1) The document discusses kinematics of particles, specifically rectilinear and curvilinear motion.
2) Rectilinear motion involves motion along a straight line that can be described using displacement, velocity, acceleration, and differential equations.
3) Curvilinear motion occurs along a curved path in a plane and uses vector analysis to describe position, displacement, and velocity.
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document outlines the key concepts and objectives covered in Chapter 1 of an engineering mechanics textbook. It introduces the fundamental topics of mechanics including particles, rigid bodies, Newton's laws of motion and gravitation. It also reviews the SI system of units and procedures for dimensional analysis, significant figures and numerical calculations. The objectives are to provide an introduction to the basic concepts and quantitative methods of mechanics.
Lecture 3 mohr’s circle and theory of failure Deepak Agarwal
The document covers Mohr's stress circle, which is a graphical method to determine normal and tangential stresses on an oblique plane for a material subjected to principal stresses and shear stresses. It also discusses different failure theories, including maximum principal stress, maximum principal strain, maximum shear stress, maximum strain energy, and maximum shear strain energy theories. The different theories predict failure based on the maximum values of stresses, strains, or strain energies for brittle versus ductile materials.
Dynamics of particles , Enginnering mechanics , murugananthanMurugananthan K
This document discusses particle dynamics and concepts such as displacement, velocity, acceleration, relative motion, Newton's second law of motion, linear momentum, angular momentum, and central forces. It provides definitions and equations for these concepts and includes 6 sample problems solving for quantities like acceleration, tension, velocity, and force using the principles of kinematics and dynamics.
The document discusses particle kinematics and concepts such as displacement, velocity, acceleration, and their relationships for rectilinear and curvilinear motion. Key concepts covered include definitions of displacement, average and instantaneous velocity, acceleration, graphical representations of position, velocity, and acceleration over time, and analytical methods for solving kinematic equations involving constant or variable acceleration. Several sample problems are provided to illustrate applying these kinematic concepts and relationships to solve for variables like time, velocity, acceleration, and displacement given relevant conditions.
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document outlines the key concepts and objectives covered in Chapter 1 of an engineering mechanics textbook. It introduces the fundamental topics of mechanics including particles, rigid bodies, Newton's laws of motion and gravitation. It also reviews the SI system of units and procedures for dimensional analysis, significant figures and numerical calculations. The objectives are to provide an introduction to the basic concepts and quantitative methods of mechanics.
Lecture 3 mohr’s circle and theory of failure Deepak Agarwal
The document covers Mohr's stress circle, which is a graphical method to determine normal and tangential stresses on an oblique plane for a material subjected to principal stresses and shear stresses. It also discusses different failure theories, including maximum principal stress, maximum principal strain, maximum shear stress, maximum strain energy, and maximum shear strain energy theories. The different theories predict failure based on the maximum values of stresses, strains, or strain energies for brittle versus ductile materials.
Dynamics of particles , Enginnering mechanics , murugananthanMurugananthan K
This document discusses particle dynamics and concepts such as displacement, velocity, acceleration, relative motion, Newton's second law of motion, linear momentum, angular momentum, and central forces. It provides definitions and equations for these concepts and includes 6 sample problems solving for quantities like acceleration, tension, velocity, and force using the principles of kinematics and dynamics.
The document discusses particle kinematics and concepts such as displacement, velocity, acceleration, and their relationships for rectilinear and curvilinear motion. Key concepts covered include definitions of displacement, average and instantaneous velocity, acceleration, graphical representations of position, velocity, and acceleration over time, and analytical methods for solving kinematic equations involving constant or variable acceleration. Several sample problems are provided to illustrate applying these kinematic concepts and relationships to solve for variables like time, velocity, acceleration, and displacement given relevant conditions.
Watch Video of this presentation on Link: https://youtu.be/bHKaPBgDk6g
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For Video, Visit our YouTube Channel (link is given below).
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1) To analyze accelerations, positions must first be found to calculate velocities by differentiation and accelerations by further differentiation.
2) Acceleration has two components - tangential and centripetal. For uniform motion only centripetal acceleration exists, and for straight-line motion only tangential acceleration exists.
3) The Coriolis component arises for points moving on rotating links and is perpendicular to the link and proportional to the product of linear and angular velocities.
The document discusses the concepts of buoyancy, stability, and equilibrium of submerged and floating bodies in fluids. It states that:
1. According to Archimedes' principle, the buoyant force on a submerged body equals the weight of the fluid displaced and acts vertically upwards through the centroid of the displaced volume. For a floating body in equilibrium, the buoyant force must balance the weight of the body.
2. A submerged body will be in stable, unstable, or neutral equilibrium depending on whether its center of gravity is below, above, or coincident with the center of buoyancy, respectively.
3. For a floating body, stability depends on the relative positions of its metac
Varignon's principle of moments states that the total moment of the forces acting on a rigid body is equal to the time rate of change of its angular momentum about any point. This principle is useful for analyzing rotational motion and dynamics problems involving torques. It provides a relationship between the net torque on a rigid body and how its angular momentum changes with respect to time.
This document discusses key concepts in fluid dynamics, including:
(i) Fluid kinematics describes fluid motion without forces/energies, examining geometry of motion through concepts like streamlines and pathlines.
(ii) Fluids can flow steadily or unsteadily, uniformly or non-uniformly, laminarly or turbulently depending on properties of the flow and fluid.
(iii) The continuity equation states that mass flow rate remains constant for an incompressible, steady flow through a control volume according to the principle of conservation of mass.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
Kinematic analysis of mechanisms analytical methodsajitkarpe1986
This document discusses analytical methods for position, velocity, and acceleration analysis of mechanisms like slider crank mechanisms. It covers topics like loop closure equations, Chase solutions, and using vector and complex algebra methods. Numerical examples are provided to analyze parameters like crank position, piston velocity, acceleration, and angular velocities of links in slider crank engines. Different joint mechanisms like Hooke's joint and double Hooke's joint are also discussed.
This document discusses oblique shock waves that occur in supersonic flows when the flow direction changes. It provides the governing equations for analyzing oblique shock waves using conservation of mass, momentum, and energy across a control volume. The equations show that an oblique shock acts like a normal shock in the direction normal to the wave. Relations are developed to determine the post-shock Mach number, static properties, and stagnation properties in terms of the shock angle and pre-shock Mach number using normal shock tables. An example problem applies these relations to analyze an oblique shock occurring at a sharp concave corner.
This document discusses concepts related to determining the center of gravity, center of mass, and centroid of objects. It begins with definitions of these terms and how to calculate them using integrals for continuous objects and the summation of moments for discrete systems. Several examples are then provided to demonstrate calculating these values for different shapes, including areas, lines, volumes, and composite bodies. The expected outcome is the ability to determine the center of gravity, center of mass, and centroid for various objects.
This document discusses kinetics of particles and Newton's laws of motion. It introduces the concept of equation of motion relating the forces acting on a particle to its acceleration. Equations of motion are developed for rectangular, normal-tangential, and cylindrical coordinate systems. Examples are provided to demonstrate solving equations of motion for particles undergoing accelerated motion under various force conditions in different coordinate systems.
Pressure distribution along convergent- divergent NozzleSaif al-din ali
SAIF ALDIN ALI MADIN
سيف الدين علي ماضي
S96aif@gmail.com
This aim of this practical was to investigate compressible flow in a
convergent-divergent nozzle. Different flow patterns that influence
the results of the investigation are also explored. The different
pressure distributions that occur at varying lengths in the nozzle
were also recorded and analyzed
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Unit 4- balancing of rotating masses, Dynamics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
This document discusses the law of gearing in three main points:
1) The common normal at the point of contact between gear teeth must always pass through the pitch point. This is the fundamental condition for designing gear teeth profiles.
