The document provides a summary of key concepts from chapters 1-3 of a physics textbook. It discusses measurement units in the International System of Units (SI), including standards for time, length, and mass. It also covers physical quantities, vectors, kinematics concepts like displacement, velocity, acceleration, and equations of motion for constant acceleration. The summary is provided in 3 sentences or less highlighting the essential information covered.
The document provides an overview of tensor calculus and its notations. It discusses two methods for representing tensors: direct notation which treats tensors as invariant objects, and index notation which uses tensor components. The direct notation is preferred. Basic operations for vectors and second rank tensors are defined, including addition, scalar multiplication, dot products, cross products, and properties. Polar and axial vectors are distinguished. Guidelines are given for tensor calculus notation and rules used throughout the work.
The document summarizes key concepts in vector analysis presented in a physics presentation:
Vectors have both magnitude and direction, unlike scalars which only have magnitude. Common vector quantities include displacement, velocity, force. Vectors can be added using the parallelogram law or triangle law. The dot product of two vectors produces a scalar, while the cross product produces a vector perpendicular to the two input vectors. Vector concepts like resolution, equilibrium of forces, and area/volume calculations utilize dot and cross products.
The document defines dot products and related concepts for vectors in R3. It shows that the dot product of two vectors v and w is defined as v1w1 + v2w2 + v3w3. It proves properties of dot products, including that the dot product is commutative and distributive over vector addition and scalar multiplication. It then introduces the angle between two vectors and proves that the dot product is equal to the product of the vector magnitudes and the cosine of the angle between them.
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
Vectors have both magnitude and direction, while scalars only have magnitude. Common vectors include displacement, velocity, force, and momentum. Vectors can be added using the triangle law of parallelogram law. The resultant vector is the single vector that represents the total effect of multiple vectors acting on a point. Equilibrium occurs when the net force on an object is zero. Concurrent forces pass through a common point, while the equilibrant force produces equilibrium when acting with other forces in the system.
1) The document describes vectors and motion in two dimensions, including defining vectors, coordinate systems, vector addition and subtraction using graphical and algebraic methods, and vector components.
2) It provides an example of finding the displacement vectors for each day of a hiker's trip, then using those to find the overall displacement vector for the entire trip.
3) The hiker walked 25km southeast on the first day. On the second day, she walked 40km in a direction 60 degrees north of east. The overall displacement vector was calculated to be 37.7km along the x-axis and 16.9km along the y-axis, or 41.3km at an angle of 24.1 degrees north
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
The document contains a list of 6 group members with their names and student identification numbers. The group members are:
1. Ridwan bin shamsudin, student ID: D20101037472
2. Mohd. Hafiz bin Salleh, student ID: D20101037433
3. Muhammad Shamim Bin Zulkefli, student ID: D20101037460
4. Jasman bin Ronie, student ID: D20101037474
5. Hairieyl Azieyman Bin Azmi, student ID: D20101037426
6. Mustaqim Bin Musa, student ID:
The document provides an overview of tensor calculus and its notations. It discusses two methods for representing tensors: direct notation which treats tensors as invariant objects, and index notation which uses tensor components. The direct notation is preferred. Basic operations for vectors and second rank tensors are defined, including addition, scalar multiplication, dot products, cross products, and properties. Polar and axial vectors are distinguished. Guidelines are given for tensor calculus notation and rules used throughout the work.
The document summarizes key concepts in vector analysis presented in a physics presentation:
Vectors have both magnitude and direction, unlike scalars which only have magnitude. Common vector quantities include displacement, velocity, force. Vectors can be added using the parallelogram law or triangle law. The dot product of two vectors produces a scalar, while the cross product produces a vector perpendicular to the two input vectors. Vector concepts like resolution, equilibrium of forces, and area/volume calculations utilize dot and cross products.
The document defines dot products and related concepts for vectors in R3. It shows that the dot product of two vectors v and w is defined as v1w1 + v2w2 + v3w3. It proves properties of dot products, including that the dot product is commutative and distributive over vector addition and scalar multiplication. It then introduces the angle between two vectors and proves that the dot product is equal to the product of the vector magnitudes and the cosine of the angle between them.
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
Vectors have both magnitude and direction, while scalars only have magnitude. Common vectors include displacement, velocity, force, and momentum. Vectors can be added using the triangle law of parallelogram law. The resultant vector is the single vector that represents the total effect of multiple vectors acting on a point. Equilibrium occurs when the net force on an object is zero. Concurrent forces pass through a common point, while the equilibrant force produces equilibrium when acting with other forces in the system.
1) The document describes vectors and motion in two dimensions, including defining vectors, coordinate systems, vector addition and subtraction using graphical and algebraic methods, and vector components.
2) It provides an example of finding the displacement vectors for each day of a hiker's trip, then using those to find the overall displacement vector for the entire trip.
3) The hiker walked 25km southeast on the first day. On the second day, she walked 40km in a direction 60 degrees north of east. The overall displacement vector was calculated to be 37.7km along the x-axis and 16.9km along the y-axis, or 41.3km at an angle of 24.1 degrees north
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
The document contains a list of 6 group members with their names and student identification numbers. The group members are:
1. Ridwan bin shamsudin, student ID: D20101037472
2. Mohd. Hafiz bin Salleh, student ID: D20101037433
3. Muhammad Shamim Bin Zulkefli, student ID: D20101037460
4. Jasman bin Ronie, student ID: D20101037474
5. Hairieyl Azieyman Bin Azmi, student ID: D20101037426
6. Mustaqim Bin Musa, student ID:
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
- A particle starts from the point with position vector (3i + 7j) m and then moves with constant velocity (2i – j) ms-1. The question asks to find the position vector of the particle 4 seconds later.
- Substituting the values into the displacement equation gives the final position vector as (12i + 3j) m.
- A second particle is given a position vector of (2i + 4j) m at time t = 0 and a position vector of (12i + 16j) m five seconds later. Using the displacement equation gives the velocity of the particle as (2i + 4j) ms-1.
