1) Work is defined as the product of the force applied and the distance moved in the direction of the force. Work can cause a change in energy but the total energy in a system remains constant.
2) There are four types of work depending on whether the force and/or distance is constant or variable. Work by a conservative force depends only on the start and end points and not the path taken.
3) Kinetic energy is the energy of motion while potential energy is stored energy due to an object's position or state. Mechanical energy is the sum of an object's kinetic and potential energies.
In tis slide, an introduction to string theory has been given. Apart from that, a simple proof of 26 dimensions of bosonic string theory is given (following Zwiebach's approach).
I explained this presentation in two parts (on my YouTube channel). Here are the links
_______________________________________________
Part 1
https://www.youtube.com/watch?v=QQA4JQ6Y-eo&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp
_______________________________________________
Part 2
https://www.youtube.com/watch?v=vhLCtLn79jE&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp&index=2
_______________________________________________
In tis slide, an introduction to string theory has been given. Apart from that, a simple proof of 26 dimensions of bosonic string theory is given (following Zwiebach's approach).
I explained this presentation in two parts (on my YouTube channel). Here are the links
_______________________________________________
Part 1
https://www.youtube.com/watch?v=QQA4JQ6Y-eo&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp
_______________________________________________
Part 2
https://www.youtube.com/watch?v=vhLCtLn79jE&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp&index=2
_______________________________________________
The slides are designed for my guided study in MSc CUHK.
It is about the brief description on classical mechanics and quantum mechanics .
Some Slides I got from the slideshare clipboards for better illustration of the ideas in Physics. Thanks to slideshare, I make a milestone on presenting one of the prominent fields in modern physics.
Helmholtz equation (Motivations and Solutions)Hassaan Saleem
Solutions of Helmholtz equation in cartesian, cylindrical and spherical coordinates are discussed and the applications to the problem of a quantum mechanical particle in a cubical box is discussed.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Qausi Conformal Curvature Tensor on 푳푪푺 풏-Manifoldsinventionjournals
In this paper, we focus on qausi-conformal curvature tensor of 퐿퐶푆 푛 -manifolds. Here we study quasi-conformally flat, Einstein semi-symmetric quasi -conformally flat, 휉-quasi conformally flat and 휙-quasi conformally flat 퐿퐶푆 푛 -manifolds and obtained some interesting results
The slides are designed for my guided study in MSc CUHK.
It is about the brief description on classical mechanics and quantum mechanics .
Some Slides I got from the slideshare clipboards for better illustration of the ideas in Physics. Thanks to slideshare, I make a milestone on presenting one of the prominent fields in modern physics.
Helmholtz equation (Motivations and Solutions)Hassaan Saleem
Solutions of Helmholtz equation in cartesian, cylindrical and spherical coordinates are discussed and the applications to the problem of a quantum mechanical particle in a cubical box is discussed.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Qausi Conformal Curvature Tensor on 푳푪푺 풏-Manifoldsinventionjournals
In this paper, we focus on qausi-conformal curvature tensor of 퐿퐶푆 푛 -manifolds. Here we study quasi-conformally flat, Einstein semi-symmetric quasi -conformally flat, 휉-quasi conformally flat and 휙-quasi conformally flat 퐿퐶푆 푛 -manifolds and obtained some interesting results
Differential Geometry for Machine LearningSEMINARGROOT
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
Elasticity, Plasticity and elastic plastic analysisJAGARANCHAKMA2
It is actually the basis of structural engineering to study elasticity and plasticity analysis. So people who are also studying in various fields of structure and need to analyze finite element analysis also need to study this basis.
