PHY101 Lec1-22_Midterm_handouts (1) PDF - Physics Introduction

Physics-PHY101-Lecture #01 INTRODUCTION TO PHYSICS & THIS COURSE 1.1. Introduction Welcome to Physics. We will embark on a long journey that will consist of 45 lectures. Nevertheless, I hope will find it interesting and enjoyable. As I continued to study and research in this subject...

Physics-PHY101-Lecture #01 INTRODUCTION TO PHYSICS & THIS COURSE 1.1. Introduction Welcome to Physics. We will embark on a long journey that will consist of 45 lectures. Nevertheless, I hope will find it interesting and enjoyable. As I continued to study and research in this subject my interest in this subject grew. But before we come to physics, I want to remind all students that physics is the branch of science and must be aware of the fact that our modern world is based on concepts from science, modern machines/equipment rely on science and inventions such as telephones, satellites, etc. are based on science. But in reality, science is a way of thinking that only accepts the rule of reasoning. In which the judgment of truth and falsehood is based on the sets of results from an experiment. So as mentioned earlier physics is the branch of science that is often termed as the “queen” of science and is rightly said to be the greatest science. The history of science is as old as the story of mankind. When did it start? It is probably tens of thousands of years old. Every civilization has contributed to it. It may have started from the time of Babylon, but after that, the Greeks created great perfection in it, and then Chinese, Hindu and Islamic civilizations advanced it. Physics in its present form is not that much old. It started about three and a half hundred years ago, and it was a time when a great scientific revolution took place in Europe, which is called the Scientific Revolution, and this was the time when great scientists like Newton revolutionized it which is the reason why our present-day world is so different from previous eras. Physics is related to every worldly thing (actually everything in the whole universe) with its main purpose being to fully understand the whole physical world/universe and it means that everything, no matter how big or small it is. For example, if we look at our solar system (as shown in Figure 1.1), the sun is at the center and planets revolve around it. Human beings have thought a lot over many fundamental questions like, how heat is produced in the sun and then how it reaches the Earth. Why and how does the Earth rotate around the sun? Contrary to this, in general, if anything moves it moves in a straight line but the earth continues to move around the sun in a circular path. Figure 1.1. Schematic of Solar System. Sun at the center and planets revolving around the Sun. Atom was considered to be the smallest thing in the world making up all the matter. The atom was considered to be a basic building block but now we know that it can be further broken down as it has a center called the nucleus and electrons revolve around the nucleus as shown in Figure 1.2(a). Upon closer inspection of the nucleus Protons and neutrons can be observed. Interestingly protons and neutrons can be further subdivided into particles called quarks as shown in Figure 1.2(b). (a) (b) Figure 1.2. (a) Atomic Structure showing nucleus at the center having Proton and Neutron. Electrons revolve around the nucleus. (b) Protons can be subdivided into quarks. So, on the one hand physics is related to the sun, the solar system, galaxies, and the universe itself, and on the other hand, it is also related to atoms and subatomic particles, and lastly all kinds of entities that come in between these two extremes. These entities can be classified as forms of matter which are as follows 1. Solids 2. Liquids 3. Gases The first form of matter is called solids in which atoms can't move very far from each other for example table salt. The sodium atom in the salt cannot move away from the chlorine atoms. Atoms are fixed to each other and immobile as shown in Figure 1.3. Figure 1.3. Structure of common table Salt (Sodium Chloride, NaCl) Matter can be in a liquid state and when the substance is in the liquid state, the atoms or molecules that are there can move quite far from each other but still there is some kind of attraction among them confining them. For example, if we put water in a vessel, then this water takes the shape of the vessel and this is a characteristic of the matter when it is in a liquid state. Lastly, there is another situation in which matter can be in a gas state. Well, if the matter is in a gaseous state, then its atoms and molecules can move around freely anywhere and far away from each other. Gases like oxygen, hydrogen, and nitrogen make up our atmosphere and without oxygen, we cannot even breathe. The purpose of this course is to introduce students to the vast subject of physics. Although even experts in the field only know a fraction of it, we will do our best to learn and gain knowledge related to physics in all 45 lectures. The main goal of this course is to teach problem-solving to students. Science is not just about solving existing problems, but also about understanding and solving new and challenging problems. To achieve this goal, students must listen to all video lectures, read all materials (such as handouts and referred books) given for this course, and complete the assignments (samples) provided after going through the materials. It's essential to pay attention to the assignments since they will help students to apply the knowledge they have gained. Studying physics has multiple purposes, one of which is to gain knowledge for future courses. In every field of engineering, ranging from bridge construction to designing electric components, fundamental physics knowledge is required. However, it is crucial to develop the habit of critical thinking and evaluation based on intellect. Science relies on reason, logic, and experience to determine the truth or falsehood of a claim. Physics is often considered a difficult subject because it involves mathematics. Calculus is required, but we will develop as much calculus as we need. Knowledge of algebra, trigonometry, linear equations, or quadratic equations is necessary prerequisite knowledge. 