Basic Quantities<\/h1>\r\nThe following four quantities are used when we talk about mechanics.\u00a0 They are often called \"base units\" as other units can be made by combining the base units.\r\n\r\nLength.<\/strong> Length is used to locate the position of a point in space.\u00a0 \u00a0It describes the size of a certain physical system.\u00a0 \u00a0Once you have length, you can have area and volume.\r\n\r\nTime.<\/strong> Time is the indefinite continued progress of existence and events that occur in an apparently irreversible succession from the past, through the present, into the future.\r\n\r\nMass.<\/strong> Mass is a measure of the quantity of matter.\u00a0 It is defined by its inertia, its resistance to being accelerated, and by the gravitational attractions that occurs between two bodies.\r\n\r\nForce.<\/strong> Forces are considered to be pushes or pulls exerted by one body on another body.\u00a0 It can occur due to direct contact when two bodies touch or there can be action at a distance such as the gravitational, magnetic and electric forces.\u00a0 Forces are vectors and is characterized by its magnitude, direction and the point where it is applied.\r\nModels or Idealizations<\/h1>\r\nModels, or idealizations, are used in science in order to simplify applications of theories.\u00a0 Here are three main idealizations we will use in this textbooks.\u00a0 There is an aphorism, often attributed to George Box a statistician that goes \"all models are wrong, but some are useful.\"\r\n\r\nParticle.<\/strong>\u00a0 A particle has a mass, but the size can be neglected.\u00a0 This often depends on the system.\u00a0 As we stand on the Earth we are aware that is it not a particle. Yet when we consider the Earth orbiting around the Sun we treat it as a particle because with a radius of 6378 km and the semi-major axis of the orbit being 149,596,000 km the means that the Earth's radius is only 0.004% of the orbital distance so it is reasonable to consider it a particle when we deal with orbits. We reduce things to particles to simplify the analysis as the geometry of the object will not be involved in the analysis.\r\n\r\nRigid Body:<\/strong>\u00a0 A rigid body can be considered to be a combination of a large number of particles where all the particles remain at the same distance from each other, even when a large force or load is applied to the body.\u00a0 The shape does not change.\u00a0 In most cases the actual deformations that occur when a body is forced is relatively small.\r\n\r\nConcentrated Force.<\/strong>\u00a0 A concentrated force is the effect of a load which is assumed to act at point on a rigid body.\u00a0 We can represent the total load by a concentrated force provided the area over which the load is applied is very small compared to the overall size of the body.\u00a0 Think of a wheel in contact with the ground.\r\nNewton's Three Laws of Motion<\/h1>\r\nThese laws are valid for non-accelerating reference frames.\u00a0 \u00a0First formulated in Philosophi\u00e6 Naturalis Principia Mathematica, <\/i>often referred to as just The Principia.\u00a0<\/em> it was first published in July of 1687 and is considered one of the most important works in the history of science.\u00a0 These laws have been experimentally verified for many hundreds of years. While Einstein's laws modify these laws at extreme masses and speed, the study of engineering mechanics deals with Newton's Laws.\r\nFirst law<\/h2>\r\nOften called the law of inertia.\u00a0 A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net<\/strong> external force. The property of a body to remain at rest or to remain in motion with constant velocity is called inertia<\/strong>.\u00a0 We say that a body is in equilibrium when the net force acting on it is zero.\r\n\r\n[caption id=\"attachment_211\" align=\"aligncenter\" width=\"1871\"]![\"first](\"https:\/\/pressbooks.bccampus.ca\/bcengrphys3\/wp-content\/uploads\/sites\/926\/2020\/12\/newton-first.jpg\")
These force vectors are drawn such that the magnitude of the vectors is proportional to its length so in both cases the net force is zero. Credit: Jennifer Kirkey 2020 CCBY[\/caption]\r\nSecond Law<\/h2>\r\nNewton\u2019s second law of motion<\/span><\/strong> is closely related to Newton\u2019s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton\u2019s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration<\/span><\/strong>. Newton\u2019s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration<\/em>. You must define the boundaries of the system before you can determine which forces are external<\/em>. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton\u2019s laws.\r\nTo obtain an equation for Newton\u2019s second law, we first write the relationship of acceleration and net external force as the proportionality<\/p>\r\n\r\n
[latex] {a}\\propto F_{\\textbf{net}} ,[\/latex]<\/div>\r\nwhere the symbol [latex] \\propto [\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\r\nIn equation form, Newton\u2019s second law of motion is<\/p>\r\n\r\n
[latex]\\boldsymbol{\\vec{\\textbf{F}}_{\\textbf{net}}=m}\\vec{\\textbf{a}}.[\/latex]<\/div>\r\nRemember that mass,\u00a0<\/strong> m,\u00a0 <\/strong><\/em>is the quantity of matter in a substance.\u00a0 We often use bold text as opposed to the vector symbol, and net force is often just written as F to give F<\/strong> = ma<\/strong>.<\/div>\r\n<\/div>\r\n\r\n\r\n[caption id=\"attachment_212\" align=\"aligncenter\" width=\"1024\"]![\"A](\"https:\/\/pressbooks.bccampus.ca\/bcengrphys3\/wp-content\/uploads\/sites\/926\/2020\/12\/newton-second-1024x586.jpg\")
As there is an unbalanced force, there is a net force acting on the body so there is an acceleration. Credit: Jennifer Kirkey 2020 CCBY[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\nThe same net external force applied to a car produces a much smaller acceleration than when applied to a basketball, as illustrated below.<\/div>\r\n<\/div>\r\n\r\n\r\n[caption id=\"attachment_196\" align=\"aligncenter\" width=\"1000\"]![\"A](\"https:\/\/pressbooks.bccampus.ca\/bcengrphys3\/wp-content\/uploads\/sites\/926\/2020\/12\/44230a76249c989838ed3776b13f50f15a74690c.jpg\")
The same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Photo credit: OpenStax College Physics.[\/caption]\r\n\r\n<\/div>\r\nThird Law<\/h2>\r\nWhenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.\r\n\r\n \r\n\r\n[caption id=\"attachment_213\" align=\"aligncenter\" width=\"1024\"]![\"Two](\"https:\/\/pressbooks.bccampus.ca\/bcengrphys3\/wp-content\/uploads\/sites\/926\/2020\/12\/newton-third-1024x695.jpg\")
Action - reaction forces on two bodies that are touching.[\/caption]\r\n\r\nWe can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em>wall on feet<\/sub><\/strong>. Note that the swimmer pushes in the direction opposite to that in which she wishes to move. The reaction to her push is thus in the desired direction.\r\n\r\n \r\n\r\n[caption id=\"attachment_198\" align=\"aligncenter\" width=\"875\"]![\"A](\"https:\/\/pressbooks.bccampus.ca\/bcengrphys3\/wp-content\/uploads\/sites\/926\/2020\/12\/98abd08fa724a4afeb0e3114154eab1bd6311d37.jpg\")
When the swimmer exerts a force \ud835\udc05feet on wall<\/sub> on the wall, she accelerates in the direction opposite to that of her push. This means the net external force on her is in the direction opposite to \ud835\udc05feet on wall<\/sub>. This opposition occurs because, in accordance with Newton\u2019s third law of motion, the wall exerts a force \ud835\udc05wall on feet<\/sub> on her, equal in magnitude but in the direction opposite to the one she exerts on it. The line around the swimmer indicates the system of interest. Note that \ud835\udc05feet on wall<\/sub> does not act on this system (the swimmer) and, thus, does not cancel \ud835\udc05wall on feet.<\/sub> Thus the free-body diagram shows only \ud835\udc05wall on feet,<\/sub> \ud835\udc30, the gravitational force, and \ud835\udc01\ud835\udc05, the buoyant force of the water supporting the swimmer\u2019s weight. The vertical forces \ud835\udc30 and \ud835\udc01\ud835\udc05 cancel since there is no vertical motion. Photo credit: OpenStax College Physics[\/caption]\r\nNewton's Law of Gravitational Attraction<\/h1>\r\nAfter he had formulated his three laws of motion, Newton began to think about gravity. His law states that each mass reaches out through space and attracts (forces toward) every other mass to itself.\u00a0 In equation form\r\n