- For a third particle
The dot product of two vectors a and b is a scalar value defined as |a||b|cosθ, where |a| and |b| are the lengths of the vectors and θ is the angle between them. The dot product will be 0 if the vectors are perpendicular, positive if the angle between them is less than 90 degrees, and negative if the angle is greater than 90 degrees. If one vector is a unit vector, the dot product equals the length of the other vector projected onto the direction of the unit vector.
The document discusses various topics related to vectors including:
- Definitions of vectors, scalars, magnitude and direction
- Equality of vectors and types of vectors
- Addition and subtraction of vectors using triangle law and parallelogram law
- Multiplication of a vector by a scalar
- Scalar (dot) product and properties
- Vector (cross) product and properties
- Applications to work done, moments and areas
The document provides explanations, properties, examples and formulas for key vector algebra concepts.
The document discusses using the dot product to determine the angle between two vectors and to determine the projection of a vector along a specified line. It provides definitions and examples of using the dot product to calculate the angle between two vectors and to find the parallel and perpendicular components of a vector. An example problem demonstrates applying these concepts to find the angle between a force vector and a pole and the magnitude of the force projection along the pole.
1. The document discusses three geometry problems involving vectors and their solutions:
2. It shows that the midpoint lines of a parallelogram trisect its diagonal lines.
3. It proves that if two pairs of opposite edges in a tetrahedron are perpendicular, then the third pair is also perpendicular.
4. It demonstrates that the midpoint line between two sides of a triangle is parallel to the third side and half its length.
The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar value that geometrically equals the magnitudes of A and B multiplied by the cosine of the angle between them.
This document discusses vector concepts including:
- Two methods for adding vectors graphically: the tip-to-tail method and parallelogram method.
- Decomposing a vector into perpendicular components using trigonometric functions.
- Analyzing forces on an inclined plane by decomposing weight into components parallel and perpendicular to the plane.
- The normal force on an object on an inclined plane equals the perpendicular component of the object's weight.
The document provides information about an introductory physics module including the topics, instructors, and meeting times. It then summarizes various core concepts in physics measurements and quantities including: physical quantities and their magnitude and units; base and derived quantities and units; supplementary quantities of plane angle and solid angle; and significant figures in measurements. Dimensional analysis and the benefits of checking for dimensional homogeneity in equations is also outlined. Vectors, scalar and vector products, and torque are defined.
Vectors can be used to describe the position of a point in space using a coordinate system with an origin and axes. A vector quantity has both magnitude and direction, while a scalar only has magnitude. Vectors can be added graphically by drawing them tip to tail or algebraically using their components. A vector's components are its projections onto the x- and y-axes and are found using trigonometric functions of the vector's angle. The problem provides the displacements of a hiker over two days and uses vector addition to find the total displacement, expressing it in terms of its components and as a single vector in unit vector form.
This document introduces vectors and their properties. It defines a vector as having both magnitude and direction, represented by bold letters with arrows. Scalar quantities only have magnitude. The key vector operations are addition, by placing vectors tip to tail, and scalar multiplication. Laws of vector algebra include commutativity, associativity and distributivity. Dot and cross products are also introduced, with the dot product yielding a scalar and cross product a vector. Several problems demonstrate applying concepts like finding angles between vectors and using vector identities.
This document provides an overview of vectors and their applications in physics. It defines vectors and differentiates them from scalars, discusses vector notation and representation, and covers key concepts like addition, subtraction, and multiplication of vectors. Examples are given of vector quantities like displacement, velocity and force. The document also explains vector operators like gradient, divergence and curl, which allow converting between scalar and vector quantities, and outlines how calculus is important in physics for studying change.
The document discusses key concepts regarding vector quantities including:
1) Vectors can be represented graphically with arrows to indicate both magnitude and direction.
2) A vector can be described by its components in different directions or bases.
3) The scalar (dot) product of two vectors results in a scalar and indicates whether vectors are parallel, while the vector (cross) product produces another vector perpendicular to the two.
4) Important vector relationships include the direction cosines that specify a vector's direction, and calculating the angle between two vectors.
The dot product of two vectors A and B is defined as AB cosθ, where θ is the angle between the vectors. It is denoted as A.B and results in a scalar quantity. The cross product of two vectors A and B is defined as AB sinθ with a direction perpendicular to A and B in a right-handed system. It is denoted as A x B and results in a vector quantity. Both the dot and cross products were developed in the 19th century and have various applications in physics and mathematics.
1. A scalar is a physical quantity that has only magnitude, such as mass, length, and time. A vector is a physical quantity that has both magnitude and direction, such as displacement, velocity, and force.
2. Vectors can be classified based on their orientation as parallel, anti-parallel, or perpendicular. They can also be added using geometric methods like the triangle law or analytical methods using components.
3. Multiplying a vector by a scalar multiplies the magnitude by the scalar value. Vector multiplication can produce a scalar using the dot product or a vector using the cross product. The dot product returns the magnitudes of the vectors and their cosine angle.
This document discusses power series representations of functions and their radii of convergence. It provides examples of finding the power series for functions like 1/(1+x) and sin(x), as well as the general process for determining the power series representation of a function and the radius of convergence based on the interval of convergence. It also discusses rules for differentiating and integrating term-by-term for power series whose functions have a given radius of convergence.
Scalars and vectors are different types of quantities. Scalars only have magnitude and not direction, while vectors have both magnitude and direction. Common scalar quantities include speed, temperature, and time. Common vector quantities include displacement, velocity, acceleration, force, and electric field. Vectors can be represented using arrow diagrams with a specified scale and direction. The resultant of two or more vectors can be found geometrically by drawing them head to tail or mathematically using trigonometry, Pythagorean theorem, or graphical addition of vectors placed tail to head.
Physics affects technology and society in interconnected ways. Developments in physics allow new technologies to be created, which then drive further scientific progress and also impact society. For example, certain technologies rely on scientific investigations that require specific technologies, while technologies can also have unintended societal effects that spur additional scientific study. Physics is also considered the most fundamental science, as it plays a key role in advancing other disciplines and driving technological progress that transforms communication and economic development in society.