2) The angular velocity ratio between two gears must remain constant throughout meshing.
3) The angular velocity ratio is inversely proportional to the ratio of the distances of the pitch point P from the gear centers O1 and O2. The common normal intersecting the line of centers at P divides the center distance inversely proportional to the angular velocity ratio.
The document provides an overview of fluid kinematics and dynamics concepts over 12 hours. It discusses types of fluid flow such as steady, unsteady, uniform, laminar, turbulent and more. It also covers fluid motion analysis using Lagrangian and Eulerian methods. Key concepts covered include velocity, acceleration, streamlines, pathlines, continuity equation, and momentum equation. Circulation and vorticity are also defined. The document aims to equip readers with fundamental understanding of fluid motion characteristics and governing equations.
This document provides an overview of dynamics of machines including:
1. It defines force, applied force, constraint forces, and types of constrained motions like completely, incompletely, and successfully constrained motions.
2. It discusses static force analysis, dynamic force analysis, and conditions for static and dynamic equilibrium.
3. It covers concepts like inertia, inertia force, inertia torque, D'Alembert's principle, and principle of superposition.
4. It derives expressions for forces acting on the reciprocating parts of an engine while neglecting the weight of the connecting rod.
1) The document describes simple harmonic motion and cycloidal motion. Simple harmonic motion can be constructed by projecting points moving along a circle with constant angular velocity onto a diameter. Cycloidal motion is constructed using the locus of a point on a circle rolling along a straight line.
2) Formulas are provided to express the displacement, velocity, and acceleration of a follower in terms of cam rotation for simple harmonic motion. Corresponding formulas are also given for cycloidal motion.
3) Examples are given of constructing displacement diagrams for both simple harmonic motion and cycloidal motion using semicircles and a rolling circle respectively.
The document summarizes plane curvilinear motion and projectile motion. It defines key concepts like position vector, velocity, acceleration, and rectangular coordinate analysis. It provides equations to describe velocity and acceleration in x and y directions. Examples are given to demonstrate calculating displacement, velocity, acceleration from given motion equations. The last two examples solve for minimum initial velocity and angles to just clear a fence or pass through a basketball hoop.
This document discusses concepts in mechanics including kinematics, dynamics, and statics. It defines key terms like reference frames, position vectors, displacement, average speed, average velocity, and instantaneous acceleration. It also provides examples of determining trajectory, displacement, velocity, and center of mass for systems of particles.
Watch Video of this presentation on Link: https://youtu.be/bHKaPBgDk6g
For notes/articles, Visit my blog (link is given below).
For Video, Visit our YouTube Channel (link is given below).
Any Suggestions/doubts/reactions, please leave in the comment box.
Follow Us on
YouTube: https://www.youtube.com/channel/UCVPftVoKZoIxVH_gh09bMkw/
Blog: https://e-gyaankosh.blogspot.com/
Facebook: https://www.facebook.com/egyaankosh/
1) To analyze accelerations, positions must first be found to calculate velocities by differentiation and accelerations by further differentiation.
2) Acceleration has two components - tangential and centripetal. For uniform motion only centripetal acceleration exists, and for straight-line motion only tangential acceleration exists.
3) The Coriolis component arises for points moving on rotating links and is perpendicular to the link and proportional to the product of linear and angular velocities.
The document discusses the concepts of buoyancy, stability, and equilibrium of submerged and floating bodies in fluids. It states that:
1. According to Archimedes' principle, the buoyant force on a submerged body equals the weight of the fluid displaced and acts vertically upwards through the centroid of the displaced volume. For a floating body in equilibrium, the buoyant force must balance the weight of the body.
2. A submerged body will be in stable, unstable, or neutral equilibrium depending on whether its center of gravity is below, above, or coincident with the center of buoyancy, respectively.
3. For a floating body, stability depends on the relative positions of its metac
Varignon's principle of moments states that the total moment of the forces acting on a rigid body is equal to the time rate of change of its angular momentum about any point. This principle is useful for analyzing rotational motion and dynamics problems involving torques. It provides a relationship between the net torque on a rigid body and how its angular momentum changes with respect to time.
This document discusses key concepts in fluid dynamics, including:
(i) Fluid kinematics describes fluid motion without forces/energies, examining geometry of motion through concepts like streamlines and pathlines.
(ii) Fluids can flow steadily or unsteadily, uniformly or non-uniformly, laminarly or turbulently depending on properties of the flow and fluid.
(iii) The continuity equation states that mass flow rate remains constant for an incompressible, steady flow through a control volume according to the principle of conservation of mass.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
Kinematic analysis of mechanisms analytical methodsajitkarpe1986
This document discusses analytical methods for position, velocity, and acceleration analysis of mechanisms like slider crank mechanisms. It covers topics like loop closure equations, Chase solutions, and using vector and complex algebra methods. Numerical examples are provided to analyze parameters like crank position, piston velocity, acceleration, and angular velocities of links in slider crank engines. Different joint mechanisms like Hooke's joint and double Hooke's joint are also discussed.
This document discusses oblique shock waves that occur in supersonic flows when the flow direction changes. It provides the governing equations for analyzing oblique shock waves using conservation of mass, momentum, and energy across a control volume. The equations show that an oblique shock acts like a normal shock in the direction normal to the wave. Relations are developed to determine the post-shock Mach number, static properties, and stagnation properties in terms of the shock angle and pre-shock Mach number using normal shock tables. An example problem applies these relations to analyze an oblique shock occurring at a sharp concave corner.
This document discusses concepts related to determining the center of gravity, center of mass, and centroid of objects. It begins with definitions of these terms and how to calculate them using integrals for continuous objects and the summation of moments for discrete systems. Several examples are then provided to demonstrate calculating these values for different shapes, including areas, lines, volumes, and composite bodies. The expected outcome is the ability to determine the center of gravity, center of mass, and centroid for various objects.
This document discusses kinetics of particles and Newton's laws of motion. It introduces the concept of equation of motion relating the forces acting on a particle to its acceleration. Equations of motion are developed for rectangular, normal-tangential, and cylindrical coordinate systems. Examples are provided to demonstrate solving equations of motion for particles undergoing accelerated motion under various force conditions in different coordinate systems.
Pressure distribution along convergent- divergent NozzleSaif al-din ali
SAIF ALDIN ALI MADIN
سيف الدين علي ماضي
S96aif@gmail.com
This aim of this practical was to investigate compressible flow in a
convergent-divergent nozzle. Different flow patterns that influence
the results of the investigation are also explored. The different
pressure distributions that occur at varying lengths in the nozzle
were also recorded and analyzed
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Unit 4- balancing of rotating masses, Dynamics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
This document discusses the law of gearing in three main points:
1) The common normal at the point of contact between gear teeth must always pass through the pitch point. This is the fundamental condition for designing gear teeth profiles.
2) The angular velocity ratio between two gears must remain constant throughout meshing.
3) The angular velocity ratio is inversely proportional to the ratio of the distances of the pitch point P from the gear centers O1 and O2. The common normal intersecting the line of centers at P divides the center distance inversely proportional to the angular velocity ratio.
The document provides an overview of fluid kinematics and dynamics concepts over 12 hours. It discusses types of fluid flow such as steady, unsteady, uniform, laminar, turbulent and more. It also covers fluid motion analysis using Lagrangian and Eulerian methods. Key concepts covered include velocity, acceleration, streamlines, pathlines, continuity equation, and momentum equation. Circulation and vorticity are also defined. The document aims to equip readers with fundamental understanding of fluid motion characteristics and governing equations.
This document provides an overview of dynamics of machines including:
1. It defines force, applied force, constraint forces, and types of constrained motions like completely, incompletely, and successfully constrained motions.