1.2. Main Areas of Physics: Following are the main areas of physics 1. Classical Mechanics 2. Electricity and Magnetism 3. Thermal Physics 4. Quantum Mechanics First of all, classical mechanics is the field of physics, which is attributed to Isaac Newton. It is related to objects, the movement of objects, momentum, force, and energy, these are all concepts that come under classical mechanics. After this, we will talk about electricity and magnetism. Now, electricity and magnetism are not so different from each other and this is a discovery of James Clerk Maxwell from about 200 years ago. And now all modern technologies in the world (telephone, radio, television etc.) are based on these discoveries. So, we will study electricity and magnetism after classical mechanics. When we have this much background, then we will come to thermal physics, i.e. the physics of heat. Concepts like temperature and entropy will be discussed. In these three areas, classical mechanics, electricity and magnetism and thermal physics, we will spend a lot of time on them. But there is another field, which is more related to the physics of atoms. We will not be able to spend much time on this because to understand it and to study it correctly, you need special mathematics. So, my advice to those people who want to study physics further is that you should study mathematics separately. 1.3.Dimension: Just like a house is built with bricks, in every field of physics, everything is built with three types of bricks. These are called dimensions. Dimension is a concept which is about 200 years old and it was proposed by a French scientist named Joseph Fourier. Now, let us focus on the three most important fundamental dimensions1. 1. Time T 2. Length L 3. Mass M First of all, there is a dimension of time. What is the reality of time? Now, philosophers have been discussing this for centuries, but scientists and physicists are not very interested in these discussions. In physics, we want to know how to measure a thing or a dimension. There is a method to measure time which is called a clock. Now, clocks can be of different types. For example, my pulse is also a clock although not an accurate one. When I stand here, my pulse moves at one speed but when I run, my pulse increases. But there is a better clock, a pendulum. Now, the pendulum moves to and fro. And every time it moves to and fro, the time duration is the same. The earth rotates around its axis and when it rotates, it comes back exactly to the same position after 24 hours. Similarly, when the earth rotates around the sun, we say that one year has passed. The best clock is called the atomic clock which is based on the regular vibration of an atom. With this, we can measure time with better accuracy than one part in a billion. To measure length, normally we need a ruler but its accuracy is not very good. In modern ways, we measure with the wavelength of atoms. So, accuracy is much better as compared to the ruler. The third fundamental dimension is mass, which tells us how much matter is present in a body. For example, if I have an apple and I add another apple to it, the mass will double (assuming both apples are identical). The more mass 1 Fundamental and Derived Dimensions there is, the more difficult it will be to move. For example, if I put equal force on a rickshaw and a taxi, the rickshaw will move faster and the taxi will move slower. So, mass tells us how much resistance to motion there is. Now, let us look at dimensional quantities, which are made up of mass, length and time. For example, the dimension of the area is LxL or L2. For a volume, it will be L3. For example, for a box, its length multiplied by width multiplied by height will give its volume having the dimension of L3. We can make another thing from this, which is called density. The dimensions of density will be mass per unit volume or M / L3. In the same way, we find out the dimensions of frequency. What is frequency? For example, there is a pendulum that completes 2 cycles in a second. So, its frequency is 2. Or, if we consider alternating current (AC) of electricity, which changes its polarity 50 times every second. So, its frequency is 50. Frequency has a dimension of 1/T or T-1. The dimensions of speed are L/T or LT-1. But there are some quantities which have no dimensions such as angle. The angle between two adjacent fingers is dimensionless. Degrees are units not the dimension of the angle2. In short, there are seven fundamental dimensions as given in the table. And everything in physics, every other quantity can be constructed in terms of these. Fundamental set dimension is given below in form of table. Fundamental Quantity Dimension SI Unit Length [L] Meter (m) Time [T] Second (s) Mass [M] Kilogram (kg) Temperature [θ] Kelvin (K) Electric Current [I] Ampere (A) 2 Dimensions refer to the fundamental physical properties that describe a quantity. They are abstract and universal, not tied to any specific measurement system. In physics, there are several fundamental dimensions, such as length (L), mass (M), and time (T). Units, on the other hand, are specific measurements that quantify the magnitude of a physical quantity. Units are used to express how much of a particular dimension is present in a given quantity. Dimensions represent the abstract properties of a quantity, while units are specific measurements that combine a numerical value with dimensions to quantify that property. The combination of dimensions and units provides a complete description of a physical quantity in a specific measurement system.  