 + 𝐴2si n 𝜔𝑡 𝑐𝑜𝑠Φ2 + 𝐴2cos ωt 𝑠𝑖𝑛 Φ2
𝐴1, 𝐴2, 𝜙1, 𝜙2 are constants and 𝜔𝑡 is variable.
𝑥 = (𝐴1 cos Φ1 + 𝐴2 cos Φ2) sin 𝜔𝑡 + (𝐴1si𝑛 Φ1 + 𝐴2 𝑠𝑖𝑛 Φ2) cos 𝜔𝑡
As 𝐴1, 𝐴2, 𝜙1 𝑎𝑛𝑑 𝜙2 are constants, we can combine them in terms of another convient
constants R and 𝛿 as
R cos 𝛿 = 𝐴1 𝑐𝑜𝑠 𝜙1 + 𝐴2 𝑐𝑜𝑠 ϕ2
and R sin 𝛿 = 𝐴1 𝑠𝑖𝑛 𝜙1 + 𝐴2 𝑠𝑖𝑛 ϕ2
∴ 𝑥 = 𝑅 (si n 𝜔𝑡 co s 𝛿 + co s 𝜔𝑡 si n 𝛿)
∴ 𝑥 = 𝑅 𝑠𝑖𝑛 (ωt + δ)
Resultant amplitude,
Initial phase (δ) of the resultant motion:
Special cases: (i) If the two S.H.M.s are in phase,
(ii) If the two S.H.M.s are 90° out of phase,
(iii) If the two S.H.M.s are 180° out of phase,
Energy of a Particle Performing S.H.M.
Total energy of the particle performing an S.H.M. is thus the sum of its kinetic and potential
energies.
Fig.: Energy in an S.H.M.
Velocity of the particle in S.H.M. is,
9. 9
where x is the displacement of the particle performing S.H.M. and A is the amplitude of
S.H.M.
Thus, the kinetic energy,
This is the kinetic energy at displacement x. At time t, it is
Thus, with time, it varies as 𝑐𝑜𝑠2
𝜃.
The external work done (dW) during this displacement is
The total work done on the particle to displace it from O to P is,
This should be the potential energy (P.E.) 𝐸𝑃 of the particle at displacement x.
Thus, with time, it varies as 𝑠𝑖𝑛2
𝜃.
The total energy of the particle is the sum of its kinetic energy and potential energy.
As m, ω and A are constant, the total energy of the particle at any point P is constant.
If n is the frequency in S.H.M., 𝜔 = 2𝜋𝑛.
Thus, the total energy in S.H.M. is directly proportional to (a) the mass of the particle (b) the
square of the amplitude (c) the square of the frequency (d) the force constant, and inversely
proportional to square of the period.
Special cases: (i) At the mean position, x = 0 and velocity is maximum. Hence,
and potential energy
(ii) At the extreme positions, the velocity of the particle is zero and 𝑥 = ±𝐴. Hence
and kinetic energy
(iii) If K E.= P. E.,
10. 10
Thus at 𝑥 =
±𝐴
√2
, the K.E. = P.E. =
𝐸
2
for a particle performing linear S.H.M.
(iv) At x =
±𝐴
2
, P.E. =
1
2
𝑘𝑥2
=
1
4
(
1
2
𝑘𝐴2) =
𝐸
4
∴ 𝐾. 𝐸. = 3(𝑃. 𝐸)
Thus, at 𝑥 =
±𝐴
2
, the energy is 25% potential and 75% kinetic.
Fig.: Energy in S.H.M.
Simple Pendulum
An ideal simple pendulum is a heavy particle suspended by a massless, inextensible, flexible
string from a rigid support.
The distance between the point of suspension and centre of gravity of the bob is called the
length of the pendulum.
Fig.: Simple pendulum
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
(ii) Weight mg, in the vertically downward direction.
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 = −𝑚 𝑔 sin 𝜃
As θ is very small (θ <10°), sin𝜃 ≅ 𝜃𝑐
∴ 𝐹 ≅ −𝑚 𝑔 𝜃
The small angle 𝜃 =
𝑥
𝐿
∴ 𝐹 = −𝑚 𝑔
𝑥
𝐿
As m, g and L are constant, F ∝- x
The period T of oscillation of a pendulum =
2 𝜋
𝜔
𝑇 =
2𝜋
√𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝐹 = −𝑚 𝑔
𝑥
𝐿
∴ 𝑚 𝑎 = −𝑚 𝑔
𝑥
𝐿
∴ 𝑎 = − 𝑔
𝑥
𝐿
11. 11
∴
𝑎
𝑥
= −
𝑔
𝐿
=
𝑔
𝐿
(𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒)
Substituting in the expression for T,
𝑇 = 2 𝜋 √
𝐿
𝑔
The Equation gives the expression for the time period of a simple pendulum.