2. It discusses static force analysis, dynamic force analysis, and conditions for static and dynamic equilibrium.
3. It covers concepts like inertia, inertia force, inertia torque, D'Alembert's principle, and principle of superposition.
4. It derives expressions for forces acting on the reciprocating parts of an engine while neglecting the weight of the connecting rod.
1) The document describes simple harmonic motion and cycloidal motion. Simple harmonic motion can be constructed by projecting points moving along a circle with constant angular velocity onto a diameter. Cycloidal motion is constructed using the locus of a point on a circle rolling along a straight line.
2) Formulas are provided to express the displacement, velocity, and acceleration of a follower in terms of cam rotation for simple harmonic motion. Corresponding formulas are also given for cycloidal motion.
3) Examples are given of constructing displacement diagrams for both simple harmonic motion and cycloidal motion using semicircles and a rolling circle respectively.
The document summarizes plane curvilinear motion and projectile motion. It defines key concepts like position vector, velocity, acceleration, and rectangular coordinate analysis. It provides equations to describe velocity and acceleration in x and y directions. Examples are given to demonstrate calculating displacement, velocity, acceleration from given motion equations. The last two examples solve for minimum initial velocity and angles to just clear a fence or pass through a basketball hoop.
This document discusses concepts in mechanics including kinematics, dynamics, and statics. It defines key terms like reference frames, position vectors, displacement, average speed, average velocity, and instantaneous acceleration. It also provides examples of determining trajectory, displacement, velocity, and center of mass for systems of particles.
- The document discusses kinematic concepts such as position, displacement, velocity, and acceleration for particles moving along a straight path. It defines these concepts using equations of motion.
- Rectilinear motion is analyzed by creating graphs of position vs. time, velocity vs. time, and acceleration vs. time. The slopes of these graphs are used to define velocity, acceleration, and how they relate to each other.
- Integrals of the kinematic equations are used to determine relationships between position, velocity, acceleration, and time.
GEN PHYSICS 1 WEEK 2 KINEMATICS IN ONE DIMENSION.pptxAshmontefalco4
This document provides an overview of kinematics in one dimension, including graphical representations of motion, motion along a straight line, and motion with constant acceleration. It discusses topics like displacement, velocity, acceleration, and their relationships. Examples are provided to demonstrate calculating variables like position, velocity and acceleration from graphs or equations of motion. Kinematic formulas are also introduced for problems involving constant acceleration.
This document discusses key terms and equations related to rectilinear motion. Rectilinear motion refers to motion along a straight line. Kinematics deals with the motion of bodies without considering forces. Important concepts discussed include displacement, average and instantaneous velocity, acceleration, distance traveled, and equations of motion. Graphical representations of motion using velocity-time graphs are also presented for different scenarios including uniform velocity, variable velocity from rest to a final velocity, and variable velocity between two points.
Curvilinear motion occurs when a particle moves along a curved path.
Since this path is often described in three dimensions, vector analysis will
be used to formulate the particle's position, velocity, and acceleration
1) The document presents three theoretical physics problems involving spinning balls, charged particles in loops, and laser cooling of atoms.
2) Problem 1 considers a spinning ball falling and rebounding, calculating the rebound angle, horizontal distance traveled, and minimum spin rate. Problem 2 analyzes relativistic effects on charged particles in a moving loop in an electric field.
3) Problem 3 describes using lasers to cool atoms by resonant absorption. It calculates the laser frequency needed, velocity range absorbed, direction change upon emission, maximum velocity decrease, number of absorption events to slow to zero velocity, and distance traveled during cooling.
C3L1_Tangent Lines and Velocity_G12A.pptxkaran11dhawan
This document provides rules and learning objectives for an online Grade 12 mathematics class. It discusses tangent lines to curves and how to find the equation of a tangent line using limits. It provides examples of using derivatives to find average and instantaneous velocity. The document explains how to calculate average velocity from displacement over time and how the sign of instantaneous velocity indicates direction of motion. It discusses the units of instantaneous rate of change and how the limit represents the slope of the tangent line.
This chapter discusses kinematics of linear motion, including:
1) It defines kinematics as the study of motion without considering forces, and describes linear and projectile motion.
2) It introduces key concepts such as displacement, speed, velocity, acceleration and their relationships. Equations for these quantities under constant and uniformly accelerated motion are provided.
3) It describes motion under constant acceleration due to gravity, known as freely falling bodies, and provides the relevant equations.
The document discusses kinematics of particles, including rectilinear and curvilinear motion. It defines key concepts like displacement, velocity, and acceleration. It presents equations for calculating these values for rectilinear motion under different conditions of acceleration, such as constant acceleration, acceleration as a function of time, velocity, or displacement. Graphical interpretations are also described. An example problem is worked through to demonstrate finding velocity, acceleration, and displacement at different times for a particle moving in a straight line.
Learn Online Courses of Subject Introduction to Civil Engineering and Engineering Mechanics. Clear the Concepts of Introduction to Civil Engineering and Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/introduction-to-civil-engineering-and-engineering-mechanics
Chapter 2 introduces the concepts of kinematics including reference frames, displacement, velocity, acceleration, and motion with constant acceleration. Equations are derived that relate displacement, velocity, acceleration, and time for objects undergoing constant acceleration. Near the Earth's surface, the acceleration due to gravity is approximately 9.80 m/s2, so these equations can be applied to analyze falling or projected objects using this value of acceleration.
1) The document describes curvilinear motion and how to analyze the motion of objects moving along curved paths using rectangular components.
2) It provides examples of how to determine the velocity and acceleration of planes in formation and a roller coaster car moving along a fixed helical path using their x, y, and z coordinates.
3) The document also gives an example problem solving for the collision point and speeds of two particles moving along curved paths given their position vectors as functions of time.
1) The document discusses curvilinear motion and describes how to analyze motion along curved paths using normal (n) and tangential (t) coordinate axes located at the particle.
2) Velocity is always tangent to the path and acceleration has both normal and tangential components.
3) Two examples are provided to illustrate the analysis procedure for determining velocity and acceleration directions and magnitudes at specific points along curved paths.
This document discusses kinematics in normal and tangential coordinates for curvilinear motion. It describes how to calculate the tangential and normal components of velocity and acceleration for a particle moving along a curved trajectory. The tangential component represents changes in speed, while the normal component represents changes in direction. Equations are provided to calculate acceleration along the tangent (at) and normal (an) directions in terms of velocity, radius of curvature, and derivatives. Examples are given for constant acceleration and trajectories defined as functions of position.
This document discusses curvilinear motion and kinematics. It introduces position vectors, path coordinates, velocity vectors, and acceleration vectors for particles moving in three-dimensional space. Key concepts covered include defining the position vector r(t) from a reference point to the particle, the instantaneous velocity vector v as the time derivative of r(t), and the acceleration vector a as the time derivative of v. When working in Cartesian coordinates, the derivatives of vector components are simply the derivatives of the individual x, y, z components.
1) Acceleration is a measure of how quickly velocity changes. It can represent a change in speed, direction, or both.
2) The slope of a velocity-time graph represents acceleration. The area under the graph represents displacement.
3) For objects experiencing uniform acceleration, displacement can be calculated using: s = ut + (1/2)at^2
This document presents a course on air standard cycles and actual thermodynamic cycles for engines. The course objectives are to explain gas power cycles, efficiency, mean effective pressure, and describe the working principles of constant pressure cycles and actual/theoretical P-V diagrams of internal combustion engines. Various thermodynamic processes involved in engine cycles are introduced, including constant pressure, constant volume, isothermal, and reversible adiabatic processes. The air standard Otto, Diesel, and Dual cycles are described and compared to the actual cycles of spark ignition, early compression ignition, and modern compression ignition engines. The ideal Brayton cycle for gas turbines is also introduced and compared to the actual gas turbine cycle.