[latex] F_{gravity} = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nwhere\r\n\r\nF<\/em> = force of gravitation between the two particles\r\n\r\nG<\/em> = universal constant of gravitation.\u00a0 \u00a0Using the SI or Metric system G = 66.73(10-12<\/sup>) m3<\/sup>\/(kg \u00b7s2<\/sup>).\u00a0 It was measured experimentally for the first time in 1798 by Henry Cavendish.\r\n\r\nm1<\/sub>, m2<\/sub> = the masses of the two particles\r\n\r\nr = distance between the two particles.\r\nWeight and the Gravitational Force<\/h1>\r\n\r\nWhen an object is dropped, it accelerates toward the centre of Earth. Newton\u2019s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight W<\/em><\/span><\/strong>. Weight can be denoted as a vector [latex]\\vec{\\textbf{W}}[\/latex] because it has a direction; down<\/em> is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is often denoted as W, <\/strong><\/em>though F gravity<\/sub><\/strong> will often be used.\u00a0 \u00a0Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration g<\/strong><\/em>.<\/p>\r\n\r\n<\/section>\r\n[latex] F_{gravity} = W = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nIf you take m2<\/sub> = MEarth<\/sub> = 5.972 x 1024<\/sup> kg, and r = distance between a particle on the surface of the Earth and the centre of the Earth = 6378 km = 6.378 Mm at the equator,\u00a0 then letting\r\n\r\n[latex] g = G \\frac { m_{Earth} } {r^2} [\/latex]\u00a0 will give you\r\nW = m g\u00a0<\/em><\/p>\r\n\r\n\r\nWhen the net external force on an object is its weight, we say that it is in free-fall<\/span><\/strong>. That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air acting on the object.<\/p>\r\nThe acceleration due to gravity g<\/strong><\/em> varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. The equatorial radius of the Earth is 6378 km while the polar radius is 6357 km.\u00a0 The International Bureau of Weights and Measures has a standard value for g adopted in 1901 that is based on sea level and at a latitude of 45 degrees equal to 9.80665 m\/s<\/span>2<\/sup>.<\/span>\u00a0<\/span><\/p>\r\n\r\n<\/section>https:\/\/www.bipm.org\/en\/CGPM\/db\/3\/2\/#:~:text=The%20value%20adopted%20in%20the,the%20laws%20of%20some%20countries.<\/a>\r\n\r\nWeight varies dramatically if one leaves Earth\u2019s surface. On the Moon, for example, the acceleration due to gravity is only 1.67 m\/s2<\/sup><\/strong>. A 1.0-kg mass thus has a weight of 9.8 N on Earth and only about 1.7 N on the Moon.\r\nThe broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large body<\/em>, such as Earth, the Moon, the Sun, and so on. This is the most common and useful definition of weight in physics. It differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of \u201cweightlessness\u201d and \u201cmicrogravity,\u201d they are really referring to the phenomenon we call \u201cfree-fall\u201d in physics. We shall use the above definition of weight, and we will make careful distinctions between free-fall and actual weightlessness.<\/p>\r\nIt is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much \u201cstuff\u201d) and does not vary in classical physics, whereas weight is the gravitational force and does vary depending on gravity. It is tempting to equate the two, since most of our examples take place on Earth, where the weight of an object only varies a little with the location of the object. Furthermore, the terms mass<\/em> and weight<\/em> are used interchangeably in everyday language; for example, our medical records often show our \u201cweight\u201d in kilograms, but never in the correct units of newtons.<\/p>\r\n\r\n\r\n\r\n\r\nCOMMON MISCONCEPTIONS: MASS VS. WEIGHT\r\n<\/span><\/h3>\r\nMass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the \u201cslug\u201d in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object (m<\/strong><\/em>) multiplied by the acceleration due to gravity (g<\/strong><\/em>). Like any other force, weight is measured in terms of newtons (or pounds in English units).<\/p>\r\nAssuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change<\/em> when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is 1.67 m\/s2<\/sup><\/strong> (which is much less than the acceleration due to gravity on Earth, 9.80 m\/s2<\/sup><\/strong>). If you measured your weight on Earth and then measured your weight on the Moon, you would find that you \u201cweigh\u201d much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are \u201closing weight,\u201d they really mean that they are losing \u201cmass\u201d (which in turn causes them to weigh less).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section> \r\nReview of Vectors and Scalars from OpenStax College Physics<\/h1>\r\n
The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (\u2212) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector\u2019s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector.<\/p>\r\n
Some physical quantities, like distance, either have no direction or none is specified. A scalar<\/span> <\/strong>is any quantity that has a magnitude, but no direction. For example, a 20 \u00b0C temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km\/h speed limit, a person\u2019s 1.8 m height, and a distance of 2.0 m are all scalars\u2014quantities with no specified direction. Note, however, that a scalar can be negative, such as a -20 \u00b0C temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.<\/p>\r\n\r\n\r\nCoordinate Systems for One-Dimensional Motion<\/h2>\r\n
In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however,\u00a0 it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.<\/p>\r\n\r\n<\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"200\"]![\"An](\"https:\/\/pressbooks.bccampus.ca\/phys1107introductorygeneralphysics\/wp-content\/uploads\/sites\/896\/2020\/01\/Figure_02_02_00b-1.jpg\")
Figure:<\/strong>\u00a0It is usually convenient to consider motion upward or to the right as positive (+)<\/strong> and motion downward or to the left as negative (\u2212)<\/strong>.[\/caption]\r\n\r\n\u00a0<\/span><\/figure>\r\n\r\n\r\n\r\nCheck Your Understanding<\/h3>\r\n\r\n1:<\/strong> A person\u2019s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\nSection Summary<\/h2>\r\n\r\n \t- A vector is any quantity that has magnitude and direction.<\/li>\r\n \t
- A scalar is any quantity that has magnitude but no direction.<\/li>\r\n \t
- Displacement and velocity are vectors, whereas distance and speed are scalars.<\/li>\r\n \t
- In one-dimensional motion, direction is specified by a plus or minus sign to signify left or right, up or down, and the like.<\/li>\r\n<\/ul>\r\n<\/section>
\r\n\r\nConceptual Questions<\/h3>\r\n\r\n\r\n1: <\/strong>A student writes, \u201cA bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d What is wrong with the student\u2019s statement? What has the student actually described? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n2: <\/strong>What is the speed of the bird in the question above where it is stated \"A bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n3: <\/strong>Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n4: <\/strong>A weather forecast states that the temperature is predicted to be -5 \u00b0C the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<\/div>\r\n\r\nSolutions<\/h3>\r\nCheck Your Understanding<\/strong>\r\n\r\n \t- Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant).<\/li>\r\n<\/ol>\r\nConceptual Questions\r\n
\r\n \t- Speed is a scalar so it is correct to say \"speed of 10 m\/s\".\u00a0 The student has described the velocity vector as they gave both magnitude (speed of 10 m\/s) and direction of \"-\".<\/li>\r\n \t
- Speed is a scalar so it is correct to say \"speed of 10 m\/s\".<\/li>\r\n \t
- Acceleration is a vector as velocity is a vector.<\/li>\r\n \t
- Scalar.\u00a0 Scalars can be negative as well as positive.<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"
Let us start off with a brief review of certain fundamental concepts and principles.<\/p>\n
In this course it is assumed that you have completed a basic physics class in your secondary school.\u00a0 If you need more information, I recommend that you refer to two open source textbooks.\u00a0 OpenStax College Physics is an algebra-based book.\u00a0 \u00a0 https:\/\/openstax.org\/details\/books\/college-physics\u00a0<\/a> \u00a0OpenStax University Physics is a calculus-based textbook.\u00a0 https:\/\/openstax.org\/details\/books\/university-physics-volume-1\u00a0<\/a> I will be using some examples and images from them in this book but I encourage you to look them over before and maybe during this course.<\/p>\nBasic Quantities<\/h1>\n
The following four quantities are used when we talk about mechanics.\u00a0 They are often called “base units” as other units can be made by combining the base units.<\/p>\n
Length.<\/strong> Length is used to locate the position of a point in space.\u00a0 \u00a0It describes the size of a certain physical system.\u00a0 \u00a0Once you have length, you can have area and volume.<\/p>\nTime.<\/strong> Time is the indefinite continued progress of existence and events that occur in an apparently irreversible succession from the past, through the present, into the future.<\/p>\nMass.<\/strong> Mass is a measure of the quantity of matter.\u00a0 It is defined by its inertia, its resistance to being accelerated, and by the gravitational attractions that occurs between two bodies.<\/p>\nForce.<\/strong> Forces are considered to be pushes or pulls exerted by one body on another body.\u00a0 It can occur due to direct contact when two bodies touch or there can be action at a distance such as the gravitational, magnetic and electric forces.\u00a0 Forces are vectors and is characterized by its magnitude, direction and the point where it is applied.<\/p>\nModels or Idealizations<\/h1>\n
Models, or idealizations, are used in science in order to simplify applications of theories.\u00a0 Here are three main idealizations we will use in this textbooks.\u00a0 There is an aphorism, often attributed to George Box a statistician that goes “all models are wrong, but some are useful.”<\/p>\n