This document contains a worksheet on physical quantities, units of measurement, and measurement tools. It covers:
1) Base quantities in the SI system and their units, including length, mass, time, temperature, and prefixes.
2) Instruments for measuring length, including micrometers, vernier calipers, and rulers. It provides examples of measuring thickness and diameter.
3) Time measurement using stopwatches and the period of a pendulum. Seconds are the SI unit for time.
4) Distinguishing between scalar and vector quantities like displacement, velocity, force. Examples of resolving forces into components and calculating resultant forces.
The document provides information about motion under uniform acceleration including:
- Four kinematic equations that describe motion with constant acceleration in one dimension.
- Derivations of the equations from the concept of displacement being equal to area under the velocity-time graph.
- An example problem using one of the equations to calculate deceleration from initial velocity, final velocity, and displacement.
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
- A particle starts from the point with position vector (3i + 7j) m and then moves with constant velocity (2i – j) ms-1. The question asks to find the position vector of the particle 4 seconds later.
- Substituting the values into the displacement equation gives the final position vector as (12i + 3j) m.
- A second particle is given a position vector of (2i + 4j) m at time t = 0 and a position vector of (12i + 16j) m five seconds later. Using the displacement equation gives the velocity of the particle as (2i + 4j) ms-1.
- For a third particle
The dot product of two vectors a and b is a scalar value defined as |a||b|cosθ, where |a| and |b| are the lengths of the vectors and θ is the angle between them. The dot product will be 0 if the vectors are perpendicular, positive if the angle between them is less than 90 degrees, and negative if the angle is greater than 90 degrees. If one vector is a unit vector, the dot product equals the length of the other vector projected onto the direction of the unit vector.
The document discusses various topics related to vectors including:
- Definitions of vectors, scalars, magnitude and direction
- Equality of vectors and types of vectors
- Addition and subtraction of vectors using triangle law and parallelogram law
- Multiplication of a vector by a scalar
- Scalar (dot) product and properties
- Vector (cross) product and properties
- Applications to work done, moments and areas
The document provides explanations, properties, examples and formulas for key vector algebra concepts.
The document discusses using the dot product to determine the angle between two vectors and to determine the projection of a vector along a specified line. It provides definitions and examples of using the dot product to calculate the angle between two vectors and to find the parallel and perpendicular components of a vector. An example problem demonstrates applying these concepts to find the angle between a force vector and a pole and the magnitude of the force projection along the pole.
1. The document discusses three geometry problems involving vectors and their solutions:
2. It shows that the midpoint lines of a parallelogram trisect its diagonal lines.
3. It proves that if two pairs of opposite edges in a tetrahedron are perpendicular, then the third pair is also perpendicular.
4. It demonstrates that the midpoint line between two sides of a triangle is parallel to the third side and half its length.
The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar value that geometrically equals the magnitudes of A and B multiplied by the cosine of the angle between them.
This document discusses vector concepts including:
- Two methods for adding vectors graphically: the tip-to-tail method and parallelogram method.
- Decomposing a vector into perpendicular components using trigonometric functions.
- Analyzing forces on an inclined plane by decomposing weight into components parallel and perpendicular to the plane.
- The normal force on an object on an inclined plane equals the perpendicular component of the object's weight.
The document provides information about an introductory physics module including the topics, instructors, and meeting times. It then summarizes various core concepts in physics measurements and quantities including: physical quantities and their magnitude and units; base and derived quantities and units; supplementary quantities of plane angle and solid angle; and significant figures in measurements. Dimensional analysis and the benefits of checking for dimensional homogeneity in equations is also outlined. Vectors, scalar and vector products, and torque are defined.
Vectors can be used to describe the position of a point in space using a coordinate system with an origin and axes. A vector quantity has both magnitude and direction, while a scalar only has magnitude. Vectors can be added graphically by drawing them tip to tail or algebraically using their components. A vector's components are its projections onto the x- and y-axes and are found using trigonometric functions of the vector's angle. The problem provides the displacements of a hiker over two days and uses vector addition to find the total displacement, expressing it in terms of its components and as a single vector in unit vector form.
This document introduces vectors and their properties. It defines a vector as having both magnitude and direction, represented by bold letters with arrows. Scalar quantities only have magnitude. The key vector operations are addition, by placing vectors tip to tail, and scalar multiplication. Laws of vector algebra include commutativity, associativity and distributivity. Dot and cross products are also introduced, with the dot product yielding a scalar and cross product a vector. Several problems demonstrate applying concepts like finding angles between vectors and using vector identities.
This document provides an overview of vectors and their applications in physics. It defines vectors and differentiates them from scalars, discusses vector notation and representation, and covers key concepts like addition, subtraction, and multiplication of vectors. Examples are given of vector quantities like displacement, velocity and force. The document also explains vector operators like gradient, divergence and curl, which allow converting between scalar and vector quantities, and outlines how calculus is important in physics for studying change.
The document discusses key concepts regarding vector quantities including:
1) Vectors can be represented graphically with arrows to indicate both magnitude and direction.
2) A vector can be described by its components in different directions or bases.
3) The scalar (dot) product of two vectors results in a scalar and indicates whether vectors are parallel, while the vector (cross) product produces another vector perpendicular to the two.
4) Important vector relationships include the direction cosines that specify a vector's direction, and calculating the angle between two vectors.
The dot product of two vectors A and B is defined as AB cosθ, where θ is the angle between the vectors. It is denoted as A.B and results in a scalar quantity. The cross product of two vectors A and B is defined as AB sinθ with a direction perpendicular to A and B in a right-handed system. It is denoted as A x B and results in a vector quantity. Both the dot and cross products were developed in the 19th century and have various applications in physics and mathematics.
1. A scalar is a physical quantity that has only magnitude, such as mass, length, and time. A vector is a physical quantity that has both magnitude and direction, such as displacement, velocity, and force.
2. Vectors can be classified based on their orientation as parallel, anti-parallel, or perpendicular. They can also be added using geometric methods like the triangle law or analytical methods using components.
3. Multiplying a vector by a scalar multiplies the magnitude by the scalar value. Vector multiplication can produce a scalar using the dot product or a vector using the cross product. The dot product returns the magnitudes of the vectors and their cosine angle.
