This document provides information about oscillations and simple harmonic motion (SHM). It defines oscillation as periodic motion that repeats after a definite time interval. SHM is described as oscillatory motion where the restoring force is directly proportional to displacement from the equilibrium position. The key characteristics of SHM include:
- The differential equation relating displacement, velocity, and acceleration.
- Expressions for displacement, velocity, and acceleration as functions of time and constants.
- Definitions and calculations of important terms like amplitude, period, frequency, phase.
- Conditions required for motion to be considered SHM.
- Examples of SHM and calculations related to restoring force and period.
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
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 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
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.
Gravity Gravitation English Presentation
Tugas Fisika
Tugas Bahasa Inggris
oleh :
Kelas 12 IPA 6 SMA Negeri 1 Yogyakarta tahun 2014
Semangat!!!!!!! SUKSES
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
Rotational dynamics as per class 12 Maharashtra State Board syllabusRutticka Kedare
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 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
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 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.
Gravity Gravitation English Presentation
Tugas Fisika
Tugas Bahasa Inggris
oleh :
Kelas 12 IPA 6 SMA Negeri 1 Yogyakarta tahun 2014
Semangat!!!!!!! SUKSES
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
Rotational dynamics as per class 12 Maharashtra State Board syllabusRutticka Kedare
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 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
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 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.
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
Similar to CLASS XII - CHAPTER 5: OSCILLATION (PHYSICS - MAHARASHTRA STATE BOARD) (20)
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
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.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
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.
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?
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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.
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.
For more information, visit-www.vavaclasses.com
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.
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. 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.
8. 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
9. 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
10. 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)
11. 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)
12. 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
13. 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
14. 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
15. 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
16. ∴ 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
17. 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)
19. 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)
20. 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
21. 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
22. 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
23. 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
24. 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.
25. 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
26. 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
27. 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
28. 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.
29. 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.
30. 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