Particle.<\/strong>\u00a0 A particle has a mass, but the size can be neglected.\u00a0 This often depends on the system.\u00a0 As we stand on the Earth we are aware that is it not a particle. Yet when we consider the Earth orbiting around the Sun we treat it as a particle because with a radius of 6378 km and the semi-major axis of the orbit being 149,596,000 km the means that the Earth’s radius is only 0.004% of the orbital distance so it is reasonable to consider it a particle when we deal with orbits. We reduce things to particles to simplify the analysis as the geometry of the object will not be involved in the analysis.<\/p>\nRigid Body:<\/strong>\u00a0 A rigid body can be considered to be a combination of a large number of particles where all the particles remain at the same distance from each other, even when a large force or load is applied to the body.\u00a0 The shape does not change.\u00a0 In most cases the actual deformations that occur when a body is forced is relatively small.<\/p>\nConcentrated Force.<\/strong>\u00a0 A concentrated force is the effect of a load which is assumed to act at point on a rigid body.\u00a0 We can represent the total load by a concentrated force provided the area over which the load is applied is very small compared to the overall size of the body.\u00a0 Think of a wheel in contact with the ground.<\/p>\nNewton’s Three Laws of Motion<\/h1>\n
These laws are valid for non-accelerating reference frames.\u00a0 \u00a0First formulated in Philosophi\u00e6 Naturalis Principia Mathematica, <\/i>often referred to as just The Principia.\u00a0<\/em> it was first published in July of 1687 and is considered one of the most important works in the history of science.\u00a0 These laws have been experimentally verified for many hundreds of years. While Einstein’s laws modify these laws at extreme masses and speed, the study of engineering mechanics deals with Newton’s Laws.<\/p>\nFirst law<\/h2>\n
Often called the law of inertia.\u00a0 A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net<\/strong> external force. The property of a body to remain at rest or to remain in motion with constant velocity is called inertia<\/strong>.\u00a0 We say that a body is in equilibrium when the net force acting on it is zero.<\/p>\nThese force vectors are drawn such that the magnitude of the vectors is proportional to its length so in both cases the net force is zero. Credit: Jennifer Kirkey 2020 CCBY<\/figcaption><\/figure>\nSecond Law<\/h2>\n
Newton\u2019s second law of motion<\/span><\/strong> is closely related to Newton\u2019s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton\u2019s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration<\/span><\/strong>. Newton\u2019s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration<\/em>. You must define the boundaries of the system before you can determine which forces are external<\/em>. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton\u2019s laws.<\/p>\nTo obtain an equation for Newton\u2019s second law, we first write the relationship of acceleration and net external force as the proportionality<\/p>\n
[latex]{a}\\propto F_{\\textbf{net}} ,[\/latex]<\/div>\nwhere the symbol [latex]\\propto[\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\nIn equation form, Newton\u2019s second law of motion is<\/p>\n
[latex]\\boldsymbol{\\vec{\\textbf{F}}_{\\textbf{net}}=m}\\vec{\\textbf{a}}.[\/latex]<\/div>\nRemember that mass,\u00a0<\/strong> m,\u00a0 <\/strong><\/em>is the quantity of matter in a substance.\u00a0 We often use bold text as opposed to the vector symbol, and net force is often just written as F to give F<\/strong> = ma<\/strong>.<\/div>\n<\/div>\n\nAs there is an unbalanced force, there is a net force acting on the body so there is an acceleration. Credit: Jennifer Kirkey 2020 CCBY<\/figcaption><\/figure>\n<\/div>\n<\/div>\nThe same net external force applied to a car produces a much smaller acceleration than when applied to a basketball, as illustrated below.<\/div>\n<\/div>\n\nThe same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Photo credit: OpenStax College Physics.<\/figcaption><\/figure>\n<\/div>\nThird Law<\/h2>\n
Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.<\/p>\n
<\/p>\nAction – reaction forces on two bodies that are touching.<\/figcaption><\/figure>\nWe can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em>
Newton's Three Laws of Motion<\/h1>\r\nThese laws are valid for non-accelerating reference frames.\u00a0 \u00a0First formulated in Philosophi\u00e6 Naturalis Principia Mathematica, <\/i>often referred to as just The Principia.\u00a0<\/em> it was first published in July of 1687 and is considered one of the most important works in the history of science.\u00a0 These laws have been experimentally verified for many hundreds of years. While Einstein's laws modify these laws at extreme masses and speed, the study of engineering mechanics deals with Newton's Laws.\r\nFirst law<\/h2>\r\nOften called the law of inertia.\u00a0 A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net<\/strong> external force. The property of a body to remain at rest or to remain in motion with constant velocity is called inertia<\/strong>.\u00a0 We say that a body is in equilibrium when the net force acting on it is zero.\r\n\r\n[caption id=\"attachment_211\" align=\"aligncenter\" width=\"1871\"] These force vectors are drawn such that the magnitude of the vectors is proportional to its length so in both cases the net force is zero. Credit: Jennifer Kirkey 2020 CCBY[\/caption]\r\nSecond Law<\/h2>\r\nNewton\u2019s second law of motion<\/span><\/strong> is closely related to Newton\u2019s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton\u2019s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration<\/span><\/strong>. Newton\u2019s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration<\/em>. You must define the boundaries of the system before you can determine which forces are external<\/em>. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton\u2019s laws.\r\nTo obtain an equation for Newton\u2019s second law, we first write the relationship of acceleration and net external force as the proportionality<\/p>\r\n\r\n
[latex] {a}\\propto F_{\\textbf{net}} ,[\/latex]<\/div>\r\nwhere the symbol [latex] \\propto [\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\r\nIn equation form, Newton\u2019s second law of motion is<\/p>\r\n\r\n
[latex]\\boldsymbol{\\vec{\\textbf{F}}_{\\textbf{net}}=m}\\vec{\\textbf{a}}.[\/latex]<\/div>\r\nRemember that mass,\u00a0<\/strong> m,\u00a0 <\/strong><\/em>is the quantity of matter in a substance.\u00a0 We often use bold text as opposed to the vector symbol, and net force is often just written as F to give F<\/strong> = ma<\/strong>.<\/div>\r\n<\/div>\r\n\r\n\r\n[caption id=\"attachment_212\" align=\"aligncenter\" width=\"1024\"] As there is an unbalanced force, there is a net force acting on the body so there is an acceleration. Credit: Jennifer Kirkey 2020 CCBY[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\nThe same net external force applied to a car produces a much smaller acceleration than when applied to a basketball, as illustrated below.<\/div>\r\n<\/div>\r\n\r\n\r\n[caption id=\"attachment_196\" align=\"aligncenter\" width=\"1000\"] The same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Photo credit: OpenStax College Physics.[\/caption]\r\n\r\n<\/div>\r\nThird Law<\/h2>\r\nWhenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.\r\n\r\n \r\n\r\n[caption id=\"attachment_213\" align=\"aligncenter\" width=\"1024\"] Action - reaction forces on two bodies that are touching.[\/caption]\r\n\r\nWe can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em>wall on feet<\/sub><\/strong>. Note that the swimmer pushes in the direction opposite to that in which she wishes to move. The reaction to her push is thus in the desired direction.\r\n\r\n \r\n\r\n[caption id=\"attachment_198\" align=\"aligncenter\" width=\"875\"] When the swimmer exerts a force \ud835\udc05feet on wall<\/sub> on the wall, she accelerates in the direction opposite to that of her push. This means the net external force on her is in the direction opposite to \ud835\udc05feet on wall<\/sub>. This opposition occurs because, in accordance with Newton\u2019s third law of motion, the wall exerts a force \ud835\udc05wall on feet<\/sub> on her, equal in magnitude but in the direction opposite to the one she exerts on it. The line around the swimmer indicates the system of interest. Note that \ud835\udc05feet on wall<\/sub> does not act on this system (the swimmer) and, thus, does not cancel \ud835\udc05wall on feet.<\/sub> Thus the free-body diagram shows only \ud835\udc05wall on feet,<\/sub> \ud835\udc30, the gravitational force, and \ud835\udc01\ud835\udc05, the buoyant force of the water supporting the swimmer\u2019s weight. The vertical forces \ud835\udc30 and \ud835\udc01\ud835\udc05 cancel since there is no vertical motion. Photo credit: OpenStax College Physics[\/caption]\r\nNewton's Law of Gravitational Attraction<\/h1>\r\nAfter he had formulated his three laws of motion, Newton began to think about gravity. His law states that each mass reaches out through space and attracts (forces toward) every other mass to itself.\u00a0 In equation form\r\n