RADIAL HEAT CONDUCTION SOLVED USING THE INTEGRAL EQUATION .pdfWasswaderrick3
We look at the case of radial heat flow. Again, in radial heat flow, the temperature profiles that satisfy the boundary and initial conditions are the exponential and hyperbolic functions as derived in literature of conduction in fins. We use the technique of transforming the PDE into an integral equation. But in the case of radial heat flow, we have to multiply through by r the heat equation and then introduce integrals. We do this to avoid introducing integrals of the form of the exponential integral whose solutions cannot be expressed in the form of a simple mathematical function. We look at the case of a semi-infinite hollow cylinder for both insulated and non-insulated cases and then find the solution. We also look at cases of finite radius hollow cylinders subject to given boundary conditions. We notice that the solutions got for finite radius hollow cylinders do not reduce to those of semi-infinite hollow cylinders. We conclude by saying that this same analysis can be extended to spherical co-ordinates heat conduction.
A derivation of the Schwarzchild solution is presented with all relevant information. I have used this slides to teach Schwarzchild solution at my youtube channel. Here is the link
https://www.youtube.com/watch?v=ixhgvnGQZHM&t=1635s
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
Autopilot per Apps
Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
2. Work and Energy
Principle of Conservation of Energy
“Energy can neither be created nor
destroyed in any process. It can be
converted from one form to another or
transferred from one body to another,
but the total amount remains constant”
Work
;the product of a force times the
distance through which it acts
𝑊 = 𝐹𝑠
***𝐹 and 𝑠 are both vectors, but 𝑊 is
scalar
Multiplication of vectors
; Dot product (scalar)
; Cross product (vector)
Dot product
Suppose
𝐴 = 𝑎 𝑥 𝑖 + 𝑎 𝑦 𝑗 + 𝑎 𝑧 𝑘
𝐵 = 𝑏 𝑥 𝑖 + 𝑏 𝑦 𝑗 + 𝑏 𝑧 𝑘
𝐴 ∙ 𝐵 = 𝑎 𝑥 𝑏 𝑥 + 𝑎 𝑦 𝑏 𝑦 + 𝑎 𝑧 𝑏 𝑧
𝐴 ∙ 𝐵 = 𝐴 𝐵 𝑐𝑜𝑠𝜃
𝐴
𝐵
𝜃
𝐴 ∙ 𝐵 = 𝐴 𝐵 𝑐𝑜𝑠𝜃
𝐹
𝜃 𝐹 𝑐𝑜𝑠𝜃
𝐹 𝑠𝑖𝑛𝜃
∆𝑟
From the diagram,
𝑊 = 𝐹∆𝑟𝑐𝑜𝑠𝜃
𝑊 = 𝐹 ∆𝑟 𝑐𝑜𝑠𝜃
𝑊 = 𝐹 ∙ ∆𝑟