(i) The amplitude of oscillations is very small.
(ii) The length of the string is large and
(iii) During the oscillations, the bob moves along a single vertical plane.
Frequency of oscillation n of the simple pendulum is
𝑛 =
1
𝑇
=
1
2𝜋
√
𝑔
𝐿
(a) The period of a simple pendulum is directly proportional to the square root of its length.
(b) The period of a simple pendulum is inversely proportional to the square root of
acceleration due to gravity.
(c) The period of a simple pendulum does not depend on its mass.
(d) The period of a simple pendulum does not depend on its amplitude.
These conclusions are also called the 'laws of simple pendulum'.
Second’s Pendulum
A simple pendulum whose period is two seconds is called second’s pendulum.
𝑃𝑒𝑟𝑖𝑜𝑑 𝑇 = 2 𝜋√
𝐿
𝑔
∴ 𝐹𝑜𝑟 𝑎 𝑠𝑒𝑐𝑜𝑛𝑑′
𝑠𝑝𝑒𝑛𝑑𝑢𝑙𝑢𝑚, 2 = 2 𝜋 √
𝐿𝑠
𝑔
where 𝐿𝑆 is the length of second’s pendulum, having period T = 2 s.
∴ 𝐿𝑠 =
𝑔
𝜋2
Experimentally, if 𝐿𝑆 is known, it can be used to determine acceleration due to gravity g at
that place.
Angular S.H.M. and its Differential Equation
If the disc is slightly twisted about the axis along the wire, and released, it performs rotational
motion partly in clockwise and anticlockwise sense. Such oscillations are called angular
oscillations or torsional oscillations.
Fig.: Torsional (angular) oscillations
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
The constant of proportionality c is the restoring torque per unit angular displacement.
If I is the moment of inertia of the body, the torque acting on the body is, 𝜏 = 𝐼𝛼
Where α is the angular acceleration. Using this in Equation, 𝐼𝛼 = −𝑐𝜃
∴ 𝐼
𝑑2
𝜃
𝑑𝑡2
+ 𝑐 𝜃 = 0
𝛼 =
𝑑2
𝜃
𝑑𝑡2
= −
𝑐 𝜃
𝐼
Since c and I are constants. This oscillatory motion is called angular S.H.M.
12. 12
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,
𝑇 =
2𝜋
𝜔
𝑇 =
2𝜋
√𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
Magnet Vibrating in Uniform Magnetic Field
If a bar magnet is freely suspended in the plane of a uniform magnetic field, it remains in
equilibrium with its axis parallel to the direction of the field.
Fig.: Magnet vibrating in a uniform magnetic field
The magnitude of this torque is 𝜏 = 𝜇 𝐵 sin 𝜃
If 𝜃 𝑖𝑠 𝑠𝑚𝑎𝑙𝑙, sin𝜃 ≅ 𝜃𝑐
∴ 𝜏 = 𝜇 𝐵 𝜃
For clockwise angular displacement θ, the restoring torque is in the anticlockwise direction.
∴ 𝜏 = 𝐼 𝛼 = − 𝜇 𝐵 𝜃
where I is the moment of inertia of the bar magnet and α is its angular acceleration.
∴ 𝛼 = − (
𝜇 𝐵
𝐼
) 𝜃
Since μ, B and I are constants.
𝑇 =
2𝜋
√𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑇 =
2𝜋
√
𝛼
𝜃
∴ 𝑇 = 2𝜋 √
𝐼
𝜇𝐵
Damped Oscillations
Fig.: A damped oscillator
Periodic oscillations of gradually decreasing amplitude are called damped harmonic
oscillations and the oscillator is called a damped harmonic oscillator.
13. 13
Where b is the damping constant and negative sign indicates that 𝐹𝑑 opposes the velocity.
For spring constant k, the force on the block from the spring is 𝐹𝑠 = −𝑘𝑥.
A𝑒
−𝑏𝑡
2𝑚 is the amplitude of the damped harmonic oscillations.
Fig.: Displacement against time graph
The term cos (𝜔′
𝑡 + 𝜙) shows that the motion is still an S.H.M.
The damping increases the period (slows down the motion) and decreases the amplitude.
Free Oscillations, Forced Oscillations and Resonance
Free Oscillations: If an object is allowed to oscillate or vibrate on its own, it does so with its
natural frequency.
which is called its natural frequency and the oscillations are free oscillations.
Fig: Forced oscillations
For unequal natural frequencies on either side, the energy absorbed is less. The peak
occurs when the forced frequency matches with the natural frequency.
Fig: Resonant frequency