This document discusses power series representations of functions and their radii of convergence. It provides examples of finding the power series for functions like 1/(1+x) and sin(x), as well as the general process for determining the power series representation of a function and the radius of convergence based on the interval of convergence. It also discusses rules for differentiating and integrating term-by-term for power series whose functions have a given radius of convergence.
Scalars and vectors are different types of quantities. Scalars only have magnitude and not direction, while vectors have both magnitude and direction. Common scalar quantities include speed, temperature, and time. Common vector quantities include displacement, velocity, acceleration, force, and electric field. Vectors can be represented using arrow diagrams with a specified scale and direction. The resultant of two or more vectors can be found geometrically by drawing them head to tail or mathematically using trigonometry, Pythagorean theorem, or graphical addition of vectors placed tail to head.
Physics affects technology and society in interconnected ways. Developments in physics allow new technologies to be created, which then drive further scientific progress and also impact society. For example, certain technologies rely on scientific investigations that require specific technologies, while technologies can also have unintended societal effects that spur additional scientific study. Physics is also considered the most fundamental science, as it plays a key role in advancing other disciplines and driving technological progress that transforms communication and economic development in society.
This document contains a worksheet on physical quantities, units of measurement, and measurement tools. It covers:
1) Base quantities in the SI system and their units, including length, mass, time, temperature, and prefixes.
2) Instruments for measuring length, including micrometers, vernier calipers, and rulers. It provides examples of measuring thickness and diameter.
3) Time measurement using stopwatches and the period of a pendulum. Seconds are the SI unit for time.
4) Distinguishing between scalar and vector quantities like displacement, velocity, force. Examples of resolving forces into components and calculating resultant forces.
The document provides information about motion under uniform acceleration including:
- Four kinematic equations that describe motion with constant acceleration in one dimension.
- Derivations of the equations from the concept of displacement being equal to area under the velocity-time graph.
- An example problem using one of the equations to calculate deceleration from initial velocity, final velocity, and displacement.
This is a summary of the topic "Physical quantities, units and measurement" in the GCE O levels subject: Physics. Students taking either the combined science (chemistry/physics) or pure Physics will find this useful. These slides are prepared according to the learning outcomes required by the examinations board.
This document discusses scalar and vector quantities in physics. It defines scalars as physical quantities that have magnitude but no direction, while vectors have both magnitude and direction. Examples are given such as distance, time and mass for scalars, and displacement, velocity and force for vectors. The document then explains how to add scalar and vector quantities, noting that vectors are represented by arrows and can be added graphically by placing the arrows head to tail. It provides examples of adding vectors in the same and opposite directions. Finally, it presents a homework problem on calculating distance and displacement.
This document discusses the vernier caliper, a tool used to measure internal, external, and depth dimensions. It describes the main parts of the vernier caliper including the main scale, vernier scale, inside and outside jaws, screw clamp, and depth probe. The document explains how to properly read measurements from the vernier scale by identifying where the zero points of the main and vernier scales align. Examples are given of measuring lengths accurately to within 1/10, 1/20 and 1/50 of a millimeter. Slides provide exercises for students to practice using the vernier caliper to take measurements.
The role of science and technology in developmentJanette Balagot
The document discusses the role of science and technology in development. It states that development is a multidimensional process that involves changes to economic, social, administrative, and belief systems. Science and technology can improve welfare but may also contribute to environmental degradation and dehumanization if not implemented properly. For effective application, science and technology must be integrated into national concepts and ways of life, directed toward reducing inequalities, and implemented within a framework of social and economic rights. Universities, education, research, and infrastructure support are also needed.
Science is defined as the human attempt to understand the natural world through discovering facts and relationships to develop theories, while technology is defined as the human attempt to change the world by creating useful products using the findings of science. Science drives technology through scientific breakthroughs that enable new technologies, and technology drives science by allowing experiments that were not previously possible and shaping the questions scientists investigate based on available technologies. Examples of the interconnections between science and technology include engineers using scientific knowledge to develop products, some scientific experiments requiring enabling technologies to be possible, and technology not being able to advance without the underlying scientific discoveries.
1. Vectors have both magnitude and direction, while scalars only have magnitude.
2. Common vector quantities include velocity and force, while common scalars include mass and time.
3. Vectors can be represented by arrows in diagrams or with signs to indicate direction in equations. The resultant vector represents the total effect of multiple vectors.
This document discusses the use of a micrometer screw gauge for measurements and contains exercises for calculating its least count and reading measurements taken with it. The exercises guide the student to determine the least count, read measurements indicated on the gauge, and calculate the reading for a given gauge.
This document defines technology and discusses the differences between technology and science. It provides examples of different fields of science such as biology, chemistry, physics, geology and their experts. Technology is defined as humans modifying nature to meet needs and wants through processes like invention, innovation and problem solving. Science seeks to understand nature through inquiry and exploration, while technology seeks to change it. Both science and technology are important but have different goals.
Studying Simon Sinek: Start With the Golden CircleChiara Ojeda
Most of us communicate from the outside in, but author Simon Sinek believes true inspiration comes when we start with why. This brief introduction to The Golden Circle is a teaching tool used in class.
Vectors have both magnitude and direction, represented by arrows. The sum of two vectors is obtained by placing the tail of one vector at the head of the other. If the vectors are at right angles, their dot product is zero, while their cross product is maximum. Scalar multiplication scales the magnitude but not the direction of a vector.
Introductory Physics - Physical Quantities, Units and MeasurementSutharsan Isles
This document provides an introduction to physical quantities, units, and measurement in physics. It begins with definitions of key terminology like physical property, scalar and vector quantities, and standard form notation. It then discusses the International System of Units (SI) including the seven base units, common prefixes, and how to convert between multiples and submultiples of units. The document also covers derived SI units and examples of converting between derived units. It emphasizes the importance of understanding whether a quantity is scalar or vector.