This document outlines the assignment instructions for a mechanical vibration group project. Students must form groups of two, select two questions from five given problems, and submit their assignment by April 08, 2021. Papers submitted within two days of the due date will have a 10% deduction applied.
This document discusses various nontraditional machining processes that remove material using mechanical, thermal, electrical, or chemical energy rather than sharp cutting tools. It covers mechanical processes like ultrasonic machining and water jet cutting, electrochemical processes like electrochemical machining, thermal processes like electric discharge machining and laser beam machining, and chemical machining. For each process, it provides details on working principles, applications, advantages, and limitations.
This document summarizes sorting and recursion concepts in Python. It discusses common sorting algorithms like selection sort and bubble sort. It provides code examples to sort a list using these algorithms and the built-in sorted() and list.sort() functions. It then explains recursion through examples like calculating factorials and Fibonacci numbers recursively. Code snippets are given to implement recursive functions to calculate these values. Finally, it discusses using recursion to check for palindromes in a string.
Manufacture of pig iron involves feeding iron ore, coke and limestone into the top of a blast furnace continuously. Air is blown into the bottom so that chemical reactions take place, melting the iron which is tapped from the bottom as molten pig iron. Pig iron is an impure form of iron containing carbon and other impurities. It is classified into grey pig iron which is high in silicon and forms graphite, and white pig iron which is low in silicon and forms cementite.
The document discusses the history and development of artificial intelligence over the past 70 years. It outlines some of the key milestones in AI research from the early work in the 1950s to modern advances in machine learning using neural networks. The document suggests AI has made significant progress but still has many challenges to overcome to match human-level intelligence.
1) The document discusses heat, work, and the first law of thermodynamics. It defines heat and work as the two types of energy transfer across boundaries of closed systems.
2) The first law of thermodynamics, also called the law of conservation of energy, states that the total energy of a system remains constant, with increases in internal energy equal to net heat and work transfers.
3) Specific examples are provided to illustrate the first law for closed systems undergoing various processes like heating, cooling, and adiabatic changes with and without work. Formulas are derived for calculating internal energy changes based on the first law.
1) This document discusses heat, work, and the first law of thermodynamics. It defines heat and work as the two ways energy can transfer across the boundary of a closed system, with heat transferring due to a temperature difference and work occurring from a force acting through a distance.
2) The first law of thermodynamics states that the change in a system's internal energy is equal to the net heat transferred to the system plus the net work done by the system. This is illustrated with examples of processes involving only heat transfer, where the energy change equals the net heat.
3) Different types of thermodynamic processes are examined, including isobaric, isochoric, isothermal, and poly
This document discusses properties of pure substances. It defines a pure substance as having a fixed chemical composition throughout, such as water or nitrogen. A pure substance can exist in different phases like solid, liquid, or gas. The properties of a pure substance, like density and specific volume, depend on its phase and conditions of temperature and pressure. Phase change processes, like boiling and condensation, involve absorbing or releasing heat. Property diagrams are used to understand relationships between temperature, pressure, and phase for a pure substance. An ideal gas is defined by the ideal gas law, but real gases deviate from this behavior at high pressures.
This document provides an overview of key concepts in thermodynamics. It defines thermodynamics as the science of energy and discusses the first and second laws of thermodynamics. The document outlines different types of thermodynamic systems (closed, open, isolated), properties (intensive, extensive), states, equilibrium, and processes (steady flow, quasi-static). It also defines temperature, forms of energy (kinetic, potential, internal), and dimensional analysis. Application areas covered include propulsion, HVAC, computers and various engineering systems.
TIME DIVISION MULTIPLEXING TECHNIQUE FOR COMMUNICATION SYSTEMHODECEDSIET
Time Division Multiplexing (TDM) is a method of transmitting multiple signals over a single communication channel by dividing the signal into many segments, each having a very short duration of time. These time slots are then allocated to different data streams, allowing multiple signals to share the same transmission medium efficiently. TDM is widely used in telecommunications and data communication systems.
### How TDM Works
1. **Time Slots Allocation**: The core principle of TDM is to assign distinct time slots to each signal. During each time slot, the respective signal is transmitted, and then the process repeats cyclically. For example, if there are four signals to be transmitted, the TDM cycle will divide time into four slots, each assigned to one signal.
2. **Synchronization**: Synchronization is crucial in TDM systems to ensure that the signals are correctly aligned with their respective time slots. Both the transmitter and receiver must be synchronized to avoid any overlap or loss of data. This synchronization is typically maintained by a clock signal that ensures time slots are accurately aligned.
3. **Frame Structure**: TDM data is organized into frames, where each frame consists of a set of time slots. Each frame is repeated at regular intervals, ensuring continuous transmission of data streams. The frame structure helps in managing the data streams and maintaining the synchronization between the transmitter and receiver.
4. **Multiplexer and Demultiplexer**: At the transmitting end, a multiplexer combines multiple input signals into a single composite signal by assigning each signal to a specific time slot. At the receiving end, a demultiplexer separates the composite signal back into individual signals based on their respective time slots.
### Types of TDM
1. **Synchronous TDM**: In synchronous TDM, time slots are pre-assigned to each signal, regardless of whether the signal has data to transmit or not. This can lead to inefficiencies if some time slots remain empty due to the absence of data.
2. **Asynchronous TDM (or Statistical TDM)**: Asynchronous TDM addresses the inefficiencies of synchronous TDM by allocating time slots dynamically based on the presence of data. Time slots are assigned only when there is data to transmit, which optimizes the use of the communication channel.
### Applications of TDM
- **Telecommunications**: TDM is extensively used in telecommunication systems, such as in T1 and E1 lines, where multiple telephone calls are transmitted over a single line by assigning each call to a specific time slot.
- **Digital Audio and Video Broadcasting**: TDM is used in broadcasting systems to transmit multiple audio or video streams over a single channel, ensuring efficient use of bandwidth.
- **Computer Networks**: TDM is used in network protocols and systems to manage the transmission of data from multiple sources over a single network medium.
### Advantages of TDM
- **Efficient Use of Bandwidth**: TDM all
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
2. Learning outcomes
Able to explain rectilinear motion and calculate dynamics problem using analytical and graphical
method.
Able to explain Plane Curvilinear Motion and calculate dynamics problem using analytical.
3. 2. INTRODUCTION
KINEMATICS :is the branch of dynamics which describes the motion of bodies without reference to
the forces which either cause the motion or are generated as a result of the motion.
Kinematics is often described as the “geometry of motion.”
Knowledge of kinematics is a prerequisite to kinetics.
A particle is a body whose physical dimensions are so small compared
with the radius of curvature of its path.
Here we may treat the motion of the particle as that of a point.
Figure 2.1. Shows a particle P moving along some general path in space.
4. If the particle is confined to a specified path, as with a bead sliding along a fixed wire, its motion is
said to be constrained.
If there are no physical guides, the motion is said to be unconstrained.
E.g. A small rock tied to the end of a string and whirled in a circle undergoes constrained motion
until the string breaks, after this instant its motion is become unconstrained.
The position of particle P at any time t can be described by specifying its:
Rectangular coordinates x, y, z,
Its cylindrical coordinates r, 𝜃, z, or
Its spherical coordinates R, 𝜃, ɸ.
5. The motion of P can also be described by measurements along the tangent t and normal n to the
curve.
The direction of n lies in the local plane of the curve.
These last two measurements are called path variables.
The motion of particles (or rigid bodies) can be described by using coordinates measured from
fixed reference axes (absolute-motion analysis) or
Using coordinates measured from moving reference axes (relative-motion analysis).
We restrict our attention in the first part of this chapter to the case of plane motion, where all
movement occurs in or can be represented as occurring in a single plane.
6. 2.2. RECTILINEAR MOTION
The position of P at any instant of time t can be specified by its distance s measured from some
convenient reference point O fixed on the line.