[latex] F_{gravity} = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nwhere\r\n\r\nF<\/em> = force of gravitation between the two particles\r\n\r\nG<\/em> = universal constant of gravitation.\u00a0 \u00a0Using the SI or Metric system G = 66.73(10-12<\/sup>) m3<\/sup>\/(kg \u00b7s2<\/sup>).\u00a0 It was measured experimentally for the first time in 1798 by Henry Cavendish.\r\n\r\nm1<\/sub>, m2<\/sub> = the masses of the two particles\r\n\r\nr = distance between the two particles.\r\nWeight and the Gravitational Force<\/h1>\r\n\r\nWhen an object is dropped, it accelerates toward the centre of Earth. Newton\u2019s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight W<\/em><\/span><\/strong>. Weight can be denoted as a vector [latex]\\vec{\\textbf{W}}[\/latex] because it has a direction; down<\/em> is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is often denoted as W, <\/strong><\/em>though F gravity<\/sub><\/strong> will often be used.\u00a0 \u00a0Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration g<\/strong><\/em>.<\/p>\r\n\r\n<\/section>\r\n[latex] F_{gravity} = W = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nIf you take m2<\/sub> = MEarth<\/sub> = 5.972 x 1024<\/sup> kg, and r = distance between a particle on the surface of the Earth and the centre of the Earth = 6378 km = 6.378 Mm at the equator,\u00a0 then letting\r\n\r\n[latex] g = G \\frac { m_{Earth} } {r^2} [\/latex]\u00a0 will give you\r\nW = m g\u00a0<\/em><\/p>\r\n\r\n\r\nWhen the net external force on an object is its weight, we say that it is in free-fall<\/span><\/strong>. That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air acting on the object.<\/p>\r\nThe acceleration due to gravity g<\/strong><\/em> varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. The equatorial radius of the Earth is 6378 km while the polar radius is 6357 km.\u00a0 The International Bureau of Weights and Measures has a standard value for g adopted in 1901 that is based on sea level and at a latitude of 45 degrees equal to 9.80665 m\/s<\/span>2<\/sup>.<\/span>\u00a0<\/span><\/p>\r\n\r\n<\/section>https:\/\/www.bipm.org\/en\/CGPM\/db\/3\/2\/#:~:text=The%20value%20adopted%20in%20the,the%20laws%20of%20some%20countries.<\/a>\r\n\r\nWeight varies dramatically if one leaves Earth\u2019s surface. On the Moon, for example, the acceleration due to gravity is only 1.67 m\/s2<\/sup><\/strong>. A 1.0-kg mass thus has a weight of 9.8 N on Earth and only about 1.7 N on the Moon.\r\nThe broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large body<\/em>, such as Earth, the Moon, the Sun, and so on. This is the most common and useful definition of weight in physics. It differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of \u201cweightlessness\u201d and \u201cmicrogravity,\u201d they are really referring to the phenomenon we call \u201cfree-fall\u201d in physics. We shall use the above definition of weight, and we will make careful distinctions between free-fall and actual weightlessness.<\/p>\r\nIt is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much \u201cstuff\u201d) and does not vary in classical physics, whereas weight is the gravitational force and does vary depending on gravity. It is tempting to equate the two, since most of our examples take place on Earth, where the weight of an object only varies a little with the location of the object. Furthermore, the terms mass<\/em> and weight<\/em> are used interchangeably in everyday language; for example, our medical records often show our \u201cweight\u201d in kilograms, but never in the correct units of newtons.<\/p>\r\n\r\n\r\n\r\n\r\nCOMMON MISCONCEPTIONS: MASS VS. WEIGHT\r\n<\/span><\/h3>\r\nMass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the \u201cslug\u201d in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object (m<\/strong><\/em>) multiplied by the acceleration due to gravity (g<\/strong><\/em>). Like any other force, weight is measured in terms of newtons (or pounds in English units).<\/p>\r\nAssuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change<\/em> when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is 1.67 m\/s2<\/sup><\/strong> (which is much less than the acceleration due to gravity on Earth, 9.80 m\/s2<\/sup><\/strong>). If you measured your weight on Earth and then measured your weight on the Moon, you would find that you \u201cweigh\u201d much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are \u201closing weight,\u201d they really mean that they are losing \u201cmass\u201d (which in turn causes them to weigh less).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section> \r\nReview of Vectors and Scalars from OpenStax College Physics<\/h1>\r\n
The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (\u2212) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector\u2019s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector.<\/p>\r\n
Some physical quantities, like distance, either have no direction or none is specified. A scalar<\/span> <\/strong>is any quantity that has a magnitude, but no direction. For example, a 20 \u00b0C temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km\/h speed limit, a person\u2019s 1.8 m height, and a distance of 2.0 m are all scalars\u2014quantities with no specified direction. Note, however, that a scalar can be negative, such as a -20 \u00b0C temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.<\/p>\r\n\r\n\r\nCoordinate Systems for One-Dimensional Motion<\/h2>\r\n
In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however,\u00a0 it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.<\/p>\r\n\r\n<\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"200\"] Figure:<\/strong>\u00a0It is usually convenient to consider motion upward or to the right as positive (+)<\/strong> and motion downward or to the left as negative (\u2212)<\/strong>.[\/caption]\r\n\r\n\u00a0<\/span><\/figure>\r\n\r\n\r\n\r\nCheck Your Understanding<\/h3>\r\n\r\n1:<\/strong> A person\u2019s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\nSection Summary<\/h2>\r\n\r\n \t- A vector is any quantity that has magnitude and direction.<\/li>\r\n \t
- A scalar is any quantity that has magnitude but no direction.<\/li>\r\n \t
- Displacement and velocity are vectors, whereas distance and speed are scalars.<\/li>\r\n \t
- In one-dimensional motion, direction is specified by a plus or minus sign to signify left or right, up or down, and the like.<\/li>\r\n<\/ul>\r\n<\/section>
\r\n\r\nConceptual Questions<\/h3>\r\n\r\n\r\n1: <\/strong>A student writes, \u201cA bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d What is wrong with the student\u2019s statement? What has the student actually described? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n2: <\/strong>What is the speed of the bird in the question above where it is stated \"A bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n3: <\/strong>Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n4: <\/strong>A weather forecast states that the temperature is predicted to be -5 \u00b0C the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<\/div>\r\n\r\nSolutions<\/h3>\r\nCheck Your Understanding<\/strong>\r\n\r\n \t- Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant).<\/li>\r\n<\/ol>\r\nConceptual Questions\r\n
\r\n \t- Speed is a scalar so it is correct to say \"speed of 10 m\/s\".\u00a0 The student has described the velocity vector as they gave both magnitude (speed of 10 m\/s) and direction of \"-\".<\/li>\r\n \t
- Speed is a scalar so it is correct to say \"speed of 10 m\/s\".<\/li>\r\n \t
- Acceleration is a vector as velocity is a vector.<\/li>\r\n \t
- Scalar.\u00a0 Scalars can be negative as well as positive.<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"
Let us start off with a brief review of certain fundamental concepts and principles.<\/p>\n
In this course it is assumed that you have completed a basic physics class in your secondary school.\u00a0 If you need more information, I recommend that you refer to two open source textbooks.\u00a0 OpenStax College Physics is an algebra-based book.\u00a0 \u00a0 https:\/\/openstax.org\/details\/books\/college-physics\u00a0<\/a> \u00a0OpenStax University Physics is a calculus-based textbook.\u00a0 https:\/\/openstax.org\/details\/books\/university-physics-volume-1\u00a0<\/a> I will be using some examples and images from them in this book but I encourage you to look them over before and maybe during this course.<\/p>\nBasic Quantities<\/h1>\n
The following four quantities are used when we talk about mechanics.\u00a0 They are often called “base units” as other units can be made by combining the base units.<\/p>\n
Length.<\/strong> Length is used to locate the position of a point in space.\u00a0 \u00a0It describes the size of a certain physical system.\u00a0 \u00a0Once you have length, you can have area and volume.<\/p>\nTime.<\/strong> Time is the indefinite continued progress of existence and events that occur in an apparently irreversible succession from the past, through the present, into the future.<\/p>\nMass.<\/strong> Mass is a measure of the quantity of matter.\u00a0 It is defined by its inertia, its resistance to being accelerated, and by the gravitational attractions that occurs between two bodies.<\/p>\nForce.<\/strong> Forces are considered to be pushes or pulls exerted by one body on another body.\u00a0 It can occur due to direct contact when two bodies touch or there can be action at a distance such as the gravitational, magnetic and electric forces.\u00a0 Forces are vectors and is characterized by its magnitude, direction and the point where it is applied.<\/p>\nModels or Idealizations<\/h1>\n
Models, or idealizations, are used in science in order to simplify applications of theories.\u00a0 Here are three main idealizations we will use in this textbooks.\u00a0 There is an aphorism, often attributed to George Box a statistician that goes “all models are wrong, but some are useful.”<\/p>\n