This document provides an introduction to and overview of the topics of science and technology that will be studied. It includes definitions of science, technology, and the field of science and technology studies. It also summarizes the history of science and technology as the study of how humanity's understanding of the natural world has changed over time. Additionally, it outlines India's progress in the fields of science and technology, noting its development of skills and technologies to modernize society as well as its growth in areas like energy and research publications. However, it states that India is still lagging behind countries like the US in areas such as research investment and researchers per capita.
1) The document discusses various topics related to motion in a plane including scalar and vector quantities, vectors and their properties, resolution of vectors, projectile motion, and uniform circular motion.
2) Key concepts explained are position and displacement vectors, addition and subtraction of vectors, constant acceleration motion in a plane using components, and the trajectory, time of flight, and range for projectile motion with both horizontal and angled projection.
3) Circular motion is defined as movement along a circular path that can be uniform or non-uniform, and angular displacement is the angle through which an object rotates.
The document defines scalars and vectors. Scalars are physical quantities that only require a magnitude, while vectors require both magnitude and direction. It then discusses various types of vectors, including displacement vectors, unit vectors, the null vector, proper vectors, and the negative of a vector. It explains how to represent vectors graphically and mathematically. Finally, it covers vector operations such as addition, subtraction, and multiplication of vectors, as well as the dot product and properties of the dot product.
There are two kinds of physical quantities: scalar and vector. Scalars have magnitude and unit, while vectors have magnitude, direction, and unit. Examples of scalars include mass and temperature, while examples of vectors include force, velocity, and magnetic field. Vectors are represented by arrows and can be added using the parallelogram or polygon method to find the resultant vector. Vector components allow vectors to be broken down into their x and y components.
This document provides information about the Classical Mechanics course SPHA021 including:
- The minimum pass mark is 50% and exam weighting is 40% while tests, practicals and assignments make up the remaining 60%.
- Attendance is important for understanding the material.
- A study guide is available for all chapters at the bookshop.
- The course outline covers topics like rigid body dynamics, simple harmonic motion, Lagrangian and Hamiltonian dynamics. Recommended textbooks are also listed. There will be two assignments, two tests and quizzes throughout the semester.
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
Electromagnetic fields: Review of vector algebraDr.SHANTHI K.G
This document provides an introduction to electromagnetic fields and vector algebra concepts. It begins with an overview of vector algebra topics like vector addition, multiplication of vectors by scalars, and dot and cross products. It then discusses orthogonal coordinate systems, focusing on Cartesian coordinates. The document provides examples and solved problems for various vector algebra concepts. It aims to review key vector algebra that will be used as a mathematical tool for electromagnetic concepts.
The document defines vectors and their components. It discusses direction, magnitude, types of vectors including collinear, concurrent and coplanar vectors. It covers vector addition and subtraction using graphical and analytical methods. Vector multiplication by scalars is explained. Components and rectangular decomposition of vectors are also summarized. Finally, vectors in 3D space and their operations are defined.
Vectors have both magnitude and direction, while scalars only have magnitude. Common vector quantities include displacement, velocity, acceleration, force, and electric and magnetic fields. Vector notation allows physical laws to be written compactly. Vectors can be added, subtracted, and multiplied by scalars. The dot product yields a scalar and the cross product yields a vector perpendicular to the two original vectors. Uniform circular motion results in constant speed but centripetal acceleration directed radially inward.
Okay, here are the steps to solve this problem using component addition of vectors:
1) The first displacement is 120 m due north, so its x-component is 0 and its y-component is 120.
2) The second displacement is 72 m due west, so its x-component is -72 and its y-component is 0.
3) Add the x-components: 0 + -72 = -72
4) Add the y-components: 120 + 0 = 120
5) Use the Pythagorean theorem to find the magnitude of the resultant: R = √(-72)2 + 1202 = 132 m
6) The resultant displacement is due west (because the x-component
1. The document discusses vector analysis and its applications in electromagnetics. It covers topics like scalar and vector quantities, vector algebra, coordinate systems, and Maxwell's equations.
2. Coordinate systems discussed include Cartesian, cylindrical, and spherical coordinates. Position vectors and base vectors are used to define locations and directions in these systems.
3. Examples show how to represent points using coordinates, draw and write position vectors, and calculate the magnitude and unit vector of a position vector. Determining distances between points using position vectors is also mentioned.
This document provides an overview of electromagnetic fields and discusses scalars, vectors, and coordinate systems. It begins by defining electromagnetics and listing common EM devices. It then discusses scalars, vectors, unit vectors, and how to add and subtract vectors. It also covers position vectors, dot and cross products, and vector components. The document finishes by explaining Cartesian, cylindrical, and spherical coordinate systems, and how to transform between coordinate systems. It defines constant coordinate surfaces for each system.
This document provides an overview of foundation and mathematics science studies. It includes sections on physics, mathematics, and a physics chapter on physical quantities, units, and vectors. The physics section covers introduction to physics, kinetic motion, and Newton's laws. The mathematics section includes number systems, equations, inequalities, polynomials, and sequences and series. The physics chapter objectives are to describe basic and derived quantities and their SI units, and define scalar and vector quantities. It covers adding and subtracting vectors graphically, resolving vectors into components, and multiplying vectors using dot and cross products.
The document defines scalar and vector quantities, and discusses vector operations including addition, subtraction, and multiplication. It provides examples of scalar quantities like mass and temperature that only have magnitude, and vector quantities like force and velocity that have both magnitude and direction. Vector diagrams are used to represent vectors as arrows, and rules for vector addition and subtraction are covered. Dot and cross products are also summarized, including properties and examples of calculating these products of vectors.
Scalars have magnitude but no direction, such as temperature, mass, and time. Vectors have both magnitude and direction, represented by an arrow. Vectors can be specified by their magnitude and direction, or by their x and y components. The magnitude of a vector is found using the Pythagorean theorem, and the direction can be found using tangent. Vectors can be added using the triangle or parallelogram method, and multiplied by scalars. The dot product yields a scalar and the cross product yields a vector perpendicular to the original vectors.
This document provides an overview of key concepts in physics, including:
- Physics is the science that describes the basic components of the universe and forces. It underpins other sciences.
- Physical quantities have numerical values and units, and can be basic or derived. Basic quantities include length, mass, and time.