*** 𝑊 = 𝐹 ∙ ∆𝑟 is more accurate
***In reality, 𝐹 is not applied constantly
and ∆𝑟 is not always a straight line
Concept
***For a curve path, see the curve as the
combination of increadibly tiny straight
lines, 𝑑𝑟
4 types of Work
i. 𝐹; constant magnitude and direction
ii. 𝐹; inconstant magnitude and constant
direction
iii. 𝐹; constant magnitude and inconstant
direction
iv. 𝐹; inconstant magnitude and direction
𝑊 = 𝐹 ∙ ∆𝑟
𝑑𝑊 = 𝐹 ∙ 𝑑𝑟
𝑊𝑡𝑜𝑡𝑎𝑙 = 𝑑𝑊 = 𝐹 ∙ 𝑑𝑟
3. Uniform Force
;both magnitude and direction are
constant
𝑊𝐴→𝐵 = 𝑑𝑊
𝐵
𝐴
= 𝐹 ∙
𝐵
𝐴
𝑑𝑟
Since, 𝐹 is uniform,
𝑊𝐴→𝐵 = 𝐹 ∙ 𝑑𝑟
𝐵
𝐴
𝑊𝐴→𝐵 = 𝐹 ∙ (𝑟𝐵 − 𝑟𝐴)
𝑊𝐴→𝐵 = 𝐹 ∙ ∆𝑟𝐴→𝐵
***𝑊 is independent on the path, but is
depend on the displacement when 𝐹 is
uniform. A uniform force is considered
“Conservative Force”
𝑊 > 0; work done by the system
𝑊 < 0; work done on the system
𝑟𝐴
𝑥
𝑦
𝑜
𝑟𝐵
∆𝑟
𝐵
𝐴
𝑊𝐴→𝐵 = 𝐹 ∙ ∆𝑟𝐴→𝐵
Non-uniform Force
;inconstant magnitude and constant
direction
Ex; Spring
At (a), x=0, equilibrium point
At (b), an external force (𝐹𝑒𝑥𝑡) pulls an
object to the right, restoring force (𝐹𝑠)
will be experienced by the object also
∴ 𝐹𝑒𝑥𝑡 = 𝐹𝑠
At (b),𝐹𝑒𝑥𝑡 = 𝑘𝑥1
At (c),𝐹𝑒𝑥𝑡 = 𝑘𝑥2
𝑑𝑊 = 𝐹 ∙ 𝑑𝑟
𝑊 𝑏 →(𝑐) = 𝑑𝑊
(𝑐)
(𝑏)
= 𝐹𝑒𝑥𝑡 ∙
(𝑐)
𝑏
𝑑𝑟
= 𝐹𝑒𝑥𝑡
(𝑐)
𝑏
𝑑𝑟 𝑐𝑜𝑠𝜃
= 𝑘𝑥
𝑐
𝑏
𝑑𝑥 (𝑐𝑜𝑠0°)
m
m
m
(𝑐)
(𝑏)
(𝑎)
𝑥1
𝑥 = 0
𝐹𝑒𝑥𝑡
𝐹𝑠
𝐹𝑠 = 𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒
𝑥2
4. 𝑊 𝑏 →(𝑐) = 𝑘𝑥
𝑐
𝑏
𝑑𝑥 (𝑐𝑜𝑠0°)
𝑊 𝑏 →(𝑐) =
1
2
𝑘𝑥2
2
−
1
2
𝑘𝑥1
2
Since,
1
2
𝑘𝑥2
is Elastic Potential Energy
*** 𝑊𝐴→𝐵 = 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑏𝑦 𝐹𝑒𝑥𝑡
***𝑊 𝐹𝑠
= 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 𝑏𝑦𝐹𝑠
Non-uniform Force
;constant magnitude and inconstant
direction
Ex; Circular motion
𝑑𝑊 = 𝐹 ∙ 𝑑𝑟 = 𝐹 𝑑𝑟 𝑐𝑜𝑠𝜃
Since at every point, 𝜃 = 90°
𝑊 = 0
𝑊𝐴→𝐵 = 𝐸𝑃𝐸 𝐵 − 𝐸𝑃𝐸𝐴
𝑊 𝐹𝑒𝑥𝑡
= −𝑊 𝐹𝑠
𝑣0
𝑣0
𝐹𝐴
𝐹𝐵
𝐴
𝐵
Non-uniform Force
;inconstant magnitude and direction
***non-conservative force; work done by
force depends on the paths taken by a
particle
Ex; a particle moves from A to B in AEB
and ADCB paths. The equation of force
in the field is 𝐹 = −𝑦𝑖 + 𝑥𝑗 Newton.
Find the work done in the 2 paths.
The external force acting on the particle
need to be equal to the force of the field
and has opposite direction to cancel out.