Amount of Substance [N] Mole (mol) Luminous Intensity [J] Candela (cd) 1.4.Units As per the previous example, there is an angle between my two adjacent fingers, and measure this angle in terms of degrees or radians. So now we will need a system of units. There are two different unit systems. The first one is called the MKS system. MKS means meter, kilogram, second system. where the length is measured in meters. The mass is measured in kilograms. And the time is measured in seconds. Then there is another system called the CGS system. CGS means centimeter, gram, second system. Here, the length is measured in centimeters. The mass is measured in grams. And the time is measured in seconds, like MKS. Approximate lengths in meters (m) of a few entities are given below in the table Distance Length (m) Tall Person 2 × 100 Cricket Ground 3 × 102 Radius of Earth 6.4 × 106 Earth to Sun 1.5 × 1011 Radius of Universe 1 × 1026 Thickness of Paper 1 × 10−4 Diameter of Hydrogen atom 1 × 10−10 Diameter of Proton 1 × 10−15 For very large lengths, scientific notation is required and a power of 10 is used. So, for example, the size of the earth is 6.4 × 106 meters and the distance of the sun from the Earth is about 1.5 × 1011 meters. Similarly, the universe is very vast, but today we know that it is not unlimited, it is limited and its radius is about 1 × 1026 meters. Now, on the other hand, there are many small things in the world. For example, the thickness of a piece of paper is about 10−4 meters and atom, for example, a hydrogen atom, its radius is about 10−10 meters. The proton in it is 100,000 times smaller than that and its size is 10−15 meters. So, these are different scales of length. Similarly, there are different scales of time and mass. The approximate time for particular events is given below. Event Time (s) Light travels from Earth to the moon 1.3 × 100 One hour 3.6 × 103 One year 3.2 × 107 Age of universe 5 × 1017 Open/close eyelid 1 × 100 One cycle of radio wave 1 × 10−8 Light moves at a very fast speed, but light also needs time to go from Earth to the moon and this is about 1.3 seconds, i.e. 1.3 × 100 s. In an hour, there are 3600 seconds, i.e. 3.6 × 103 s. In a year, there are 3.2 × 107 s. Today, we also know that the universe was formed about 15 billion years ago and if written in seconds, then it becomes about 5 × 1017 s. On the other hand, there are such events in which the duration is very short. For example, if you blink your eyes, you can do this twice in a second, i.e. the frequency is 2 per second. For a radio wave (electromagnetic wave) in which the descent and ascent happens 108 in a second, i.e. its time duration is 1 × 10−8 seconds. There are also different scales of mass as listed in the table below.  Object Mass (kg) Student 7 × 101 Car 1 × 103 Ship 1 × 106 Erath 6 × 1024 Sun 2 × 1030 Milky Way (Galaxy) 4 × 1041 Dust Particle 1 × 10−9 Oxygen Atom 3 × 10−25 Electron 9 × 10−31 For example, the weight of an ordinary person is about 70 kilograms, the weight of a car is about 1000 kilograms, while the weight of a ship is 1000 times more, i.e. 1 × 106 kg. It is worth noting that powers of 10 are used to write very large as well as very small quantities. Earth’s weight is 6 × 1024 kg and the sun are about 1 million times heavier; its weight is 2 × 1030 kg and if we calculate the weight of our galaxy, then that is about 4 × 1041 kg. Dust particle barely weighs 1 × 10−9 kg. An atom of oxygen is much lighter than this as its weight is about 3 × 10−25 kg and an electron is 1000 times lighter than that, and its weight is 9 × 10−31 kg. 1.5.Conversion of Units Going from a one-unit system (like MKS or CGS) to another system is called conversion of units. There are some useful conversion factors i.e. 1 inch is equal to 2.54 centimeters and 1 meter is equal to 3.28 feet. Sometimes it is required to convert other quantities as well. For example, converting miles per hour (mi/hr) into meters per second (m/s). 1 mile per hour (mi/hr) can be easily converted to meter per second (m/s) using only two following conversions and cross multiplications:  5280 𝑓𝑡 1 𝑚𝑖 = 5280 𝑓𝑡 → 1 = 1 𝑚𝑖 1𝑚 1 𝑚 = 3.28 𝑓𝑡 → 1 = 3.28 𝑓𝑡 1 ℎ𝑟 1 ℎ𝑟 = 3600 𝑠 → 1 = 3600 𝑠 𝑚𝑖 So, multiplying all the above unities by 1 ℎ𝑟 𝑚𝑖 𝑚𝑖 5280 𝑓𝑡 1𝑚 1 ℎ𝑟 1. 1.1.1 = (1 ).( ).( ).( ) ℎ𝑟 ℎ𝑟 1 𝑚𝑖 3.28 𝑓𝑡 3600 𝑠 By cancelling units, we get 𝑚𝑖 5280 𝑚 𝑚 1 = = 0.477 ℎ𝑟 3.28 × 3600 𝑠 𝑠 This was also an example of dimensional analysis. Even though units were of miles per hour on the left side and meters per second on the right side, both were length over time [L/T]. So, the dimensions were equal in both cases. So, it becomes very important that the dimensions on the left side should be on the right side in every equation. If this is not the case, then it means that the equation cannot be correct. For example, let’s assume the following equation 𝑑 = 𝑣𝑡 2 Where 𝑑 is distance, 𝑣 is velocity and 𝑡 is time. Dimension on the left side of the equation is [L] and on the right side, it is [LT-1] [T2] = [LT] which is not equal to the left side so this equation is incorrect. Explain whether the following equation(s) could possibly be right or wrong based on Dimensional analysis. 𝑣 2 =𝑢2 + 𝑎𝑡 𝑣 2 =𝑢2 + 3𝑎2 𝑡 2 Here v and u are velocities, a is acceleration, and t is time. 1.6. Rules of Dimensions Dimensions (M, L, and T) can be considered as algebraic quantities. Likewise, symbols and dimensions can be added and subtracted but rules differ from conventional algebraic rules. Multiplication and division are applicable to dimensions. For example, in the case of length divided by length [L/L] dimension will cancel and the result will be a dimensionless quantity. Dimension for speed can be simplified to [LT-1] based on division. So [M], [L] and [T] can be divided and multiplied but cannot be added or subtracted from each other. For example, [L] + [T] is physically not realizable. Even [T2] cannot be added to [T]. So, addition and subtraction of the same dimension is only possible. 