Particle.<\/strong>\u00a0 A particle has a mass, but the size can be neglected.\u00a0 This often depends on the system.\u00a0 As we stand on the Earth we are aware that is it not a particle. Yet when we consider the Earth orbiting around the Sun we treat it as a particle because with a radius of 6378 km and the semi-major axis of the orbit being 149,596,000 km the means that the Earth’s radius is only 0.004% of the orbital distance so it is reasonable to consider it a particle when we deal with orbits. We reduce things to particles to simplify the analysis as the geometry of the object will not be involved in the analysis.<\/p>\nRigid Body:<\/strong>\u00a0 A rigid body can be considered to be a combination of a large number of particles where all the particles remain at the same distance from each other, even when a large force or load is applied to the body.\u00a0 The shape does not change.\u00a0 In most cases the actual deformations that occur when a body is forced is relatively small.<\/p>\nConcentrated Force.<\/strong>\u00a0 A concentrated force is the effect of a load which is assumed to act at point on a rigid body.\u00a0 We can represent the total load by a concentrated force provided the area over which the load is applied is very small compared to the overall size of the body.\u00a0 Think of a wheel in contact with the ground.<\/p>\nNewton’s Three Laws of Motion<\/h1>\n
These laws are valid for non-accelerating reference frames.\u00a0 \u00a0First formulated in Philosophi\u00e6 Naturalis Principia Mathematica, <\/i>often referred to as just The Principia.\u00a0<\/em> it was first published in July of 1687 and is considered one of the most important works in the history of science.\u00a0 These laws have been experimentally verified for many hundreds of years. While Einstein’s laws modify these laws at extreme masses and speed, the study of engineering mechanics deals with Newton’s Laws.<\/p>\nFirst law<\/h2>\n
Often called the law of inertia.\u00a0 A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net<\/strong> external force. The property of a body to remain at rest or to remain in motion with constant velocity is called inertia<\/strong>.\u00a0 We say that a body is in equilibrium when the net force acting on it is zero.<\/p>\nThese force vectors are drawn such that the magnitude of the vectors is proportional to its length so in both cases the net force is zero. Credit: Jennifer Kirkey 2020 CCBY<\/figcaption><\/figure>\nSecond Law<\/h2>\n
Newton\u2019s second law of motion<\/span><\/strong> is closely related to Newton\u2019s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton\u2019s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration<\/span><\/strong>. Newton\u2019s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration<\/em>. You must define the boundaries of the system before you can determine which forces are external<\/em>. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton\u2019s laws.<\/p>\nTo obtain an equation for Newton\u2019s second law, we first write the relationship of acceleration and net external force as the proportionality<\/p>\n
[latex]{a}\\propto F_{\\textbf{net}} ,[\/latex]<\/div>\nwhere the symbol [latex]\\propto[\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\nIn equation form, Newton\u2019s second law of motion is<\/p>\n
[latex]\\boldsymbol{\\vec{\\textbf{F}}_{\\textbf{net}}=m}\\vec{\\textbf{a}}.[\/latex]<\/div>\nRemember that mass,\u00a0<\/strong> m,\u00a0 <\/strong><\/em>is the quantity of matter in a substance.\u00a0 We often use bold text as opposed to the vector symbol, and net force is often just written as F to give F<\/strong> = ma<\/strong>.<\/div>\n<\/div>\n\nAs there is an unbalanced force, there is a net force acting on the body so there is an acceleration. Credit: Jennifer Kirkey 2020 CCBY<\/figcaption><\/figure>\n<\/div>\n<\/div>\nThe same net external force applied to a car produces a much smaller acceleration than when applied to a basketball, as illustrated below.<\/div>\n<\/div>\n\nThe same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Photo credit: OpenStax College Physics.<\/figcaption><\/figure>\n<\/div>\nThird Law<\/h2>\n
Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.<\/p>\n
<\/p>\nAction – reaction forces on two bodies that are touching.<\/figcaption><\/figure>\nWe can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em>
These force vectors are drawn such that the magnitude of the vectors is proportional to its length so in both cases the net force is zero. Credit: Jennifer Kirkey 2020 CCBY[\/caption]\r\nSecond Law<\/h2>\r\nNewton\u2019s second law of motion<\/span><\/strong> is closely related to Newton\u2019s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton\u2019s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration<\/span><\/strong>. Newton\u2019s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration<\/em>. You must define the boundaries of the system before you can determine which forces are external<\/em>. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton\u2019s laws.\r\nTo obtain an equation for Newton\u2019s second law, we first write the relationship of acceleration and net external force as the proportionality<\/p>\r\n\r\n
[latex] {a}\\propto F_{\\textbf{net}} ,[\/latex]<\/div>\r\nwhere the symbol [latex] \\propto [\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\r\nIn equation form, Newton\u2019s second law of motion is<\/p>\r\n\r\n
[latex]\\boldsymbol{\\vec{\\textbf{F}}_{\\textbf{net}}=m}\\vec{\\textbf{a}}.[\/latex]<\/div>\r\nRemember that mass,\u00a0<\/strong> m,\u00a0 <\/strong><\/em>is the quantity of matter in a substance.\u00a0 We often use bold text as opposed to the vector symbol, and net force is often just written as F to give F<\/strong> = ma<\/strong>.<\/div>\r\n<\/div>\r\n\r\n\r\n[caption id=\"attachment_212\" align=\"aligncenter\" width=\"1024\"] As there is an unbalanced force, there is a net force acting on the body so there is an acceleration. Credit: Jennifer Kirkey 2020 CCBY[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\nThe same net external force applied to a car produces a much smaller acceleration than when applied to a basketball, as illustrated below.<\/div>\r\n<\/div>\r\n\r\n\r\n[caption id=\"attachment_196\" align=\"aligncenter\" width=\"1000\"] The same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Photo credit: OpenStax College Physics.[\/caption]\r\n\r\n<\/div>\r\nThird Law<\/h2>\r\nWhenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.\r\n\r\n \r\n\r\n[caption id=\"attachment_213\" align=\"aligncenter\" width=\"1024\"] Action - reaction forces on two bodies that are touching.[\/caption]\r\n\r\nWe can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em>wall on feet<\/sub><\/strong>. Note that the swimmer pushes in the direction opposite to that in which she wishes to move. The reaction to her push is thus in the desired direction.\r\n\r\n \r\n\r\n[caption id=\"attachment_198\" align=\"aligncenter\" width=\"875\"] When the swimmer exerts a force \ud835\udc05feet on wall<\/sub> on the wall, she accelerates in the direction opposite to that of her push. This means the net external force on her is in the direction opposite to \ud835\udc05feet on wall<\/sub>. This opposition occurs because, in accordance with Newton\u2019s third law of motion, the wall exerts a force \ud835\udc05wall on feet<\/sub> on her, equal in magnitude but in the direction opposite to the one she exerts on it. The line around the swimmer indicates the system of interest. Note that \ud835\udc05feet on wall<\/sub> does not act on this system (the swimmer) and, thus, does not cancel \ud835\udc05wall on feet.<\/sub> Thus the free-body diagram shows only \ud835\udc05wall on feet,<\/sub> \ud835\udc30, the gravitational force, and \ud835\udc01\ud835\udc05, the buoyant force of the water supporting the swimmer\u2019s weight. The vertical forces \ud835\udc30 and \ud835\udc01\ud835\udc05 cancel since there is no vertical motion. Photo credit: OpenStax College Physics[\/caption]\r\nNewton's Law of Gravitational Attraction<\/h1>\r\nAfter he had formulated his three laws of motion, Newton began to think about gravity. His law states that each mass reaches out through space and attracts (forces toward) every other mass to itself.\u00a0 In equation form\r\n
[latex] F_{gravity} = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nwhere\r\n\r\nF<\/em> = force of gravitation between the two particles\r\n\r\nG<\/em> = universal constant of gravitation.\u00a0 \u00a0Using the SI or Metric system G = 66.73(10-12<\/sup>) m3<\/sup>\/(kg \u00b7s2<\/sup>).\u00a0 It was measured experimentally for the first time in 1798 by Henry Cavendish.\r\n\r\nm1<\/sub>, m2<\/sub> = the masses of the two particles\r\n\r\nr = distance between the two particles.\r\nWeight and the Gravitational Force<\/h1>\r\n\r\nWhen an object is dropped, it accelerates toward the centre of Earth. Newton\u2019s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight W<\/em><\/span><\/strong>. Weight can be denoted as a vector [latex]\\vec{\\textbf{W}}[\/latex] because it has a direction; down<\/em> is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is often denoted as W, <\/strong><\/em>though F gravity<\/sub><\/strong> will often be used.\u00a0 \u00a0Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration g<\/strong><\/em>.<\/p>\r\n\r\n<\/section>\r\n[latex] F_{gravity} = W = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nIf you take m2<\/sub> = MEarth<\/sub> = 5.972 x 1024<\/sup> kg, and r = distance between a particle on the surface of the Earth and the centre of the Earth = 6378 km = 6.378 Mm at the equator,\u00a0 then letting\r\n\r\n[latex] g = G \\frac { m_{Earth} } {r^2} [\/latex]\u00a0 will give you\r\nW = m g\u00a0<\/em><\/p>\r\n\r\n\r\nWhen the net external force on an object is its weight, we say that it is in free-fall<\/span><\/strong>. That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air acting on the object.