∴ 𝐹𝑒𝑥𝑡 = −𝐹
𝑊𝐴→𝐸→𝐵 = 𝐹𝑒𝑥𝑡 ∙ 𝑑𝑟
𝐵
𝐴
= −𝐹 ∙ 𝑑𝑟
𝐵
𝐴
= −
−𝑦
𝑥
∙
𝑑𝑥
𝑑𝑦
𝐵
𝐴
= 𝑦𝑑𝑥
𝐵
𝐴
− 𝑥𝑑𝑦
𝐵
𝐴
= 𝑦𝑑𝑥
𝑥=1
𝑥=1
− 1 𝑑𝑦
𝑦=−1
𝑦=1
= 0 − −1 − 1 = 2 𝐽𝑜𝑢𝑙𝑒𝑠
𝑥
𝑦
𝐴(1,1) 𝐷(2,1)
𝐵(1, −1) 𝐶(2, −1)
𝐸
0
5. 𝑊𝐴→𝐷→𝐶→𝐵 = 𝑊𝐴→𝐷 + 𝑊𝐷→𝐶 + 𝑊𝐶→𝐵
= 𝑦𝑑𝑥
𝐷
𝐴
− 𝑥𝑑𝑦
𝐷
𝐴
+ 𝑦𝑑𝑥
𝐶
𝐷
− 𝑥𝑑𝑦
𝐶
𝐷
+ 𝑦𝑑𝑥
𝐵
𝐶
− 𝑥𝑑𝑦
𝐵
𝐶
= (1)𝑑𝑥
𝑥=2
𝑥=1
− 𝑥𝑑𝑦
𝑦=1
𝑦=1
+ 𝑦𝑑𝑥
𝑥=2
𝑥=2
− (2)𝑑𝑦
𝑦=−1
𝑦=1
+ (−1)𝑑𝑥
𝑥=1
𝑥=2
− 𝑥𝑑𝑦
𝑦=−1
𝑦=−1
= (1)𝑑𝑥
𝑥=2
𝑥=1
− 0 + 0 − (2)𝑑𝑦
𝑦=−1
𝑦=1
+ (−1)𝑑𝑥
𝑥=1
𝑥=2
− 0
= (1)𝑑𝑥
𝑥=2
𝑥=1
− 2 𝑑𝑦
𝑦=−1
𝑦=1
+ (−1)𝑑𝑥
𝑥=1
𝑥=2
= 2 − 1 − −2 − 2 + −1 − −2
= 1 + 4 + 1 = 6 Joules
𝑥
𝑦
𝐴(1,1) 𝐷(2,1)
𝐵(1, −1) 𝐶(2, −1)
𝐸
0
Prove that 𝐹 = 2𝑥𝑖 + 3𝑦2
𝑗 is a
conservative force.
Let the initial and final point be 𝐴 and 𝐵
respectively
𝑊𝐴→𝐵 = 𝐹 ∙ 𝑑𝑟
𝐵
𝐴
=
2𝑥
3𝑦2 ∙
𝑑𝑥
𝑑𝑦
𝐵
𝐴
= 2𝑥𝑑𝑥
𝐵
𝐴
+ 3𝑦2
𝑑𝑦
𝐵
𝐴
= 𝑥2
+ 𝑦3
𝐴
𝐵
Since, the value of 𝑊𝐴→𝐵 depends on
only the values of 𝑥 and 𝑦 at 𝐴 and 𝐵. It
does not depend on the other points on
the path. Thus, 𝐹 is a conservative force.
***How to check whether 𝑭 is a
conservative force?
For
𝐹 = 𝐹𝑥 𝑖 + 𝐹𝑦 𝑗 + 𝐹𝑧 𝑘
𝐹 is a conservative force, when
𝜕𝐹𝑥
𝜕𝑦
=
𝜕𝐹𝑦
𝜕𝑥
And
𝜕𝐹𝑦
𝜕𝑧
=
𝜕𝐹𝑧
𝜕𝑦
And
𝜕𝐹𝑧
𝜕𝑥
=
𝜕𝐹𝑥
𝜕𝑧
Power
; average and instantaneous power
𝑃 =
𝑊
∆𝑡
𝑃 =
𝑑𝑊
𝑑𝑡
6. 𝑃 =
𝑑𝑊
𝑑𝑡
𝑃 =
𝑑𝑊
𝑑𝑡
Since, 𝑊 = 𝐹 ∙ 𝑑𝑟
𝑃 =
𝑑( 𝐹 ∙ 𝑑𝑟)
𝑑𝑡
Assume that 𝐹 is uniform,
𝑃 =
𝐹 ∙ 𝑑𝑟
𝑑𝑡
= 𝐹 ∙ 𝑣
Energy
;the capacity to do work
Mechanical Energy
2 types;
Potential Energy & Kinetic Energy
Potential Energy
(i) Elastic Potential Energy, 𝑈𝑠
For spring,
𝑊 1 →(2) =
1
2
𝑘𝑥2
2
−
1
2
𝑘𝑥1
2
(ii) Gravitational Potential Energy, 𝑈𝑔
An object initially at 𝑦𝑎 is lifted up by
𝐹 𝑎𝑝𝑝 to 𝑦 𝑏.