1.7. Accuracy Whenever we measure a quantity its accuracy cannot be 100%. For example, if a certain length is measured using a conventional ruler to be 2.95 centimeters (cm), it cannot be measured as 2.952467 cm. This is because of the fact that measuring equipment always has inaccuracy associated with it so significant figures become relevant. It is not advised to do calculations using certain equipment that misrepresents their accuracy. For example, if you are asked to measure the weight of 3 apples and then calculate the average weight. Its average weight cannot be measured at 550.234 grams. This is not possible because the equipment being used to measure weight is not 100% accurate. For this reason, we need to be aware of the fact that what level of accuracy is required for the measurement of an algebraic or an arithmetic quantity. Only an appropriate and reliable number of digits should be used for the measurement process. If length is being measured by a ruler, then there can be at most 1% accuracy, not more than that. Some general rules should be considered for calculations. For example, when two numbers are added together if one number has an accuracy of 95.9 and the other has an accuracy of 39.32, then you add them together and it becomes 135.22. But it would be more appropriate to call its accuracy 135. So, there is no point in adding the decimal figures to it. It would not be of any benefit. When subtracting when multiplying and when dividing, for example, multiplying, one number has an accuracy of one figure of decimal i.e. 105.8 and the other has an accuracy of 31.4. If we multiply both, then we have six figures. But we should round off this. This means that after the decimal figure, the 0.12 does not have any meaning. We should leave it. So, if we divide 105.8 by 31.4,  then we will get many decimal figures. But it would be appropriate to cut it off at 3.37 because the two numbers are known only to one decimal figure of accuracy. Therefore, 5 decimal figures after 3 do not give much information. 1.8. Orders of Magnitude: Order of magnitudes means that we want to approximately calculate a quantity. In physics, it is often not possible to do exact calculations. Either we do not have enough information or the problem is very difficult or whatever. So, it is extremely important that we can estimate. Now, we would like to discuss the order of magnitude. Order of magnitude means that when we write in powers of 10, then if it is written near the nearest power of 10. Let us give you an example. A man's weight is approximately 100 kilograms. Now, I agree that 100 kilograms is of very few people. Usually, it is half of that. But writing it as 10 to the power of 2 kilograms can be useful in some situations. Similarly, if we take the weight of a child, then it can be approximately 10 kilograms. And if you take a cricket ball, then its weight is approximately 1 kilogram. Now, here we have only talked about the closest powers of 10. So, we are not saying that this is accurate, but as mentioned earlier, whenever an estimate is required, we can use this method. Let’s have a look at an example. How many seconds are there in a person's life? Let's calculate it. If a person is 80 years old, then you can calculate it from 365 days per year. Next, there are 24 hours in a day. And there are 60 minutes in an hour. And there are 60 seconds in a minute. So, calculate by multiplying these conversion factors (i.e. 80 x 365 x 24 x 60 x 60). As a result, we will get 2.5 x 109 seconds. On the other hand, if we estimate it, then assume a person's life is 100 years for the purpose of approximation. There are 365 days in a year but we assume 100 days per year which is the closest power of 10 and so on as shown in the table below. Upon multiplication of all conversion factors, we get 109 seconds. On one hand, we did a very accurate calculation to get 80 years which is equivalent to 2.5 x 109 seconds. And on the other hand, if we make these coarse estimates, then there is not much difference. In both cases, we get 109 seconds only. Although in one calculation we get 2.5 and in the other we get unity. But this difference is very small. Approximate reasoning sometimes gives us very useful results. Conversion Factor Closest Power of 10 Closest Power of 10 (scientific notation)  80 yr 100 yr 102 yr 365 days/yr 100 days/yr 102 days/yr 24 hr/day 10 hr/day 101 hr/day 60 min/hr 100 min/hr 102 min/hr 60 s/min 100 s/min 102 s/min There once a news that armed men robbed 10 crore rupees (100 million rupees) from shopkeepers and ran away on a motorcycle. Now the question arises, can such a large amount be put on a motorcycle or not? To find out, let us do a small experiment. A balancing scale (as shown in the video lecture) is used to measure the weight of 10 currency notes each having a value of 1000 rupees. As seen in the video lecture, the weight of 10,000 rupees (10 notes x 1000 rupees/note) is 20 grams. This means that the weight of 100,000 rupees is 200 grams, which is equal to 0.2 kilograms. And the weight of 1 crore rupees is 20 kilograms. The weight of 10 crores is 200 kilograms. Now 200 kilograms is equal to the weight of 3 people. And it is not possible that such a large amount can be put on a motorcycle other than two people riding it. Therefore, there is something doubtful about the whole incident. This was a very ordinary example of scientific methodology. However, we will see in the upcoming lectures that this methodology is used in every field of physics, and it is called the scientific method. Observations are made based on which an initial thought is created. After that, the hypothesis is born in the mind. For now, we do not know whether this hypothesis is right or wrong. Now, to test it, its predictable outcome has to be compared with more observations. It has to be experimentally verified. When a law is tested again and again and it is seen again and again that it is successful, means the hypothesis combined with the observations and they connect, then it gets the status of scientific theory. We will give you an example of Newton's theory of gravitation and Newton's theory of motion. Now, we will see in the next lectures, those lectures which are related to classical mechanics and which are associated with Newton's theory, what phenomena are achieved by those. But as we know, Einstein who came after Newton proved that Newton's theory cannot be completely correct. So, does this mean that Newton's theory is wrong? No, this does not mean at all. Now, if you apply Newton's theory to any particle, as long as that particle is not moving close to the speed of light, that is, it is not moving at this extreme speed, then Newton's theory can be used very easily and effectively. For example, when we send a spacecraft to a planet far away from the Earth, then we use Newton's laws. We do not need