<\/p>\r\nThe acceleration due to gravity g<\/strong><\/em> varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. The equatorial radius of the Earth is 6378 km while the polar radius is 6357 km.\u00a0 The International Bureau of Weights and Measures has a standard value for g adopted in 1901 that is based on sea level and at a latitude of 45 degrees equal to 9.80665 m\/s<\/span>2<\/sup>.<\/span>\u00a0<\/span><\/p>\r\n\r\n<\/section>https:\/\/www.bipm.org\/en\/CGPM\/db\/3\/2\/#:~:text=The%20value%20adopted%20in%20the,the%20laws%20of%20some%20countries.<\/a>\r\n\r\nWeight varies dramatically if one leaves Earth\u2019s surface. On the Moon, for example, the acceleration due to gravity is only 1.67 m\/s2<\/sup><\/strong>. A 1.0-kg mass thus has a weight of 9.8 N on Earth and only about 1.7 N on the Moon.\r\nThe broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large body<\/em>, such as Earth, the Moon, the Sun, and so on. This is the most common and useful definition of weight in physics. It differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of \u201cweightlessness\u201d and \u201cmicrogravity,\u201d they are really referring to the phenomenon we call \u201cfree-fall\u201d in physics. We shall use the above definition of weight, and we will make careful distinctions between free-fall and actual weightlessness.<\/p>\r\nIt is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much \u201cstuff\u201d) and does not vary in classical physics, whereas weight is the gravitational force and does vary depending on gravity. It is tempting to equate the two, since most of our examples take place on Earth, where the weight of an object only varies a little with the location of the object. Furthermore, the terms mass<\/em> and weight<\/em> are used interchangeably in everyday language; for example, our medical records often show our \u201cweight\u201d in kilograms, but never in the correct units of newtons.<\/p>\r\n\r\n\r\n\r\n\r\nCOMMON MISCONCEPTIONS: MASS VS. WEIGHT\r\n<\/span><\/h3>\r\nMass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the \u201cslug\u201d in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object (m<\/strong><\/em>) multiplied by the acceleration due to gravity (g<\/strong><\/em>). Like any other force, weight is measured in terms of newtons (or pounds in English units).<\/p>\r\nAssuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change<\/em> when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is 1.67 m\/s2<\/sup><\/strong> (which is much less than the acceleration due to gravity on Earth, 9.80 m\/s2<\/sup><\/strong>). If you measured your weight on Earth and then measured your weight on the Moon, you would find that you \u201cweigh\u201d much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are \u201closing weight,\u201d they really mean that they are losing \u201cmass\u201d (which in turn causes them to weigh less).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section> \r\nReview of Vectors and Scalars from OpenStax College Physics<\/h1>\r\n
The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (\u2212) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector\u2019s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector.<\/p>\r\n
Some physical quantities, like distance, either have no direction or none is specified. A scalar<\/span> <\/strong>is any quantity that has a magnitude, but no direction. For example, a 20 \u00b0C temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km\/h speed limit, a person\u2019s 1.8 m height, and a distance of 2.0 m are all scalars\u2014quantities with no specified direction. Note, however, that a scalar can be negative, such as a -20 \u00b0C temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.<\/p>\r\n\r\n\r\nCoordinate Systems for One-Dimensional Motion<\/h2>\r\n
In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however,\u00a0 it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.<\/p>\r\n\r\n<\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"200\"] Figure:<\/strong>\u00a0It is usually convenient to consider motion upward or to the right as positive (+)<\/strong> and motion downward or to the left as negative (\u2212)<\/strong>.[\/caption]\r\n\r\n\u00a0<\/span><\/figure>\r\n\r\n\r\n\r\nCheck Your Understanding<\/h3>\r\n\r\n1:<\/strong> A person\u2019s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\nSection Summary<\/h2>\r\n\r\n \t- A vector is any quantity that has magnitude and direction.<\/li>\r\n \t
- A scalar is any quantity that has magnitude but no direction.<\/li>\r\n \t
- Displacement and velocity are vectors, whereas distance and speed are scalars.<\/li>\r\n \t
- In one-dimensional motion, direction is specified by a plus or minus sign to signify left or right, up or down, and the like.<\/li>\r\n<\/ul>\r\n<\/section>
\r\n\r\nConceptual Questions<\/h3>\r\n\r\n\r\n1: <\/strong>A student writes, \u201cA bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d What is wrong with the student\u2019s statement? What has the student actually described? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n2: <\/strong>What is the speed of the bird in the question above where it is stated \"A bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n3: <\/strong>Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n\r\n4: <\/strong>A weather forecast states that the temperature is predicted to be -5 \u00b0C the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<\/div>\r\n\r\nSolutions<\/h3>\r\nCheck Your Understanding<\/strong>\r\n\r\n \t- Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant).<\/li>\r\n<\/ol>\r\nConceptual Questions\r\n
\r\n \t- Speed is a scalar so it is correct to say \"speed of 10 m\/s\".\u00a0 The student has described the velocity vector as they gave both magnitude (speed of 10 m\/s) and direction of \"-\".<\/li>\r\n \t
- Speed is a scalar so it is correct to say \"speed of 10 m\/s\".<\/li>\r\n \t
- Acceleration is a vector as velocity is a vector.<\/li>\r\n \t
- Scalar.\u00a0 Scalars can be negative as well as positive.<\/li>\r\n<\/ol>\r\n<\/div>","rendered":"
Let us start off with a brief review of certain fundamental concepts and principles.<\/p>\n
In this course it is assumed that you have completed a basic physics class in your secondary school.\u00a0 If you need more information, I recommend that you refer to two open source textbooks.\u00a0 OpenStax College Physics is an algebra-based book.\u00a0 \u00a0 https:\/\/openstax.org\/details\/books\/college-physics\u00a0<\/a> \u00a0OpenStax University Physics is a calculus-based textbook.\u00a0 https:\/\/openstax.org\/details\/books\/university-physics-volume-1\u00a0<\/a> I will be using some examples and images from them in this book but I encourage you to look them over before and maybe during this course.<\/p>\nBasic Quantities<\/h1>\n
The following four quantities are used when we talk about mechanics.\u00a0 They are often called “base units” as other units can be made by combining the base units.<\/p>\n
Length.<\/strong> Length is used to locate the position of a point in space.\u00a0 \u00a0It describes the size of a certain physical system.\u00a0 \u00a0Once you have length, you can have area and volume.<\/p>\nTime.<\/strong> Time is the indefinite continued progress of existence and events that occur in an apparently irreversible succession from the past, through the present, into the future.<\/p>\nMass.<\/strong> Mass is a measure of the quantity of matter.\u00a0 It is defined by its inertia, its resistance to being accelerated, and by the gravitational attractions that occurs between two bodies.<\/p>\nForce.<\/strong> Forces are considered to be pushes or pulls exerted by one body on another body.\u00a0 It can occur due to direct contact when two bodies touch or there can be action at a distance such as the gravitational, magnetic and electric forces.\u00a0 Forces are vectors and is characterized by its magnitude, direction and the point where it is applied.<\/p>\nModels or Idealizations<\/h1>\n
Models, or idealizations, are used in science in order to simplify applications of theories.\u00a0 Here are three main idealizations we will use in this textbooks.\u00a0 There is an aphorism, often attributed to George Box a statistician that goes “all models are wrong, but some are useful.”<\/p>\n
Particle.<\/strong>\u00a0 A particle has a mass, but the size can be neglected.\u00a0 This often depends on the system.\u00a0 As we stand on the Earth we are aware that is it not a particle. Yet when we consider the Earth orbiting around the Sun we treat it as a particle because with a radius of 6378 km and the semi-major axis of the orbit being 149,596,000 km the means that the Earth’s radius is only 0.004% of the orbital distance so it is reasonable to consider it a particle when we deal with orbits. We reduce things to particles to simplify the analysis as the geometry of the object will not be involved in the analysis.<\/p>\nRigid Body:<\/strong>\u00a0 A rigid body can be considered to be a combination of a large number of particles where all the particles remain at the same distance from each other, even when a large force or load is applied to the body.\u00a0 The shape does not change.\u00a0 In most cases the actual deformations that occur when a body is forced is relatively small.<\/p>\nConcentrated Force.<\/strong>\u00a0 A concentrated force is the effect of a load which is assumed to act at point on a rigid body.\u00a0 We can represent the total load by a concentrated force provided the area over which the load is applied is very small compared to the overall size of the body.\u00a0 Think of a wheel in contact with the ground.<\/p>\nNewton’s Three Laws of Motion<\/h1>\n