At time 𝑡 + Δ𝑡 the particle has moved to P′ and its coordinate becomes 𝑠 + Δ𝑠, which is motion
along a straight line.
The change in the position coordinate during the interval Δ𝑡 is called the displacement Δ𝑠 of the
particle.
The displacement would be negative if the particle moved in the negative s-direction.
7. 2.2.1 Velocity and Acceleration
The average velocity of the particle during the interval Δ𝑡 is the displacement divided by the
time interval or
𝐯𝑎𝑣𝑒 =
∆𝐒
∆𝑡
… … … … … … … … … … 𝑒𝑞. 1
Thus, the velocity is the time rate of change of the position coordinate s.
The velocity is positive or negative depending on whether the corresponding displacement is
positive or negative.
As Δ𝑡 becomes smaller and approaches zero in the limit, the average velocity approaches the
instantaneous velocity of the particle, which is:
𝐯𝑖𝑛𝑠𝑡 = lim
∆𝑡→0
∆𝑆
∆𝑡
𝑜𝑟 v =
𝑑𝑆
𝑑𝑡
= 𝑆 … … … … … … … . 𝑒𝑞. 2
8. The average acceleration of the particle during the interval Δ𝑡 is the change in its velocity divided
by the time interval or
𝑎𝑎𝑣𝑒 =
∆𝐯
∆𝑡
… … … … … … … … … . 𝑒𝑞. 3
As Δ𝑡 becomes smaller and approaches zero in the limit, the average acceleration approaches the
instantaneous acceleration of the particle, which is
𝑎𝑖𝑛𝑠𝑡 = lim
∆𝑡→0
∆𝐯
∆𝑡
𝑜𝑟 𝒂 =
𝑑𝐯
𝑑𝑡
= 𝐯 𝑜𝑟 𝒂 =
𝑑2
𝑆
𝑑𝑡2
= 𝑆 … … … … … … … … . 𝑒𝑞. 4
The acceleration is positive or negative depending on whether the velocity is increasing or
decreasing.
9. If the particle is slowing down, the particle is said to be decelerating.
The sense of the vector along the path is described by a plus or minus sign.
By eliminating the time 𝑑𝑡 between Eq. 2 and Eq. 4, we obtain a differential equation relating
displacement, velocity, and acceleration.
This equation is:
𝑑𝑡 =
𝑑𝑣
𝑎
=
𝑑𝑠
𝑣
𝑣 𝑑𝑣 = 𝑎 𝑑𝑠 or 𝑠𝑑𝑣 = 𝑠 𝑑𝑠 … … … … … … … … … 𝑒𝑞. 5
Problems in rectilinear motion involving finite changes in the motion variables are solved by
integration of these basic differential relations.
10. 2.2.3. Graphical Interpretations
Interpretation of the differential equations governing rectilinear motion is considerably clarified
by representing the relationships among s, v, a, and t graphically.
By constructing the tangent to the curve at any time t, we obtain
the slope, which is the velocity:
𝑣 = 𝑑𝑠/𝑑𝑡.
Thus, the velocity can be determined at all points on the curve
and plotted against the corresponding time.
11. The area under the v-t curve during time 𝑑𝑡 is 𝑣𝑑𝑡, which from
Eq. 2 is the displacement ds.
Consequently, the net displacement of the particle during the
interval from 𝑡1 to 𝑡2is the corresponding area under the curve,
which is:
𝑠1
𝑠2
𝑑𝑠 =
𝑡1
𝑡2
𝑣𝑑𝑡 𝑜𝑟 𝑠2−𝑠1= (𝑎𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝑣 − 𝑡 𝑐𝑢𝑟𝑣𝑒)
12. 𝑣1
𝑣2
𝑑𝑣 =
𝑡1
𝑡2
𝑎𝑑𝑡 𝑜𝑟 𝑣2−𝑣1= (𝑎𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝑎 − 𝑡 𝑐𝑢𝑟𝑣𝑒)
Area under the a-t curve during time 𝑑𝑡 is 𝑎𝑑𝑡, which, from the
first of Eqs. 2 /2, is 𝑑𝑣.
Thus, the net change in velocity between 𝑡1 𝑎𝑛𝑑 𝑡2 is the
corresponding area under the curve, which is
13. 𝑣1
𝑣2
𝑣𝑑𝑣 =
𝑠1
𝑠2
𝑎𝑑𝑠 𝑜𝑟
1
2
(𝑣2
2
− 𝑣1
2
) = 𝐴𝑟𝑒𝑎 𝑢𝑛𝑑𝑒𝑟 𝑎 − 𝑠 𝑐𝑢𝑟𝑣𝑒)
When the acceleration a is plotted as a function of the position
coordinate s.
The area under the curve during a displacement ds is ads, which,
from Eq. 2 /3, is 𝑣𝑑𝑣 = 𝑑(𝑣2
/2).
Thus, the net area under the curve between position coordinates
𝑠1 and 𝑠2is:
14. 2.2.4. ANALYTICAL INTEGRATION
(a) Constant Acceleration.
When acceleration a is constant, the first of Eqs. 2 and 2 /3 can be integrated directly.
For simplicity with s = 𝑠0, v = 𝑣0, and t =0 designated at the beginning of the interval, then for a
time interval t the integrated equations become:
𝑣0
𝑣
𝑑𝑣 = 𝑎
0
𝑡
𝑑𝑡 𝑜𝑟 𝑣 = 𝑣0 +𝑎𝑡
𝑑𝑡 = 𝑑𝑠/𝑣 = 𝑑𝑣/a.
𝑣𝑑𝑣 = 𝑎𝑑𝑠.
𝑣0
𝑣
𝑣𝑑𝑣 = 𝑎
𝑠0
𝑠
𝑑𝑠 𝑜𝑟 𝑣2
= 𝑣0
2
+ 2𝑎(𝑠−𝑠0)
15. 𝑠0
𝑠
𝑑𝑠 =
0
𝑡
(𝑣0+𝑎𝑡)𝑑𝑡 𝑜𝑟 𝑠 = 𝑠0 + 𝑣0𝑡 +
1
2
𝑎𝑡2
Substitution of the integrated expression for 𝑣 into Eq. 2 and integration with respect to t gives:
These relations are necessarily restricted to the special case where the acceleration is constant only.
Caution: A common mistake is to use these equations for problems involving variable
acceleration, where they do not apply.
16. (b) Acceleration Given as a Function of Time, a =ƒ(t).
Substitution of the function into the first of Eqs. 2 gives ƒ(𝑡) = 𝑑𝑣 /𝑑𝑡. Multiplying by 𝑑𝑡
separates the variables and permits integration. Thus,
From this integrated expression for v as a function of t, the position coordinate s is obtained by
integrating Eq. 2, which, in form, would be
𝑠0
𝑠
𝑑𝑠 =
0
𝑡2
𝑣𝑑𝑡 𝑜𝑟 𝑠 = 𝑠0 +
0
𝑡2
𝑣𝑑𝑡
𝑣0
𝑣
𝑑𝑣 =
0
𝑡2
𝑓(𝑡)𝑑𝑡 𝑜𝑟 𝑣 = 𝑣0 +
0
𝑡2
𝑓(𝑡)𝑑𝑡
17. (c) Acceleration Given as a Function of Velocity, a =ƒ(v).
Substitution of the function into the first of Eqs. 2 gives:
ƒ(𝑣) = 𝑑𝑣/𝑑𝑡, which permits separating the variables and integrating. Thus,
𝑡 =
0
𝑡2
𝑑𝑡 =
𝑣0
𝑣
𝑑𝑣
𝑓(𝑣)
18. (d) Acceleration Given as a Function of Displacement, a =ƒ(s).