- Vectors have magnitude and direction, while scalars only have magnitude. Examples of each are provided.
- Methods for adding and subtracting vectors graphically and by components are described. Properties of vector operations are also summarized.
Linear algebra concepts like vectors, matrices, and linear transformations are important for recommendation systems. Vectors represent items or users, matrices represent item-user preference data. Linear algebra allows analyzing this data to identify patterns and recommend new items. Key techniques include eigendecomposition to reduce dimensionality and identify important relationships in the data, and singular value decomposition to factor matrices for recommendations. These linear algebra concepts are essential mathematical tools for building personalized recommendation models.
This document discusses physical quantities and vectors. It defines two types of physical quantities: scalar quantities which have only magnitude, and vector quantities which have both magnitude and direction. Examples of each are given. Vector quantities are represented by magnitude and direction. The document then discusses methods for adding and subtracting vectors graphically using head-to-tail and parallelogram methods. It also covers resolving vectors into rectangular components, finding the magnitude and direction of vectors, dot products of vectors which yield scalar quantities, and cross products of vectors which yield vector quantities. Examples of applying these vector concepts are provided.
This document discusses sinusoidal waves and vector functions. It defines key concepts for sinusoidal waves like amplitude, period, phase shift, and frequency. It also explains vector notation and properties, including adding vectors using the nose-to-tail and parallelogram methods. Vectors in three dimensions are represented using unit vectors i, j, and k. Vector operations like the scalar and vector products are also introduced.
1. The document discusses vectors and tensors. It defines vectors as quantities with magnitude and direction, and provides examples like position, force, and velocity.
2. Tensors are quantities that have magnitude, direction, and a plane in which they act. Rank 0 tensors are scalars, rank 1 tensors are vectors, and rank 2 tensors can be represented by matrices.
3. The document covers various types of vectors like unit vectors and displacement vectors. It also discusses vector algebra operations and different ways vectors can be represented, such as in Cartesian form.
Electric and Magnetic Fields (EEE2303)-lecture 1-3 - Vector Analysis.pptxmonaibrahim598401
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Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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Physical Quantities, Vectors, Fovce and newton's Laws
1.
2. Group “A”
Usman Abrar, Kamran Sharif,
Muneeba Idrees, Hassan Amjad, Ali Asad
3. Chapter # 01, 02, 03
Measurement
Motion in one Dimension
&
Force and Newton’s Laws
4. Chap#01 Measurement
Physical Quantities:-
“Quantities which can be measured are called Physical
Quantities.”
These require Magnitude, Unit and Sometime Direction for
their complete description.
Here we will discuss SI( System International) for
measurements…
Height of the girl, l = 1.55 m
For example
Symbol ofUnit of the
the
Numerical value
physical
physical quantity
for the
quantity magnitude of
the physical
quantity
5. System International:-
Types of physical Quantities:-
Base/ Basic Quantities .
Derived Quantities.
Basic Quantities:-
“ Quantities which are chosen as a base and many other measuring
quantities are derived from them are called basic quantities and
their units are called base units…”
They are :
6. Physical Quantities:-
Derived quantities:-
“All quantities other then “7” basic quantities are called Derived
quantities because they are derived from Base quantities and their
units are called derived units.”
For Example :-
7. System international
Units:-
“Units are symbol associated with every physical quantity
whether its basic or Derived quantity….”
Like,
Meter(m) for length ; kilogram(kg) for Mass and
second(s) for time etc…
Standards:-
“Physicals quantities must have a standard value so that
every calculation is accurate and scientist in different place
can calculate things without any ambiguity…”
Some standards are discussed here...
8. System international
The Standard of time:-
Anything which repeats its motion periodically can be set as time
standard like Oscillating Pendulum, A mass-Spring system, a
Quartz crystal etc…
But in SI time standard is defined as:
“The second is the duration of 9,192,631,770 vibrations of (specific)
radiation emitted by a Specific isotope Of the Cesium atom”
Fig shows the current national frequancy
standard, so-called Cesium foundation clock.
9. System international
The Standard of length:
SI unit of length is Meter (m)… defined as:
“The meter is the length of the path traveled
by light in vacuum during
a time interval of 1/299,792,458 of a second”
Length is also measured in cm, km, miles, feets etc…
The Standard of Mass:
SI unit Base unit for Mass is “Kg”
“The SI standard of mass is a platinum-iridium cylinder kept
at IBWM”
10. Precision And Significant Figures
Precision:-
“An indication of the scale on the measuring device that was
used.”
In other words, the more correct a measurement is, the more
accurate it is. On the other hand, the smaller the scale on the
measuring instrument, the more precise the measurement.
Fig illustrates difference
b/w Precision and accuracy.
11. Significant Figures
“The significant figures (also known as significant digits, and
often shortened to sig figs) of a number are those digits that
carry meaning contributing to its precision. This includes all
digits”
except:
leading and trailing zeros which are merely place holders
to indicate the scale of the number.
spurious digits introduced, for example, by calculations
carried out to greater precision than that of the original
data, or measurements reported to a greater precision
than the equipment supports.
12. Identifying significant figures
The rules for identifying significant figures when writing or interpreting numbers are as
follows:
All non-zero digits are considered significant. For example, 91 has two significant figures
(9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).
Zeros appearing anywhere between two non-zero digits are significant.
Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.
Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.
Trailing zeros in a number containing a decimal point are significant. For example,
12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six
significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five
significant figures since it has three trailing zeros. This convention clarifies the precision of such
numbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23
then it might be understood that only two decimal places of precision are available. Stating the
result as 12.2300 makes clear that it is precise to four decimal places (in this case, six significant
figures).
13. Identifying significant figures
The significance of trailing zeros in a number not containing a decimal
point can be ambiguous. For example, it may not always be clear if a
number like 1300 is precise to the nearest unit (and just happens
coincidentally to be an exact multiple of a hundred) or if it is only
shown to the nearest hundred due to rounding or uncertainty. Various
conventions exist to address this issue:
A bar may be placed over the last significant figure; any trailing zeros
following this are insignificant. For example, 1300 has three significant
figures (and hence indicates that the number is precise to the nearest
ten).