𝑊 = 𝐹 𝑎𝑝𝑝 ∙ ∆𝑟 = 𝑚𝑔𝑗 ∙ [(𝑦 𝑏 − 𝑦𝑎)𝑗]
𝑊 = 𝑚𝑔𝑦 𝑏 − 𝑚𝑔𝑦𝑎
𝑊 = ∆𝑈𝑔
𝑈𝑔 = 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦
***𝑈𝑔 depends on only vertical force(s).
Kinetic Energy
𝑤 = 𝐹 ∙ 𝑑𝑥
𝑥 𝑓
𝑥 𝑖
𝑃 = 𝐹 ∙ 𝑣
∆𝑟
𝑦𝑎
𝑦 𝑏
𝐹 𝑎𝑝𝑝
𝑚𝑔
0
𝑦
𝑥
𝑈𝑠 =
1
2
𝑘𝑥2
𝑈𝑔 = 𝑚𝑔𝑦
𝐹
∆𝑥
𝑣𝑖 𝑣 𝑓
8. Collisions
Law Of Conservation Of Momentum
𝑃𝑖 = 𝑃𝑓
***The law is applicable for all
collisions. Why?
; the Law is based on the 3rd law of
Newton, reaction and action forces.
According to the 3rd law of Newton,
𝐹12 = −𝐹21
𝐹12 + 𝐹21 = 0
According to the 2rd law of Newton,
𝑚1 𝑎1 + 𝑚2 𝑎2 = 0
𝑚1
𝑑𝑣1
𝑑𝑡
+ 𝑚2
𝑑𝑣2
𝑑𝑡
= 0
𝑑𝑚1 𝑣1
𝑑𝑡
+
𝑑𝑚2 𝑣2
𝑑𝑡
= 0
𝑑(𝑚1 𝑣1 + 𝑚2 𝑣2)
𝑑𝑡
= 0
𝐹12
𝑚1
𝑚2
𝐹21
𝑣1 𝑖
𝑣2 𝑖
𝑣1 𝑓
𝑣2 𝑓
𝑑(𝑃1 + 𝑃2)
𝑑𝑡
= 0
∴ 𝑃1 + 𝑃2 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑃𝑖 = 𝑃𝑓
1 dimension elastic collision
From, 𝑃𝑖 = 𝑃𝑓 and 𝐸𝑖 = 𝐸𝑓, we can
derive another formula.
or
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ
=
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑜𝑛
***The question will state whether the
collision is elastic or inelastic. However,
if the 2 masses stick together after the
collision, it is inelastic collision.
Elastic Inelastic
𝑃𝑖 = 𝑃𝑓
𝐸𝑖 = 𝐸𝑓
(Work due to non-
conservative force = 0)
𝐸𝑖 ≠ 𝐸𝑓
(Work due to non-
conservative force ≠ 0)
𝑣1𝑖
+ 𝑣1 𝑓
= 𝑣2 𝑖
+ 𝑣2 𝑓
𝑣1𝑖
− 𝑣2 𝑖
= 𝑣2 𝑓
− 𝑣1 𝑓