to use Einstein's laws. The theory of relativity created by Einstein is used only in those cases where particles’ speed is close to the speed of light. For example, in research of particle physics particle accelerators are used and electrons move at a speed close to the speed of light. Finally, I will say something which may seem a little strange to you. In this lecture, I emphasized that physics is related to the material world and that its purpose is to find out the rules and laws that are the basis of our universe. So, we are talking about materialistic things. But on the other hand, the concepts of physics are abstract and only exist in our minds. These concepts are the creation of our minds. Now, we would like to clarify this a little. In the upcoming lectures, you will hear again and again that this is a free particle, a free body, that this is a point, that this is a straight line. But there is no such thing as a free body. We say that there is nothing free on Earth because gravity pulls that body towards the Earth. So, take it a little farther from Earth, take it to the outer atmosphere. How far should we take it? Take it 1 crore miles away from the earth. Take it 100 crore miles away. But that will not be enough because there will always be a minute force that pulls that body towards the earth. And if we take it too far, then there are galaxies and other planets, therefore, a completely free body does not exist. It exists only in one place and that is in our brains. And similarly, if we talk about a dot. We assume that a dot is a very tiny thing and we can make it by putting a pencil on paper. But if we look at it under a microscope, the higher the magnification bigger the dot we will observe. Now a pencil's tip cannot be made so fine because a pencil's tip itself is made up of atoms and atoms themselves are not dots. Therefore, there is no dot in the world and there is no straight line either. The point is that the concepts of physics are in one’s mind. Therefore, these concepts are created and evolve, which is why physics is a progressive discipline. In which there is progress all the time. And this is the most interesting thing about physics. So, I hope that in the next lectures, you will feel that this is an interesting topic. That there is progress in it and that it invokes thinking. To learn this subject, you will have to work a little hard. As long as you do not solve problems by yourself. You have to believe that you can solve every problem by yourself. And the more problems you solve, your physics skills will increase accordingly. Secondly, do not assume that you will be able to solve all your problems with a single course be it this course or any other course. There is a vast ocean of knowledge in this world and we can pick up a drop of it at one time. It is a lifelong effort to understand physics and science. And we hope that you will also be a part of this effort. The journey of physics will continue in the next lecture. We will meet again at that time. With your permission, goodbye.  Physics-PHY101-Lecture #02 Kinematics "Kinematic" is derived from the Greek word "kinesis," which means to move or motion. Our world is full of motion. For example, cars move, birds can fly, and fish can move underwater. It is necessary to understand motion and its causes. The main topics of this lecture are: Displacement Velocity Acceleration Whenever we discuss the motion of a body, it is essential to locate its position. To achieve this, we need an origin point. For example, a point x has a value of x = 0 when stationary, but as it starts moving, its values begin to increase. If the position of the vector x depends on "t," then we can express its position as x(t), where x is referred to as a function of "t." (Think of a function as a factory where you input raw material and receive products, like inputting a number into a function and obtaining a result as output.) In kinematics, we require a specific function denoted as x (t). This function allows us to input a value of t and obtain the position of a moving particle. Now, let's delve into the concept of "Displacement." Displacement: As we know, the position at time t is denoted as x(t). Here, “t” can be any number, such as 9, 7, 6, etc. Similarly, when a particle moves at different times “t”, it may be denoted with different notations, such as x(t1) and x(t2), and its function may be defined as x(t). Now, when we subtract these two quantities, we will obtain a value △ 𝑥. The displacement △ 𝑥 in time interval △ 𝑡 = 𝑡2 -𝑡1 is: △ 𝑥 = 𝑥(𝑡2) - 𝑥(𝑡1 ) Sometimes we don’t want to write equation as 𝑥(𝑡1 ) and 𝑥(𝑡2 ) so we do write the above equation as: △ 𝑥 = 𝑥2 -𝑥1  This is what we call displacement. Now consider that a particle is moving in one dimension. Imagine a graph of particle’s position along the x-axis as a function of time. From the Figure. 2.1 we can see that when the particle is at position 𝑥1 the time is 𝑡1 and when at 𝑥2 the time is 𝑡2. Figure 2.1. Displacement vs. time graph of an object. If an object as shown in figure 1. is at 10 m from origin at 𝑡1 and reach at 30 m at time 𝑡2 , then, we can define displacement △ 𝑥 as: △ 𝑥 = 𝑥2 -𝑥1 =30 - 10 △ 𝑥 = 20 m But if the object at 𝑡1 is at 30 m and reach at 10 m at 𝑡2 , then the magnitude will remain same but negative sign appears due to net displacement in negative direction. △ 𝑥 = 𝑥2 -𝑥1 =10 - 30 △ 𝑥 = - 20 m This is what we call displacement, which can be both positive and negative. Speed and Velocity: Speed and velocity measure how position changes with time. There are two major concepts to consider. First, let's look at average speed: Total distance Average speed = Total time If we divide the particle's total distance travelled by the total time taken, we can determine the average speed of that particle. Average speed can vary, being either maximum or minimum. Meanwhile, average velocity can be defined as: Total displacement Average velocity = Total time 𝑥2 −𝑥1 = 𝑡2 −𝑡1 ∆𝑥 𝑣= ∆𝑡 Note that distance is always positive, while displacement can be positive or negative. It's important to clarify that this is average velocity because we are dividing displacement by time, not distance by time. This is also known as slope or gradient. For example, consider a car moving on a slope – the steeper the slope, the greater the gradient. Now, consider the Figure. 