The last significant figure of a number may be underlined; for
example, "2000" has two significant figures.
A decimal point may be placed after the number; for example "100."
indicates specifically that three significant figures are meant.
In the combination of a number and a unit of measurement the
ambiguity can be avoided by choosing a suitable unit prefix. For
example, the number of significant figures in a mass specified as
1300 g is ambiguous, while in a mass of 13 hg or 1.3 kg it is not.
14. Chapter # 02 Motion in one
dimension
Kinematics:-
“Kinematics is the branch of mechanics that describes the motion
of objects without necessarily discussing what causes the motion.”
By specifying the velocity, position and acceleration f a object, we
can describe how this object moves, including the direction of its
motion. How that direction changes with time, whether the object
speeds up or slows down and so forth…
Position, velocity, acceleration etc… can be found using vectors.
15. Vectors
Vector:-
“Vector is that quantity that requires magnitude, unit
as well as Direction for its complete description…”
For example: Displacement, velocity (v) etc…
Vector is represented by straight line with an arrow
head on its either side…
And vector quantities are represented either in Bold
or making an arrow on is symbol… A.
16. z
Properties of Vectors
Az
θ A
Representation of a Vector:- Aan a
Ax d n
d
x
Ax A cos sen θ
A Ax i Ay j Az k
Ay Asen sen θ
A A Ax2 Ay Az2
2
Az A cos θ
18. Sum of
Vectors A C
B
C
A
B Law of the polygon
R
The resulting vector is one that vector from
the origin of the first vector to the end of the last
19. Vectors
Properties
A A A
ˆ
-A
Opposite
Null 0 = A + ()-A
A
Unit vector μ
A
20. Properties of Commutative
the sum of Law
vectors
R AB BA
Difference Associative
Law
R A-B
R A (B C) ( A B) C
R A (-B) -B
A R
B A
22. Multiplication of a vector by a scalar
Given two vectors AyB
Are said to be parallel if
A B
si 0 A B
si 0 A B
si 1 A B
23.
Dot product of two
vectors
A B AB cos θ
Projection of A on B
A B A cosθ
Projection of B on A
B A B cosθ
24. i i 1
ˆ ˆ i ˆ0
ˆ j
ˆ ˆ 1
j j ˆ ˆ
i k 0
ˆ ˆ
k k 1 j ˆ
ˆk 0
A i Ax
ˆ
A ˆ Ay
j A B A XB X A YB Y A ZB Z
ˆ
A k Az
25. Vector product of two
vectors
C AB
C AB senθ
ˆˆ 0
i i ˆˆ 0
j j
ˆ ˆ
k k 0
j ˆ
iˆ ˆ k j ˆ
ˆ k iˆ
ˆ
k iˆ ˆ
j
26. Demonstrate:
C A B ( A x ˆ A y ˆ A z k) ( B x ˆ B y ˆ B z k)
i j ˆ i j ˆ
C X AY BZ AZ BY
C y Az Bx Ax Bz
C z Ax B y Ay Bx
27. Distance vs Displacement
Distance ( d )
Total length of the path travelled
Measured in meters
scalar
Displacement ( d )
Change in position (x) regardless of path
x = xf – xi
B
Measured in meters
vector
Displacement Distance
A
28. Finding displacement
1
v area l w bh
v – vo = at 2
vo
velocity
1
d vot t at
2
t
time
1
d v0t
2
at
2
29. Average Velocity
The displacement divided by the elapsed time.
Displaceme nt
Average velocity
Elapsed time
x xo x
v
t to t
SI units for velocity: meters per second (m/s)
31. Instantaneous Velocity & Speed
The instantaneous velocity indicates how fast the car moves and the
direction of motion at each instant of time.
x
v lim
t 0 t
The instantaneous speed is the magnitude of the instantaneous velocity
33. Acceleration
The notion of acceleration emerges when a change in velocity
is combined with the time during which the change occurs.
The difference between the final and initial velocity divided by the
elapsed time
vv v
a o
t to t
SI units for acceleration: meters per second per second (m/s2)
35. Example
Acceleration and Increasing Velocity
Determine the average acceleration of the plane.
vo 0 m s v 260 km h v vo
a
to 0 s t 29 s t to
260 km h 0 km h km h
a 9 .0
29 s 0 s s
37. Equations of Kinematics for Constant Acceleration
It is common to dispense with the use of boldface symbols
overdrawn with arrows for the displacement, velocity, and
acceleration vectors (AP does not show arrows on given
equations nor expect them on open-ended problems). We
will, however, continue to convey the directions with a plus
or minus sign. (AP calls elapsed time “t” where t = t – to)
v vo v vo
a a
t to t
38. Equations of Kinematics for
Constant Acceleration
v vo
a at v v o
t
AP Equation
#1
v v o at
39. Equations of Kinematics for
Constant Acceleration
If, a is constant:
x x0 1
1 v 0 v v v o at
v v o v t 2
2
x x0
v 1
t x x0 v 0 v t 1
2
x xo v o v o at t
2
AP Equation
#2
x xo vot 1 2
2
at
40. Equations of Kinematics for
Constant Acceleration
1 v vo v vo
x x0 v 0 v t t a
2 t
a
v vo
x xo 1
2
v o v AP Equation
a #3
v vo
2 2
x xo v v 2 a x x0
2 2
2a o
41. Free Fall
In the absence of air resistance, it is found that all bodies
at the same location above the Earth fall vertically with
the same acceleration. If the distance of the fall is small
compared to the radius of the Earth, then the acceleration
remains essentially constant throughout the descent.