Substituting the function into Eq. 2 and integrating give the form:
𝑣0
𝑣
𝑣𝑑𝑣 =
𝑠0
𝑠
𝑓(𝑠)𝑑𝑠 𝑜𝑟 𝑣2
= 𝑣0
2
+ 2
𝑠0
𝑠
𝑓(𝑠)𝑑𝑠
19. Example 1
The position coordinate of a particle which is confined to move along a straight line is given by
s =2𝑡3
−24t +6, where s is measured in meters from a convenient origin and tis in seconds. Determine
(a) the time required for the particle to reach a velocity of 72 m /s from its initial condition at t =0, (b)
the acceleration of the particle when v =30 m /s, and (c) the net displacement of the particle during the
interval from t =1 s to t =4 s.
20.
21. 1. The velocity of a particle which moves along the s-axis is given by v =2 − 4t +5𝑡(3/2)
, where tis in
seconds and v is in meters per second. Evaluate the position s, velocity v, and acceleration a when
t =3s. The particle is at the position 𝑠0 = 3𝑚 when t =0.
2. The acceleration of a particle is given by a =2t −10, where a is in meters per second squared and tis
in seconds. Determine the velocity and displacement as functions of time. The initial displacement
at t = 0 is 𝑠0= −4 m, and the initial velocity is 𝑣0=3 m /s.
3. From experimental data, the motion of a jet plane while traveling along a runway is defined by the
v–t graph. Construct the s–t and a–t graphs for the motion. When t =0, s =0.
22.
23.
24.
25.
26.
27.
28.
29.
30. 2.3. General Curvilinear Motion
Curvilinear motion occurs when a particle moves along a curved path which lies in a single plane.
This motion is a special case of the more general three dimensional motion.
Vector analysis will be used to formulate the particle’s position, velocity, and acceleration.
31. 2.3.1. Position
Consider a particle located at a point on a space curve defined by the path function s(t).
The position of the particle, measured from a fixed point O, will be designated by the position
vector r = r(t).
Notice that both the magnitude and direction of this vector will change as the particle moves
along the curve.
32. 2.3.2. Displacement
Suppose that during a small time interval ∆𝑡 the particle moves a distance ∆s along the curve to a
new position, defined by:
𝐫′ = 𝐫 + ∆𝐫
The displacement ∆𝐫 represents the change in the particle’s position and is determined by vector
subtraction; i.e., 𝐫′
− 𝐫 = ∆𝐫
Fig. 2.1
33. 2.3.3. Velocity
During the time ∆t the average velocity of the particle between A and A′ is defined as
𝐯𝐚𝐯𝐞 =
∆𝐫
∆t
which is a vector whose direction is that of Δr and whose magnitude is the magnitude of Δr divided
by Δt.
The average speed of the particle between A and A′ is the scalar quotient Δs/Δt.
34. The instantaneous velocity v of the particle is defined as the limiting value of the average velocity as
the time interval approaches zero. Thus,
vinst = lim
∆t→0
∆𝐫
∆𝐭
We observe that the direction of Δr approaches that of the tangent to the path as Δt approaches zero
and, thus, the velocity 𝐯 is always a vector tangent to the path.
35. We now extend the basic definition of the derivative of a scalar quantity to include a vector quantity
and write:
𝐯 =
𝐝𝐫
𝐝𝐭
= 𝐫
The derivative of a vector is itself a vector having both a magnitude and a direction.
The magnitude of v is called the speed and is the scalar:
𝒗 = 𝐯 =
𝐝𝒔
𝐝𝐭
= 𝒔
36. From Fig. 2.2. denote the velocity of the particle at A by the tangent vector v and the velocity at A′ by
the tangent v′.
• Clearly, there is a vector change in the velocity during the time Δt. The velocity v at A plus
(vectorially) the change Δv must equal the velocity at A′, so we can write:
𝐯′ − 𝐯 = 𝚫𝐯
• Inspection of the vector diagram shows that Δv depends both on the change in magnitude (length) of
v and on the change in direction of v.
• These two changes are fundamental characteristics of the derivative of a vector.
37. 2.3.4. Acceleration
The average acceleration of the particle between A and A′ is defined as Δv/Δt, which is a vector
whose direction is that of Δv.
The magnitude of this average acceleration is the magnitude of Δv divided by Δt.
The instantaneous acceleration a of the particle is defined as the limiting value of the average
acceleration as the time interval approaches zero. Thus,
𝐚𝒊𝒏𝒔𝒕 = lim
∆t→0
∆𝐯
∆t
38. 𝒂 =
𝐝𝐯
𝐝𝐭
= 𝐯 = 𝒔
By definition of the derivative, then, we write
The acceleration a, then, includes the effects of both the change in magnitude of v and the change of
direction of v.
In general, that the direction of the acceleration of a particle in curvilinear motion is neither tangent
to the path nor normal to the path.
We do observe, however, that the acceleration component which is normal to the path points toward
the centre of curvature of the path.
39. 4.3.5. RECTANGULAR COORDINATES (x-y)
This system of coordinates is particularly useful for describing motions where the x- and y-
components of acceleration are independently generated or determined.
The resulting curvilinear motion is then obtained by a vector combination of the x- and y-
components of the position vector, the velocity, and the acceleration.
𝐫 = 𝐱𝐢 + 𝐲𝐣
𝐯 = 𝐫 = 𝐱𝐢 + 𝐲𝐣
𝒂 = 𝐯 = 𝒓 = 𝒙𝐢 + 𝐲𝐣
Fig. 2.2
40. As we differentiate with respect to time, we observe that the time derivatives of the unit vectors are
zero because their magnitudes and directions remain constant.
The scalar values of the components of v and a are merely:
𝒗𝒙 = 𝒙, 𝒗𝒚 = 𝒚 and 𝒂𝒙 = 𝒗𝒙 = 𝒙 , 𝒂𝒚 = 𝒗𝒚 = 𝒚
(As drawn in Fig. 2.2., 𝒂𝒙 is in the negative x-direction, so that 𝒙 would be a negative number.)
As observed previously, the direction of the velocity is always tangent to the path, and from the
figure it is clear that:
𝑣2
= 𝑣𝑥
2
+ 𝑣𝑦
2
𝑣 = (𝑣𝑥
2
+ 𝑣𝑦
2
) 𝑡𝑎𝑛𝜃 =
𝑣𝑦
𝑣𝑥
𝑎2 = 𝑎𝑥
2 + 𝑎𝑦
2 𝑎 = (𝑎𝑥
2
+ 𝑎𝑦
2
)
41. Example1: The velocity of a particle is v = 3i + (6 - 2t)j m/s, where tis in seconds. If r = 0 when t = 0,
determine the displacement of the particle during the time interval t =1 s to t =3 s.
Example 2: At time t =10 s, the velocity of a particle moving in the x-y plane is v = 0.1i +2j m /s. By
time t =10.1s, its velocity has become −0.1i +1.8j m /s. Determine the magnitude 𝑎𝑎𝑣 of its average
acceleration during this interval and the angle 𝜃 made by the average acceleration with the positive x-
axis.
43. 2.3.6. Projectile Motion
An important application of two-dimensional kinematic theory is the problem of projectile motion.
For a first treatment of the subject:
We neglect aerodynamic drag,
The curvature and rotation of the earth, and
We assume that the altitude change is small enough so that the acceleration due to gravity can be
considered constant.
With these assumptions, rectangular coordinates are useful for the trajectory analysis.
For the axes shown in Fig. 2.3., the acceleration components are:
𝒂𝒙 = 𝟎 𝒂𝒏𝒅 𝒂𝒚 = −𝒈 = −𝟗. 𝟖𝟏m/𝒔𝟐 𝒐𝒓 − 𝟑𝟐. 𝟐 𝒇𝒕/𝒔𝟐
The horizontal component of velocity always remains constant during the motion.