2.2 (a) where the distance covered by two points creates a slope from which average velocity is calculated. Figure 2.2. Displacement vs. time graph of an object presenting (a) constant velocity, (b) Instantaneous velocity. In contrast, instantaneous velocity is determined by taking two times very close to each other, with the interval between them approaching zero. In Figure 2.2 (b). if we take two values so close to each other that they approach zero, we can call it instantaneous velocity. Now, let's discuss what is meant by "being too close." Being too close implies approaching a distance of zero. (It's important to note that approaching zero doesn’t mean you are making the distance zero.) Acceleration Acceleration measures how the velocity changes with time. As we have defined average velocity, we can also define average acceleration as: 𝑣2 −𝑣1 △𝑣 Average acceleration = = △𝑡 𝑡2 −𝑡1 We can say that average acceleration is change in velocity divided by time taken to undergo that change. It is also known as slope of graph of velocity against time. △𝑣 𝑎 = 𝑙𝑖𝑚 △𝑡 △𝑡→0 Now to understand definition better we will again construct a graph as shown in Figure. 2.3: Add more about +ve and -ve acc. Done Figure 2.3. Velocity vs. time graph of an object presenting (a) constant acceleration, (b) Instantaneous acceleration. When 𝑡1 and 𝑡2 come closer, 𝑣1 and 𝑣2 also come closer and we draw a tangent line at that point which is known as slope of tangent acceleration as shown in Figure 2.3 (b). Remember that we are approaching t to zero, not making it zero. The acceleration can be positive and negative. Moreover, it is not necessary for acceleration to be in the direction of velocity. They can have different direction as shown in Figure 2.4, in which velocity of train is decreasing as time passes which presents deacceleration. By convention we take positive x-axis in right direction and negative x-axis in left direction. If the velocity of an object is increasing in positive x-axis direction than its acceleration is positive. If the velocity of the object is increasing in negative x-axis, then its acceleration will be negative, but it never be deacceleration or retardation. The SI unit of acceleration is m s-2. Figure 2.4. The negative acceleration of a train. Constant Acceleration When an object's acceleration stays constant over time, it's referred to as having constant acceleration. Stated differently, the object's velocity changes at a constant rate. Mathematically: Let us talk about constant acceleration in more detail. For this purpose, let’s do simple calculations. For convenience, take: t1 = 0, t2 = t Then: x1 = xo and x2 = x  𝑣1 = 𝑣𝑜 and 𝑣2 = 𝑣 At time t1, the velocity of particle is 𝑣1 which is 𝑣𝑜 , and at time t2 its velocity becomes 𝑣2 = 𝑣. If acceleration is constant, then we can write the average acceleration as: v2 − v1 avg = t2 − t1 And as acceleration is constant, so, v is equals to, v = vo + at ………. (2) ⸫a = v/t, so v = at The average velocity as “it is the average of v and vo divided by 2”. 1 𝑣𝑎𝑣 = 2 (𝑣𝑜 + 𝑣) If we want to write in previous notation, then: 𝑥2 − 𝑥1 𝑣𝑎𝑣 = 𝑡2 − 𝑡1 If we substitute values from equation given above, then: 𝑥 − 𝑥𝑜 𝑣 + 𝑣𝑜 = 𝑡−0 2 𝑥−𝑥𝑜 𝑣𝑜 +𝑎𝑡+𝑣0 = 2 𝑡 𝑥−𝑥𝑜 1 = 𝑣𝑜 + 𝑎𝑡 𝑡 2 1 𝑥 = 𝑥0 + 𝑣𝑜 𝑡 + 𝑎𝑡 2 2 As we have discussed earlier that x is function of t. So, by putting t = 0 in above equation, we will have value of x as xo, which is the initial point from, we started. From all that procedure we have got two main equations:  1 𝑥 = 𝑥0 + 𝑣𝑜 𝑡 + 2 𝑎𝑡 2 ……. (A) 𝑣 = 𝑣𝑜 + 𝑎𝑡 ………. (B) Now let’s plot again them on a graph for better understanding: Figure 2.5. (a) Displacement vs. time, (b) velocity vs. time graph of an object with constant acceleration. From the graphs in Figure. 2.5 it can be observed that as the time increases particle start moving away from the origin and its velocity increases linearly, while its displacement increases quadratically. From equation (A) and (B), 𝑣 − 𝑣𝑜 𝑡=[ ] 𝑎 𝑣 − 𝑣𝑜 1 𝑣 − 𝑣𝑜 2 𝑥 = 𝑥𝑜 + 𝑣𝑜 [ ]+ 𝑎[ ] 𝑎 2 𝑎 𝑣𝑜 𝑣−𝑣𝑜2 𝑣 2 +𝑣𝑜2 −2𝑣𝑣𝑜 𝑥 − 𝑥𝑜 = + 𝑎 2𝑎 2𝑣𝑜 𝑣−2𝑣𝑜2 + 𝑣 2 +𝑣𝑜2 −2𝑣𝑣𝑜 𝑥 − 𝑥𝑜 = 2𝑎 2𝑎(𝑥 − 𝑥𝑜 ) = 𝑣 2 − 𝑣𝑜2 𝑣 2 = 𝑣02 + 2𝑎(𝑥 − 𝑥0 ) Where 𝑣 is function of 𝑥. This equation tells us when the value of x changes, the value of v also changes. Example: For better understanding, let's consider an example with cars. In the context of cars equipped with a speedometer ranging from 0 to 160 km/h, the initial state of the car is at rest, denoted as v = 0. When the car is started, its speed, let's consider it as = 70 km/h, increases to this value and moves at this speed for some time. Afterward, it changes its velocity to 60 km/h, indicating negative acceleration. If we bring the car to a stop, its velocity will gradually decrease and eventually become zero, signifying that the car is at rest. Introduction to Vectors Up until now, the discussion has focused on motion in one dimension. Now, the exploration will extend to motion in two and three dimensions. Vectors exhibit two primary properties: Magnitude Direction Consider the position vector r in two dimensions. For example, envision drawing a vector on a map of Pakistan, where one end (origin) is in Karachi, and the other end is in Islamabad. To achieve this, two essential considerations are needed: Set the origin at Karachi. Choose coordinates for distance (in kilometres) and direction (North, West, East, South) Two sets of numbers or coordinates are necessary. Subsequently, two arrows will be drawn, representing vectors originating from Islamabad and Karachi. Consider a car moving on a non-uniform surface. We will notice that’s its velocity is changing its direction. Although it is moving with constant speed, but its direction keep changing. The components of vector in two dimensions can be expressed as,  𝑟𝑋 = 𝑥 = 𝑟𝑐𝑜𝑠𝜃 𝑟𝑦 = 𝑦 = 𝑟𝑠𝑖𝑛𝜃 The position vector is written as, r = ( rX , ry ) = (x, y) Figure 2.6. The point (x, y) is in two-dimensional plane with origin (0,0). So, we write this vector as r or in two coordinates (x, y). We call these