This idealized motion is called free-fall and the acceleration
of a freely falling body is called the acceleration due to
gravity.
g 9 . 80 m s
2 2
or 32 . 2 ft s
g 10 m s
2 2
or 30 ft s
42. Freefalling bodies
I could give a boring lecture
on this and work through
some examples, but I’d
rather make it more real…
43. Free fall problems
Use same kinematic equations just substitute g for a
Choose +/- carefully to make problem as easy as possible
44. Force
Two types of forces
◦ Contact force
Force caused by physical contact
◦ Field force
Force caused by gravitational attraction between two
objects
45. Isaac Newton
Born 1642
Went to University of Cambridge in England as a student and
taught there as a professor after
Never married
Gave his attention mostly to physics and mathematics, but he
also gave his attention to religion and alchemy
Newton was the first to solve three mysteries that
intrigued the scientists
◦ Laws of Motion
◦ Laws of Planetary Orbits
◦ Calculus
46. Three Laws of Motion
Newton’s Laws of Motion are laws discovered by Physicist and
mathematician, Isaac Newton, that explains the objects’
motions depending on forces acted on them
◦ Newton’s First Law: Law of Inertia
◦ Newton’s Second Law: Law of Resultant Force
◦ Newton’s Third Law: Law of Reciprocal Action
47. Newton’s First Law
An Object at rest remains at rest, and an object in
motion continues in motion with constant velocity
(that is, constant speed in a straight line), unless it
experiences a net external force.
The tendency to resist change in motion is called
inertia
◦ People believed that all moving objects would
eventually stop before Newton came up with his
laws
48. Friction
A force that causes resistance to motion
Arises from contact between two surfaces
◦ If the force applied is smaller than the friction, then
the object will not move
If the object is not moving, then ffriction=Fapplied
◦ The object eventually slips when the applied force
is big enough
49. Friction
Friction was discovered by
Galileo Galilee when he
rolled a ball down a slope
and observed that the ball
rolls up the opposite slope
to about the same height,
and concluded that the
difference between the
initial height and the final
height is caused by friction.
Galileo also noticed that the
ball would roll almost
forever on a flat surface so
that the ball can elevate to
the same height as where it
started.
50. Two types of Friction
Static Friction
Kinetic Friction
◦ Friction that exists while ◦ The friction that exists
the object is stationary when an object is in motion
◦ If the applied force on ◦ F – fkinetic produces
an object becomes acceleration to the
greater than the direction the object is
maximum of static moving
friction, then the object ◦ If F = fkinetic, then the object
starts moving moves at constant speed
◦ Fstatic ≤ μstatic n with no acceleration
◦ fkinetic= μkineticn
◦ Kinetic friction and the
coefficient of kinetic friction
are smaller than static
friction and the static
coefficient
51.
52. Newton’s First Law
When there is no force
exerted on an object, the
motion of the object remains
the same like described in
the diagram
◦ Because the equation of
Force is F=ma, the
acceleration is 0m/s². So
the equation is
0N=m*0m/s²
◦ Therefore, force is not
needed to keep the object
in motion, when
◦ The object is in equilibrium
when it does not change its
state of motion
53. The car is traveling rightward
and crashes into a brick wall.
The brick wall acts as an
unbalanced force and stops
the car.
54. The truck stops when it But the ladder falls in front
crashes into the red car. of the truck because the
ladder was in motion with
the truck but there is
nothing stopping the
ladder when the truck
stops.
55. Inertial Frames
Any reference frame that moves with constant velocity relative to an
inertial frame is itself an inertial frame
A reference frame that moves with constant velocity relative to the
distant stars is the best approximation of an inertial frame
We can consider the Earth to be such an inertial frame, although it
has a small centripetal acceleration associated with its motion
56. Newton’s First Law – Alternative
Statement
In the absence of external forces, when viewed from an
inertial reference frame, an object at rest remains at
rest and an object in motion continues in motion with
a constant velocity
Newton’s First Law describes what happens in the
absence of a force
Does not describe zero net force
Also tells us that when no force acts on an object, the
acceleration of the object is zero
57. Inertia and Mass
“The tendency of an object to resist any attempt to change
its velocity is called inertia”
“Mass is that property of an object that specifies how much
resistance an object exhibits to changes in its velocity”
Masses can be defined in terms of the accelerations
produced by a given force acting on them:
m1 a2
m2 a1
The magnitude of the acceleration acting on an object
is inversely proportional to its mass
58. Mass vs. Weight
Mass and weight are two different quantities
Weight is equal to the magnitude of the gravitational
force exerted on the object
Weight will vary with location
Example:
wearth = 180 lb; wmoon ~ 30 lb
mearth = 2 kg; mmoon = 2 kg
59. Newton’s Second Law
The acceleration of an object is directly proportional
to the net force acting on it and inversely proportional
to its mass
Fnet
Acceleration
60. Unbalanced Force and
Acceleration
Force is equal to
acceleration multiplied by
mass
◦ When an unbalanced
force acts on an object,
there is always an
acceleration
Acceleration differs
depending on the net
force
The acceleration is
inversely related to the
mass of the object
61. Net Force
Force is a vector
◦ Because it is a vector, the net force can be determined
by subtracting the force that resists motion from the
force applied to the object.
◦ If the force is applied at an angle, then trigonometry is
used to find the force
Fnet
63. Gravitational Force
The force that exerts all objects toward the earth’s
surface is called a gravitational force.
◦ The magnitude of the gravitational force is called
weight
The acceleration due to gravity is different in each
location, but 9.80m/s² is most commonly used
Calculated with formula w=mg
64. Newton’s Third Law
If two objects interact, the force exerted on object 1
by object 2 is equal in magnitude but opposite in
direction to the force exerted on object 2 by object 1
Forces always come in pair when two objects
interact
◦ The forces are equal, but opposite in direction
Fn
Fg
65. Newton’s Third Law
As the man jumps off
the boat, he exerts
the force on the boat
and the boat exerts
the reaction force on
the man.
The man leaps forward
onto the pier, while the
boat moves away from
the pier.
66. Newton’s Third Law
Foil deflected
up
Engine pushed
forward Flow
backwardpushed backward
Flow
Foil deflected
down deflected
Foil
down
67. Applications of Newton’s Law
Assumptions
Objects can be modeled as particles
Interested only in the external forces acting on the
object
can neglect reaction forces
Initially dealing with frictionless surfaces
Masses of strings or ropes are negligible
When a rope attached to an object is pulling it, the
magnitude of that force is the tension in the rope.