44. 𝒗𝒙 = (𝒗𝒙)𝟎 and
𝒗𝒚 = (𝒗𝒚)𝟎 − 𝑔𝑡
𝒙 = 𝒙𝟎 + (𝒗𝒙)𝟎𝑡
𝑦 = 𝑦0 + (𝒗𝒚)0𝑡 −
1
2
𝑔𝑡2
𝒗𝒚
2 = (𝒗𝒚)0
2
−2𝑔(𝑦−𝑦0)
Integration of these accelerations follows the results obtained previously for constant acceleration
and yields:
Fig. 2.3
Horizontal Motion:
Vertical Motion:
45. 1. The velocity of the water jet discharging from the orifice can be
obtained from 𝑣 = 2𝑔ℎ ,where ℎ = 2𝑚 is the depth of the orifice
from the free water surface. Determine the time for a particle of
water leaving the orifice to reach point B and the horizontal distance
x where it hits the surface.
2. The projectile is launched with a velocity . Determine the range R,
the maximum height h attained, and the time of flight. Express the
results in terms of the angle and. The acceleration due to gravity is g.
46.
47.
48.
49.
50. 2.4. NORMAL AND TANGENTIAL COORDINATES (n-t)
One of the common descriptions of curvilinear motion uses path variables, which are
measurements made along the tangent t and normal n to the path of the particle.
These coordinates provide a very natural description for curvilinear motion and are frequently the
most direct and convenient coordinates to use.
The positive direction for n at any position is always taken toward the centre of curvature of the
path.
As seen from Fig. 2.4, the positive n-direction will shift from one side of the curve to the other side
if the curvature changes direction.
Fig. 2.4
51. 2.4.1. Velocity and Acceleration
We now use the coordinates n and t to describe the velocity v and acceleration a for the curvilinear
motion of a particle.
For this purpose, we introduce unit vectors 𝑒𝑛 in the n-direction and 𝑒𝑡 in the t-direction, for the
position of the particle at point A on its path.
During a differential increment of time dt, the particle moves a differential distance ds along the
curve from A to A′.
With the radius of curvature of the path at this position designated by 𝜌, we see that ds = 𝜌d𝛽,
where 𝛽 is in radians.
52. It is unnecessary to consider the differential change in 𝜌 between A and A′.
Thus, the magnitude of the velocity can be written v = ds/dt = 𝜌d𝛽/dt, and we can write the velocity
as the vector
𝐯 = 𝑣𝑒𝑡 = 𝜌𝛽𝑒𝑡
The acceleration is a vector which reflects both the change in magnitude and the change in direction
of v.
𝑎 =
𝑑v
𝑑𝑡
=
𝑑(𝑣𝑒𝑡)
𝑑𝑡
= 𝑣𝑒𝑡 + 𝑣𝑒𝑡
53. Where the unit vector 𝑒𝑡 now has a nonzero derivative because its direction changes.
54. The direction of d𝑒𝑡 is given by 𝑒𝑛. Thus, we can write d𝑒𝑡 = 𝑒𝑛 d𝛽. Dividing by d𝛽 gives
𝑒𝑛 =
𝑑𝑒𝑡
𝑑𝛽
𝑒𝑡 = 𝛽𝑒𝑛
55. With the substitution of Eq. 2 and 𝛽 from the relation v = 𝜌𝛽, for the acceleration becomes
𝑎 =
𝑣2
𝜌
𝑒𝑛 + 𝑣𝑒𝑡
𝑎𝑛 =
𝑣2
𝜌
= 𝜌𝛽2
= 𝑣𝛽
𝑎𝑡 = 𝑣 = 𝑠 =
𝑑(𝜌𝛽)
𝑑𝑡
= 𝜌𝛽 + 𝜌𝛽
𝑎 = (𝑎𝑡
2
+ 𝑎𝑛
2
)
56. 2.4.2. Circular Motion
Circular motion is an important special case of plane curvilinear motion where the radius of curvature
𝜌 becomes the constant radius 𝑟 of the circle and the angle 𝛽 is replaced by the angle 𝜃 measured from
any convenient radial reference to 𝐎𝐏, Fig. The velocity and the acceleration components for the
circular motion of the particle 𝐏 become:
𝑣 = 𝑟𝜃
𝑎𝑛 =
𝑣2
𝑟
= 𝑟𝜃2
= 𝑣𝜃
𝑎𝑡 = 𝑣 = 𝑟𝜃
57. Example: A test car starts from rest on a horizontal circular track of 80-m radius and increases its
speed at a uniform rate to reach 100 km/h in 10 seconds. Determine the magnitude a of the total
acceleration of the car 8 seconds after the start.
58.
59.
60.
61.
62. Example: The car C increases its speed at the constant rate of 1.5 𝑚/𝑠2 as it rounds the curve
shown. If the magnitude of the total acceleration of the car is 2.5 𝑚/𝑠2 at point A where the
radius of curvature is 200m, compute the speed v of the car at this point. Answer 𝑣 = 20 𝑚/𝑠
63.
64.
65.
66.
67.
68. 2.5. POLAR COORDINATES (𝒓−𝜽)
We now consider the third description of plane curvilinear motion, namely, polar coordinates where
the particle is located by the radial distance 𝑟 from a fixed point and by an angular measurement 𝜃
to the radial line.
Polar coordinates are particularly useful:
when a motion is constrained through the control of a radial distance and an angular position or
when an unconstrained motion is observed by measurements of a radial distance and an angular
position.
69. The polar coordinates r and 𝜃 which locate a particle traveling on a curved path.
An arbitrary fixed line, such as the x-axis, is used as a reference for the measurement of 𝜃.
Unit vectors 𝑒𝑟 and 𝑒𝜃 are established in the positive r- and 𝜃 directions, respectively.
The position vector r to the particle at A has a magnitude equal to the radial distance r and a
direction specified by the unit vector 𝑒𝑟.
Thus, we express the location of the particle at A by the vector
𝐫 = 𝑟𝑒𝑟
`
70. Time Derivatives of the Unit Vectors
𝑑𝑒𝑟
𝑑𝜃
= 𝑒𝜃 and
𝑑𝑒𝜃
𝑑𝜃
= −𝑒𝑟 …………………………eq 5.2
𝑒𝑟 = 𝜃𝑒𝜃 𝑎𝑛𝑑 𝑒𝜃 = −𝜃𝑒𝑟……………………………………..eq 5.3
71. Velocity
We are now ready to differentiate 𝐫 = 𝑟𝑒𝑟 with respect to time. Using the rule for differentiating
the product of a scalar and a vector gives:
𝐕 = 𝐫 = 𝒓𝑒𝑟 + 𝑟𝒆𝑟
With the substitution of 𝒆𝑟from Eq. 5.3 the vector expression for the velocity becomes
𝐕 = 𝒓𝑒𝑟 + 𝑟𝜽𝑒𝜃
𝒗𝒓 = 𝐫
𝒗𝜽 = 𝑟𝛉
𝑣 = (𝑣𝑡
2
+ 𝑣𝜃
2
)
The r-component of v is merely the rate at which the vector r stretches.
The 𝜃-component of v is due to the rotation of r.
72. Acceleration
We now differentiate the expression for v to obtain the acceleration 𝐚 = 𝐕.
Note that the derivative of 𝑟𝜽𝑒𝜃 will produce three terms, since all three factors are variable.