coordinates as components. For x = rcos(θ): From the first trigonometric identity, we have: cos(θ) = x/r Multiplying both sides by r, we get: rcos(θ) = x Therefore, x = rcos(θ) For y = rsin(θ): Similarly, from the second trigonometric identity, we have:  sin(θ) = y/r Multiplying both sides by r, we get: rsin(θ) = y Therefore, y = rsin(θ). Mathematically we write them as x and y as: 𝑟𝑋 = 𝑥 = 𝑟𝑐𝑜𝑠𝜃 𝑟𝑦 = 𝑦 = 𝑟𝑠𝑖𝑛𝜃 We are already familiar from these basic trigonometric formulas. x 2 + y 2 = r 2 (sin 2  + cos 2  ) x 2 + y 2 = r 2 (1) = r 2 Which proves that if we square two rectangular components and then add them, we get a constant number. Although the length of the vector doesn’t depend on the direction of vectors. We can write it mathematically as: Where r = |r| −1 y  = tan x Magnitude of r can be found by Pythagorean theorem. |r| = r = x2 + y 2 Where r is independent of value of angle. Vectors can be of different types. We have discussed position vector till now. For instance, we have also discussed velocity vector with example of moving car. Acceleration is a vector itself although it made up from velocity but still it is independent of it. Vector addition: We can add vectors in one dimension. For example, consider a vector in one direction with a length of 10 m and another in the opposite direction with a length of 20 m. 20 m −10 m = 10 m So, adding vectors in one dimension is an easy task. Now, let’s discuss two dimensions. We want to add two vectors A and B and resultant as C. As, C = A+B Graphically if we add these two vectors, they form a triangle with a resultant vector C. We can arrange the vectors any way we want if we maintain their length and direction. Parallelogram method for vector addition: We can also add two vectors by parallelogram method, Continuing from previous example,  Figure 2.7. We translate the vectors A and B and form the sides of a parallelogram. The diagonal between them represents their resultant vector, denoted as C. This method is known as the parallelogram method. In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin and the parallelogram constructed as follows: Figure 2.8. The resultant R is the diagonal of the parallelogram drawn from the common origin.  Component Method: The components of a vector are those vectors which, when added together, give the original vector. The sum of the components of two vectors is equal to the sum of these two vectors. When we add two vectors their components do get added. Figure 2.9. Figure presents the concept of vector addition by component method. Here vector A and B have two components along x-axis and along y-axis we get a resultant. vector R by adding these vectors components in x and y direction. Summarizing it as, Add components. 𝑹𝒙 = 𝑨𝒙 + 𝑩𝒙 𝑹𝒚 = 𝑨𝒚 + 𝑩𝒚 Bold letters present the vector nature. Then calculate the magnitude from following formula, 𝑅 = √𝑅𝑋 2 + 𝑅𝑦 2 Calculate the angle by using formula.  𝑅𝑦 𝜃 = 𝑡𝑎𝑛−1 𝑅 𝑋  | Physics | PHY101_Lecture#03 Kinematics-II Student Learning Outcomes After listening the lecture, students will be able to, Understanding Position Functions Analyzing Constants and Dimensions Exploring Derivatives and Velocity Motion in Two Dimensions Vector Operations and Applications Here are some questions for the students, 1. Is it possible for a car to have acceleration while at rest? 2. Are velocity and acceleration always in the same direction? If a car is accelerating, does it acceleration with constant acceleration? To comprehend these questions, one must understand the term 'motion.' In this lecture, we will delve into the concept of motion and explore some functions that elucidate this term. Function The function represents a machine where an input is inserted, producing an output. The emphasis is specifically on the position function, denoted as X, which is a function of time, t. 𝑥(𝑡) = 𝑐0 + 𝑐1 𝑡 + 𝑐2 𝑡 2 + 𝑐3 𝑡 3 +..... Where 𝑐0 , 𝑐1, 𝑐2, 𝑐3 are constants and remain fixed. 'Constant' refers to values that do not change over time. It is essential to focus initially on the dimensions on both the left and right side of the equation, ensuring their equality. Upon observing the dimension on the left side, it is identified as length, even if t is not explicitly stated. This implies that the function describes the movement of a particle in a straight line, indicating the distance covered over time, t. So, this is a function that says the particle is moving in a straight line and the distance it covers in time t. This is what we called 𝑥(𝑡). The dimensions must match, 𝐷𝑖𝑚[𝑐0 ] = 𝐿 𝐷𝑖𝑚[𝑐1 ] = 𝐿/𝑇 𝐷𝑖𝑚[𝑐2 ] = 𝐿/𝑇 2 𝐷𝑖𝑚[𝑐3 ] = 𝐿/𝑇 3 And obviously, when the value of t is zero, then x is C0. So, the dimension of C0 is equal to L. Now, focus on the dimension of C1. The dimension of C1 is equal to L/T. Similarly, the dimension of C2 is L/T², and the dimension of C3 is L/T³. In short, if the values of C0, C1, C2, etc., are unknown, then at any point, the value of X(t) can be calculated. If dimensions of constants, and function 𝑥(𝑡) are known, the dimensions of “t” can easily be determined. Derivatives Derivative is the invention of Newton. In differential calculus we take the differences and then dividing them by the difference the ratio of both becomes finite. Let’s try to understand the concept of derivatives, which represents the change. This change might be in position, velocity, acceleration etc. The change in position “𝑥 ” w.r.t time is written in form of derivative is, 𝑑𝑥 𝛥𝑥 = 𝑙𝑖𝑚 𝑑𝑡 𝛥𝑡→0 𝛥𝑡 Δ𝑥 𝑥(𝑡 + Δ𝑡) − 𝑥(𝑡) = 𝑙𝑖𝑚 = Δ𝑡→0 Δ𝑡 Δ𝑡 Never make the mistake of cancelling the “d” in numerator and denominator; this is not possible and doesn't hold any meaning. The meaning of Δt is clear in this definition. On one value of “t”, you take a value of 𝑥, and then on the second value of “t”, you take another value of 𝑥. Then, take the difference, which we call Δ𝑥, and divide by Δ𝑡. Here we emphasize that Δt should be very small. You might ask how small? 