These laws are valid for non-accelerating reference frames.\u00a0 \u00a0First formulated in Philosophi\u00e6 Naturalis Principia Mathematica, <\/i>often referred to as just The Principia.\u00a0<\/em> it was first published in July of 1687 and is considered one of the most important works in the history of science.\u00a0 These laws have been experimentally verified for many hundreds of years. While Einstein’s laws modify these laws at extreme masses and speed, the study of engineering mechanics deals with Newton’s Laws.<\/p>\nFirst law<\/h2>\n
Often called the law of inertia.\u00a0 A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net<\/strong> external force. The property of a body to remain at rest or to remain in motion with constant velocity is called inertia<\/strong>.\u00a0 We say that a body is in equilibrium when the net force acting on it is zero.<\/p>\nThese force vectors are drawn such that the magnitude of the vectors is proportional to its length so in both cases the net force is zero. Credit: Jennifer Kirkey 2020 CCBY<\/figcaption><\/figure>\nSecond Law<\/h2>\n
Newton\u2019s second law of motion<\/span><\/strong> is closely related to Newton\u2019s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton\u2019s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration<\/span><\/strong>. Newton\u2019s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration<\/em>. You must define the boundaries of the system before you can determine which forces are external<\/em>. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton\u2019s laws.<\/p>\nTo obtain an equation for Newton\u2019s second law, we first write the relationship of acceleration and net external force as the proportionality<\/p>\n
[latex]{a}\\propto F_{\\textbf{net}} ,[\/latex]<\/div>\nwhere the symbol [latex]\\propto[\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\nIn equation form, Newton\u2019s second law of motion is<\/p>\n
[latex]\\boldsymbol{\\vec{\\textbf{F}}_{\\textbf{net}}=m}\\vec{\\textbf{a}}.[\/latex]<\/div>\nRemember that mass,\u00a0<\/strong> m,\u00a0 <\/strong><\/em>is the quantity of matter in a substance.\u00a0 We often use bold text as opposed to the vector symbol, and net force is often just written as F to give F<\/strong> = ma<\/strong>.<\/div>\n<\/div>\n\nAs there is an unbalanced force, there is a net force acting on the body so there is an acceleration. Credit: Jennifer Kirkey 2020 CCBY<\/figcaption><\/figure>\n<\/div>\n<\/div>\nThe same net external force applied to a car produces a much smaller acceleration than when applied to a basketball, as illustrated below.<\/div>\n<\/div>\n\nThe same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Photo credit: OpenStax College Physics.<\/figcaption><\/figure>\n<\/div>\nThird Law<\/h2>\n
Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.<\/p>\n
<\/p>\nAction – reaction forces on two bodies that are touching.<\/figcaption><\/figure>\nWe can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em>
where the symbol [latex] \\propto [\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\r\n In equation form, Newton\u2019s second law of motion is<\/p>\r\n\r\n [latex] F_{gravity} = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nwhere\r\n\r\nF<\/em> = force of gravitation between the two particles\r\n\r\nG<\/em> = universal constant of gravitation.\u00a0 \u00a0Using the SI or Metric system G = 66.73(10-12<\/sup>) m3<\/sup>\/(kg \u00b7s2<\/sup>).\u00a0 It was measured experimentally for the first time in 1798 by Henry Cavendish.\r\n\r\nm1<\/sub>, m2<\/sub> = the masses of the two particles\r\n\r\nr = distance between the two particles.\r\n When an object is dropped, it accelerates toward the centre of Earth. Newton\u2019s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight W<\/em><\/span><\/strong>. Weight can be denoted as a vector [latex]\\vec{\\textbf{W}}[\/latex] because it has a direction; down<\/em> is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is often denoted as W, <\/strong><\/em>though F gravity<\/sub><\/strong> will often be used.\u00a0 \u00a0Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration g<\/strong><\/em>.<\/p>\r\n\r\n<\/section>\r\n [latex] F_{gravity} = W = G \\frac {m_1 m_2} {r^2} [\/latex]<\/p>\r\nIf you take m2<\/sub> = MEarth<\/sub> = 5.972 x 1024<\/sup> kg, and r = distance between a particle on the surface of the Earth and the centre of the Earth = 6378 km = 6.378 Mm at the equator,\u00a0 then letting\r\n\r\n[latex] g = G \\frac { m_{Earth} } {r^2} [\/latex]\u00a0 will give you\r\n W = m g\u00a0<\/em><\/p>\r\n\r\n When the net external force on an object is its weight, we say that it is in free-fall<\/span><\/strong>. That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air acting on the object.<\/p>\r\n The acceleration due to gravity g<\/strong><\/em> varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. The equatorial radius of the Earth is 6378 km while the polar radius is 6357 km.\u00a0 The International Bureau of Weights and Measures has a standard value for g adopted in 1901 that is based on sea level and at a latitude of 45 degrees equal to 9.80665 m\/s<\/span>2<\/sup>.<\/span>\u00a0<\/span><\/p>\r\n\r\n<\/section>https:\/\/www.bipm.org\/en\/CGPM\/db\/3\/2\/#:~:text=The%20value%20adopted%20in%20the,the%20laws%20of%20some%20countries.<\/a>\r\n\r\n The broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large body<\/em>, such as Earth, the Moon, the Sun, and so on. This is the most common and useful definition of weight in physics. It differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of \u201cweightlessness\u201d and \u201cmicrogravity,\u201d they are really referring to the phenomenon we call \u201cfree-fall\u201d in physics. We shall use the above definition of weight, and we will make careful distinctions between free-fall and actual weightlessness.<\/p>\r\n It is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much \u201cstuff\u201d) and does not vary in classical physics, whereas weight is the gravitational force and does vary depending on gravity. It is tempting to equate the two, since most of our examples take place on Earth, where the weight of an object only varies a little with the location of the object. Furthermore, the terms mass<\/em> and weight<\/em> are used interchangeably in everyday language; for example, our medical records often show our \u201cweight\u201d in kilograms, but never in the correct units of newtons.<\/p>\r\n\r\n Mass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the \u201cslug\u201d in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object (m<\/strong><\/em>) multiplied by the acceleration due to gravity (g<\/strong><\/em>). Like any other force, weight is measured in terms of newtons (or pounds in English units).<\/p>\r\n Assuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change<\/em> when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is 1.67 m\/s2<\/sup><\/strong> (which is much less than the acceleration due to gravity on Earth, 9.80 m\/s2<\/sup><\/strong>). If you measured your weight on Earth and then measured your weight on the Moon, you would find that you \u201cweigh\u201d much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are \u201closing weight,\u201d they really mean that they are losing \u201cmass\u201d (which in turn causes them to weigh less).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section> \r\n The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (\u2212) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector\u2019s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector.<\/p>\r\n Some physical quantities, like distance, either have no direction or none is specified. A scalar<\/span> <\/strong>is any quantity that has a magnitude, but no direction. For example, a 20 \u00b0C temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km\/h speed limit, a person\u2019s 1.8 m height, and a distance of 2.0 m are all scalars\u2014quantities with no specified direction. Note, however, that a scalar can be negative, such as a -20 \u00b0C temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.<\/p>\r\n\r\n In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however,\u00a0 it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it.<\/p>\r\n\r\n 1:<\/strong> A person\u2019s speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section> 1: <\/strong>A student writes, \u201cA bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d What is wrong with the student\u2019s statement? What has the student actually described? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 2: <\/strong>What is the speed of the bird in the question above where it is stated \"A bird that is diving for prey has a speed of <\/em>-10 m\/s.\u201d<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 3: <\/strong>Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n 4: <\/strong>A weather forecast states that the temperature is predicted to be -5 \u00b0C the following day. Is this temperature a vector or a scalar quantity? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n Let us start off with a brief review of certain fundamental concepts and principles.<\/p>\n In this course it is assumed that you have completed a basic physics class in your secondary school.\u00a0 If you need more information, I recommend that you refer to two open source textbooks.\u00a0 OpenStax College Physics is an algebra-based book.\u00a0 \u00a0 https:\/\/openstax.org\/details\/books\/college-physics\u00a0<\/a> \u00a0OpenStax University Physics is a calculus-based textbook.