Thus, continuously
𝐚 = 𝐕 = (𝒓𝑒𝑟 + 𝒓𝒆𝒓) + (𝒓𝜽𝑒𝜃 + 𝑟𝜽𝑒𝜃 + 𝑟𝜽𝒆𝜽)
𝐚 = (𝒓 − 𝒓𝜽𝟐)𝑒𝑟 + (𝑟𝜽 + 2𝒓𝜽)𝑒𝜃
𝒂𝒓 = 𝒓 − 𝒓𝜽𝟐
𝒂𝜽 = 𝑟𝜽 + 2𝒓𝜽
𝑎 = (𝑎𝑟
2
+ 𝑎𝜃
2
)
73. Geometric Interpretation
Polar coordinates developed to show the velocity vectors and their r- and 𝜃-components at position
A and at position A′ after an infinitesimal movement.
Each of these components undergoes a change in magnitude and direction.
(a) Magnitude Change of 𝐯𝒓: This change is simply the increase in length of 𝒗𝒓 or d𝒗𝒓 = d𝐫, and the
corresponding acceleration term is d𝐫/dt = 𝒓 in the positive r-direction.
74. (b) Direction Change of 𝑽𝒓: The magnitude of this change is seen from the figure to be 𝑣𝑟𝑑𝜃 = r 𝑑𝜃,
and its contribution to the acceleration becomes r𝑑𝜃/𝑑𝑡 = 𝑟 which is in the positive 𝜃-direction.
(c) Magnitude Change of 𝑽𝜽: This term is the change in length of 𝑉𝜃 or d(𝑟𝜃), and its contribution to
the acceleration is d(r𝜃)/dt = r𝜃 + r𝜃 and is in the positive 𝜃 -direction.
(d) Direction Change of 𝑽𝜽:The magnitude of this change is 𝑣𝜃𝑑𝜃 = 𝑟𝜃𝑑𝜃 and the corresponding
acceleration term is observed to be 𝑟𝜃 (d𝜃/dt)=𝑟𝜃2
in the negative r-direction.
75. Circular Motion
For motion in a circular path with r constant, the components of become simply.
76. Example: Rotation of the radially slotted arm is governed by 𝜃 = 0.2𝑡 + 0.02𝑡3
, where 𝜃 is in
radians and t is in seconds. Simultaneously, the power screw in the arm engages the slider B and
controls its distance from O according to 𝑟 = 0.2 + 0.04𝑡2
, where r is in meters and t is in seconds.
Calculate the magnitudes of the velocity and acceleration of the slider for the instant when t =3 s.
77.
78.
79. Example: Rotation of bar OA is controlled by the lead screw which imparts a horizontal velocity v
to collar C and causes pin P to travel along the smooth slot. Determine the values of 𝐫 and 𝛉, where
𝑟 = 𝑂𝑃, if h =160 mm, x =120 mm, and v =25 mm /s at the instant represented.
80.
81.
82.
83. Example: At the bottom of a loop in the vertical (𝑟−𝜃) plane at an altitude of 400 m, the
airplane P has a horizontal velocity of 600 km/h and no horizontal acceleration. The radius of
curvature of the loop is 1200 m. For the radar tracking at O, determine the recorded values of r
and 𝜃 for this instant.
84.
85.
86. RELATIVE MOTION (TRANSLATING AXES)
In the previous topics of this chapter, we have described particle motion using coordinates referred
to fixed reference axes.
However, there are many engineering problems for which the analysis of motion is done by using
measurements made with respect to a moving reference system.
These measurements, when combined with the absolute motion of the moving coordinate system,
enable us to determine the absolute motion in question. This approach is called a relative-motion
analysis.
87. Note that in the discussion:
We will confine our attention to moving reference systems which translate but do not rotate.
We will also confine our attention here to relative-motion analysis for plane motion.
88. Vector Representation
Now consider two particles A and B which may have separate curvilinear motions in a given
plane or in parallel planes.
We will arbitrarily attach the origin of a set of translating (nonrotating) axes x-y to particle B
and observe the motion of A from our moving position on B.
The position vector of A as measured relative to the frame 𝑥−𝑦 is 𝒓𝑨/𝑩 = 𝒙𝒊 + 𝒚𝒋, where the
subscript notation “A/B” means “A relative to B” or “A with respect to B.”
89. The unit vectors along the x- and y-axes are i and j, and x and y are the coordinates of A measured
in the x-y frame. The absolute position of B is defined by the vector 𝒓𝑩 measured from the origin of
the fixed axes X-Y. The absolute position of A is seen, therefore, to be determined by the vector
𝒓𝑨 = 𝒓𝑩 + 𝒓𝑨/𝑩
90. We can express the relative-motion terms in whatever coordinate system is convenient rectangular,
normal and tangential, or polar can be used for the formulations.
The selection of the moving point B for attachment of the reference coordinate system is arbitrary.
91. Example: Passengers in the jet transport A flying east at a speed of 800 km /h observe a second jet
plane B that passes under the transport in horizontal flight. Although the nose of B is pointed in the 45°
northeast direction, plane B appears to the passengers in A to be moving away from the transport at the
60°angle as shown. Determine the true velocity of B.
92.
93. Example: Car A is traveling at the constant speed of 60 km /h as it rounds the circular curve of 300m
radius and at the instant represented is at the position θ = 45°. Car B is traveling at the constant speed
of 80 km /h and passes the centre of the circle at this same instant. Car A is located with respect to car
B by polar coordinates r and θ with the pole moving with B. For this instant determine 𝒗𝑨/𝑩 and the
values of 𝒓and 𝜽 as measured by an observer in car B.
94.
95.
96. Example:- At the instant shown in Fig, cars A and B are traveling with speeds of 18 m/s and
12 m/s, respectively. Also at this instant, A has a decrease in speed of 2𝑚/𝑠2
, and B has an
increase in speed of 3𝑚/𝑠2
. Determine the velocity and acceleration of B with respect to A.
97.
98.
99. CONSTRAINED MOTION OF CONNECTED PARTICLES
In some types of problems the motion of one particle will depend on the corresponding motion of
another particle.
This dependency commonly occurs if the particles are interconnected by inextensible cords which
are wrapped around pulleys.
If the total cord length is 𝑙𝑇, then the two position coordinates are related by the equation.
100. Here 𝑙𝐶𝐷 is the length of the cord passing over arc CD.
Taking the time derivative of this expression, realizing that 𝑙𝐶𝐷 and 𝑙𝑇 remain constant,
while 𝑆𝐴 and 𝑆𝐵 measure the segments of the cord that change in length,
Time differentiation of the velocities yields the relation between the accelerations,
Note that each of the coordinate axes is
(1) measured from a fixed point (O) or fixed datum line,
(2) measured along each inclined plane in the direction of motion of each block, and
(3) has a positive sense from the fixed datums to A and to B.
101. We have chosen position coordinates which from the fixed datums
are positive to the right for 𝑠𝐴 and positive downward for 𝑠𝐵.
During the motion, the length of the red coloured segments of the
cord in Fig. a remains constant.
If l represents the total length of cord minus these segments, then the
position coordinates can be related by the equation
The time derivatives of velocity yields:
Hence, when B moves downward (+𝑠𝐵), A moves to the left (-𝑠𝐴) with twice the motion.
102. One Degree of Freedom
The system of Fig. is said to have one degree of freedom since only one variable, either x or y, is
needed to specify the positions of all parts of the system.
The motion of B is clearly the same as that of the center of its pulley, so we establish position
coordinates y and x measured from a convenient fixed datum. The total length of the cable is:
103. Two Degrees of Freedom
Here the positions of the lower cylinder and pulley C depend on the separate specifications of the two
coordinates 𝑦𝐴 and 𝑦𝐵. The lengths of the cables attached to cylinders A and B can be written,
respectively, as:
104. It is clearly impossible for the signs of all three terms to be positive simultaneously.
So, If both A and B have downward (positive) velocities, then C will have an upward (negative)
velocity.
105. Example:- If the velocity 𝑥 of block A up the incline is
increasing at the rate of 0.044 m/s each second, determine
the acceleration of B.