\u00a0 https:\/\/openstax.org\/details\/books\/university-physics-volume-1\u00a0<\/a> I will be using some examples and images from them in this book but I encourage you to look them over before and maybe during this course.<\/p>\n The following four quantities are used when we talk about mechanics.\u00a0 They are often called “base units” as other units can be made by combining the base units.<\/p>\n Length.<\/strong> Length is used to locate the position of a point in space.\u00a0 \u00a0It describes the size of a certain physical system.\u00a0 \u00a0Once you have length, you can have area and volume.<\/p>\n Time.<\/strong> Time is the indefinite continued progress of existence and events that occur in an apparently irreversible succession from the past, through the present, into the future.<\/p>\n Mass.<\/strong> Mass is a measure of the quantity of matter.\u00a0 It is defined by its inertia, its resistance to being accelerated, and by the gravitational attractions that occurs between two bodies.<\/p>\n Force.<\/strong> Forces are considered to be pushes or pulls exerted by one body on another body.\u00a0 It can occur due to direct contact when two bodies touch or there can be action at a distance such as the gravitational, magnetic and electric forces.\u00a0 Forces are vectors and is characterized by its magnitude, direction and the point where it is applied.<\/p>\n Models, or idealizations, are used in science in order to simplify applications of theories.\u00a0 Here are three main idealizations we will use in this textbooks.\u00a0 There is an aphorism, often attributed to George Box a statistician that goes “all models are wrong, but some are useful.”<\/p>\n Particle.<\/strong>\u00a0 A particle has a mass, but the size can be neglected.\u00a0 This often depends on the system.\u00a0 As we stand on the Earth we are aware that is it not a particle. Yet when we consider the Earth orbiting around the Sun we treat it as a particle because with a radius of 6378 km and the semi-major axis of the orbit being 149,596,000 km the means that the Earth’s radius is only 0.004% of the orbital distance so it is reasonable to consider it a particle when we deal with orbits. We reduce things to particles to simplify the analysis as the geometry of the object will not be involved in the analysis.<\/p>\n Rigid Body:<\/strong>\u00a0 A rigid body can be considered to be a combination of a large number of particles where all the particles remain at the same distance from each other, even when a large force or load is applied to the body.\u00a0 The shape does not change.\u00a0 In most cases the actual deformations that occur when a body is forced is relatively small.<\/p>\n Concentrated Force.<\/strong>\u00a0 A concentrated force is the effect of a load which is assumed to act at point on a rigid body.\u00a0 We can represent the total load by a concentrated force provided the area over which the load is applied is very small compared to the overall size of the body.\u00a0 Think of a wheel in contact with the ground.<\/p>\n These laws are valid for non-accelerating reference frames.\u00a0 \u00a0First formulated in Philosophi\u00e6 Naturalis Principia Mathematica, <\/i>often referred to as just The Principia.\u00a0<\/em> it was first published in July of 1687 and is considered one of the most important works in the history of science.\u00a0 These laws have been experimentally verified for many hundreds of years. While Einstein’s laws modify these laws at extreme masses and speed, the study of engineering mechanics deals with Newton’s Laws.<\/p>\n Often called the law of inertia.\u00a0 A body at rest remains at rest, or, if in motion, remains in motion at a constant velocity unless acted on by a net<\/strong> external force. The property of a body to remain at rest or to remain in motion with constant velocity is called inertia<\/strong>.\u00a0 We say that a body is in equilibrium when the net force acting on it is zero.<\/p>\n Newton\u2019s second law of motion<\/span><\/strong> is closely related to Newton\u2019s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton\u2019s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration<\/span><\/strong>. Newton\u2019s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration<\/em>. You must define the boundaries of the system before you can determine which forces are external<\/em>. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton\u2019s laws.<\/p>\n To obtain an equation for Newton\u2019s second law, we first write the relationship of acceleration and net external force as the proportionality<\/p>\n where the symbol [latex]\\propto[\/latex] means \u201cproportional to,\u201d and F<\/em>net<\/sub><\/strong> is the net external force<\/span>.<\/p>\n In equation form, Newton\u2019s second law of motion is<\/p>\n Whenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.<\/p>\n <\/p>\n We can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em> As there is an unbalanced force, there is a net force acting on the body so there is an acceleration. Credit: Jennifer Kirkey 2020 CCBY[\/caption]\r\n\r\n<\/div>\r\n The same force exerted on systems of different masses produces different accelerations. (a) A basketball player pushes on a basketball to make a pass. (The effect of gravity on the ball is ignored.) (b) The same player exerts an identical force on a stalled SUV and produces a far smaller acceleration (even if friction is negligible). (c) The free-body diagrams are identical, permitting direct comparison of the two situations. A series of patterns for the free-body diagram will emerge as you do more problems. Photo credit: OpenStax College Physics.[\/caption]\r\n\r\n<\/div>\r\nThird Law<\/h2>\r\nWhenever one body exerts a force on a second body, the first body experiences a force that is equal in magnitude and opposite in direction to the force that it exerts.\r\n\r\n \r\n\r\n[caption id=\"attachment_213\" align=\"aligncenter\" width=\"1024\"]
Action - reaction forces on two bodies that are touching.[\/caption]\r\n\r\nWe can readily see Newton\u2019s third law at work by taking a look at how people move about. Consider a swimmer pushing off from the side of a pool, as illustrated in the figure below.\u00a0 She pushes against the pool wall with her feet and accelerates in the direction opposite<\/em> to that of her push. The wall has exerted an equal and opposite force back on the swimmer. You might think that two equal and opposite forces would cancel, but they do not because they act on different systems<\/em>. In this case, there are two systems that we could investigate: the swimmer or the wall. If we select the swimmer to be the system of interest, as in the figure, then F<\/em>wall on feet<\/sub><\/strong> is an external force on this system and affects its motion. The swimmer moves in the direction of F<\/em>wall on feet<\/sub><\/strong>. In contrast, the force F<\/em>feet on wall<\/sub><\/strong> acts on the wall and not on our system of interest. Thus F<\/em>feet on wall<\/sub><\/strong> does not directly affect the motion of the system and does not cancel F<\/em>wall on feet<\/sub><\/strong>. Note that the swimmer pushes in the direction opposite to that in which she wishes to move. The reaction to her push is thus in the desired direction.\r\n\r\n \r\n\r\n[caption id=\"attachment_198\" align=\"aligncenter\" width=\"875\"] When the swimmer exerts a force \ud835\udc05feet on wall<\/sub> on the wall, she accelerates in the direction opposite to that of her push. This means the net external force on her is in the direction opposite to \ud835\udc05feet on wall<\/sub>. This opposition occurs because, in accordance with Newton\u2019s third law of motion, the wall exerts a force \ud835\udc05wall on feet<\/sub> on her, equal in magnitude but in the direction opposite to the one she exerts on it. The line around the swimmer indicates the system of interest. Note that \ud835\udc05feet on wall<\/sub> does not act on this system (the swimmer) and, thus, does not cancel \ud835\udc05wall on feet.<\/sub> Thus the free-body diagram shows only \ud835\udc05wall on feet,<\/sub> \ud835\udc30, the gravitational force, and \ud835\udc01\ud835\udc05, the buoyant force of the water supporting the swimmer\u2019s weight. The vertical forces \ud835\udc30 and \ud835\udc01\ud835\udc05 cancel since there is no vertical motion. Photo credit: OpenStax College Physics[\/caption]\r\nNewton's Law of Gravitational Attraction<\/h1>\r\nAfter he had formulated his three laws of motion, Newton began to think about gravity. His law states that each mass reaches out through space and attracts (forces toward) every other mass to itself.\u00a0 In equation form\r\n
Weight and the Gravitational Force<\/h1>\r\n
COMMON MISCONCEPTIONS: MASS VS. WEIGHT\r\n<\/span><\/h3>\r\n
Review of Vectors and Scalars from OpenStax College Physics<\/h1>\r\n
Coordinate Systems for One-Dimensional Motion<\/h2>\r\n
Figure:<\/strong>\u00a0It is usually convenient to consider motion upward or to the right as positive (+)<\/strong> and motion downward or to the left as negative (\u2212)<\/strong>.[\/caption]\r\n\r\n\u00a0<\/span><\/figure>\r\nCheck Your Understanding<\/h3>\r\n
Section Summary<\/h2>\r\n
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Conceptual Questions<\/h3>\r\n
Solutions<\/h3>\r\nCheck Your Understanding<\/strong>\r\n
\r\n \t
\r\n \t
Basic Quantities<\/h1>\n
Models or Idealizations<\/h1>\n
Newton’s Three Laws of Motion<\/h1>\n
First law<\/h2>\n
Second Law<\/h2>\n
Third Law<\/h2>\n