0.1 is not enough? no. Then 0.01 is not enough either. Even if you take 0.0001, that is still not enough. But if we make it small enough so that Δ𝑡 approaches 0." We will discuss a little detail about it now, 𝑥(𝑡) = 𝑡 𝛥𝑥 = 𝑥(𝑡 + 𝛥𝑡) − 𝑥(𝑡) = (𝑡 + 𝛥𝑡) − 𝑡 = 𝛥𝑡 𝑑𝑥 𝛥𝑥 = 𝑙𝑖𝑚 =1 𝑑𝑡 𝛥𝑡→0 𝛥𝑡 From this it is clear that the function whose value is 1 is called a linear function and the derivative of linear function is constant. Let’s calculate the derivative of “t²”, 𝑥(𝑡) = 𝑡 2 𝛥𝑥 = 𝑥(𝑡 + 𝛥𝑡) − 𝑥(𝑡) 𝛥𝑥 = (𝑡 + 𝛥𝑡)2 − 𝑡 2 = 𝑡 2 + (𝛥𝑡)2 + 2𝑡𝛥𝑡 − 𝑡 2 𝛥𝑥 = 𝛥𝑡 + 2𝑡 𝛥𝑡 𝛥𝑥 𝑑𝑥 ⇒ 𝑙𝑖𝑚 = = 2𝑡 𝛥𝑡→0 𝛥𝑡 𝑑𝑡 This case is different from the previous case. Now dx/dt the derivative of x of with respect to “t” is not constant, but it depends on function. And we know that dx/dt is known as speed or velocity. As in this case dx/dt is not constant but proportional to t. Similarly, 𝑥(𝑡) = 𝑡 3 𝛥𝑥 = (𝑡 + 𝛥𝑡)3 − 𝑡 3 = 𝑡 3 + 3𝑡 2 𝛥𝑡 + 3𝑡𝛥𝑡 2 + 𝛥𝑡 3 − 𝑡 3 𝛥𝑥 = (𝛥𝑡)2 + 3𝑡 2 + 3𝑡𝛥𝑡 𝛥𝑡  𝛥𝑥 𝑑𝑥 ⇒ 𝑙𝑖𝑚 = = 3𝑡 2 𝛥𝑡→0 𝛥𝑡 𝑑𝑡 Now generalize it for t⁴, t⁵…. etc. consider, the function with power “n”, where n=integer 𝑥(𝑡) = 𝑡 𝑛 𝑡ℎ𝑒𝑛: 𝑑𝑥 𝛥𝑥 = 𝑙𝑖𝑚 = 𝑛𝑡 𝑛−1 𝑑𝑡 𝛥𝑡→0 𝛥𝑡 n −1 Then value of dx/dt is nt. Look for previous cases when n = 1 value of dx/dt is 1. And for n = 2 value of dx/dt is 2t. And for n = 0 the value of dx/dt should be 0. The derivative of a constant is always zero. Geometrical interpretation of derivative When value of t is increased to ∆t, the value of x increases as well as ∆x. As if dx/dt is like a gradient and in gradients, when you go a bit horizontal, you also go a bit vertical, resulting in a slope. Figure 3.6. present the concept of derivatives. In Figure 3.1. Three arrows represent the change w.r.t time. The arrow pointing upward indicates a positive gradient, meaning its derivative (dx/dt) is positive or the rate of change with respect to time (t) is positive. As we progress to the right, the value of t increases, and the arrow's direction changes. It transitions from an upward orientation to a horizontal one and eventually moves downward. Consequently, the sign of the derivative changes from positive to zero and then from zero to negative. This shift in sign signifies the geometrical significance of the derivative. The second eq. of motion is, 1 x = x0 + v0t + at 2 2 Now let’s see how the derivative formulas we have drawn apply to this. Now differentiate x with respect to t 1 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 2 𝑑𝑥 1 = 0 + 𝑣0 + 𝑎(2𝑡) 𝑑𝑡 2 𝑑𝑥 ⇒𝑣= = 𝑣0 + 𝑎𝑡 𝑑𝑡 𝑑𝑣 =0+𝑎 = 𝑎 𝑑𝑡 𝑥0 , 𝑣0 is the initial displacement and velocity, respectively. Now answer the question asked by a student about how the car will accelerate when it is parked. A car at rest can be accelerating very fast 𝑣 = 𝑎𝑡 𝑑𝑣 =𝑎≠0 𝑑𝑡 The car is not moving at t = 0, then it starts to move with some acceleration. So, from here it is clear that speed is a separate thing and acceleration is separate thing. It is possible that the object is moving in a positive direction with a constant velocity, while its acceleration is directed in the negative direction. Both speed and acceleration are always considered relative to a specific origin. Now take the example of a stone. Stone always falls downward. A stone can be at rest yet accelerating. 𝑣 = −𝑔𝑡  𝑑𝑣 = −𝑔 ≠ 0 𝑑𝑡 The value of "g" does not remain constant at 9.8 m/s2. If an object goes upward, it will decrease, and as they go further up, it will decrease even more. If an object goes out into space, the value of "g" will be zero. However, it's important to note that the units of "g" are in meters per second squared. Since we live on Earth, it would be beneficial to remember this value as 9.8 m/s². A useful notation: 𝑑𝑣 𝑑 𝑑𝑥 = ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑2 𝑥 = 𝑑𝑡 2 𝑑2 𝑥 When we take second derivative of x then we write it as: 𝑑𝑡 2 𝑑𝑥 𝑑2 𝑥 Remember that speed is 𝑑𝑡 and acceleration is 𝑑𝑡 2. Motion in 2-dimension Let's discuss some characteristics of vectors. Each vector has a specific direction and a certain length. Now, there are special vectors known as unit vectors. Their characteristic is that their length is equal to one, and they only indicate directions. For example, one unit vector may point in a certain direction, while another may be perpendicular to it. A unit vector is a vector with a magnitude of 1 (no units), and it is obtained by dividing a vector by its length or magnitude. Â = Ā/A Example of unit vector are î and 𝑗̂ in 2 dimensions. The vector i and j are perpendicular to each other, and their magnitude is 1. The resultant vector is written as, 𝐴⃗ = 𝐴𝑥 𝑖̂ + 𝐴𝑌 𝑗̂ Resolution of vectors into its components The vector component along the x direction is called x-component and the component along the y direction is called y-component.  𝐴⃗ = 𝐴𝑥 𝑖̂ + 𝐴𝑌 𝑗̂ In this way, we can resolve a vector into its components as shown in Figure 3.2. Resolution of vectors means breaking down a vector into its components. We can do this in 3D too, except that it requires three vectors, î, 𝑗̂ and 𝑘̂. Figure 3.7. Illustrate the concept of addition of vectors. Velocity in 2 dimensions We have already discussed velocity in 1 dimension in detail. Velocity in 2 dimensions come from differentiating a displacement vector in 2 dimensions. 𝑟⃗ = 𝑥(𝑡)𝑖̂ + 𝑦(𝑡)𝑗̂ ⃗⃗⃗⃗⃗ 𝑑𝑟 𝑑𝑥 𝑑𝑦 𝑣⃗ = = 𝑖̂ + 𝑗̂ 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑣⃗ = 𝑣𝑥 𝑖̂ + 𝑣𝑦 𝑗̂ Acceleration in 2 dimensions Acceleration is the rate of change of velocity. When the velocity in 2-dimension is differentiated with respect to time, the acceleration is 𝑑𝑣⃗ 𝑎⃗ = 𝑑𝑡 𝑑𝑣𝑥 𝑑𝑣𝑦 = 𝑖̂ + 𝑗̂ 𝑑𝑡 𝑑𝑡 = 𝑎𝑥 𝑖̂ + 𝑎𝑦 𝑗̂ Addition of vectors Two vectors A and B can be added by head to tail rule. Mathematically, 𝐴⃗ = 𝐴𝑥 𝑖̂ + 𝐴𝑦 𝑗̂ ⃗⃗ = 𝐵𝑥 𝑖̂ + 𝐵𝑦 𝑗̂ 𝐵 𝑅⃗⃗ = 𝐴⃗ + 𝐵 ⃗⃗ = (𝐴𝑥 𝑖̂ + 𝐴𝑦 𝑗̂) + (𝐵𝑥 𝑖̂ + 𝐵𝑦 𝑗̂) = (𝐴𝑥 + 𝐵𝑥 )𝑖̂ + (𝐴𝑦 + 𝐵𝑦 )𝑗̂ = 𝑅𝑥 𝑖̂ + 𝑅𝑦 𝑗̂ Example 𝐴⃗ = 6𝑖̂ + 5𝑗̂ ⃗⃗ = 8𝑖̂ + 7𝑗̂ 𝐵 What is the magnitude of of 2𝐴⃗ – 𝐵 ⃗⃗? Letting R = 2𝐴⃗ – 𝐵⃗⃗ = 2 ( 6𝑖̂ + 5𝑗̂) – ( 8𝑖̂ + 7𝑗̂) = ( 12 - 8 )î + ( 10 – 7 ) 𝑗̂ R = 4î + 3𝑗̂ The magnitude is, 𝑅 = √𝑅𝑥2 + 𝑅𝑦2 = √42 + 32 = 5 Consider two vectors 𝐴⃗ and 𝐵 ⃗⃗ making an angle θ with each other as shown in Figure 3.3, The scalar product of 𝐴⃗ and 𝐵 ⃗⃗ is defined as: A.B = AB cos  We can also write it as: A.B = ( A )( B cos  ) = (length of A) × (projection of B on A) Scalar product of unit vectors The dot product of vector with itself is always 1, and dot product of two mutually perpendicular vectors is always zero. iˆ.iˆ = ˆj. ˆj = (1)(1) cos ( 0 ) = 1 iˆ. ˆj = (1)(1) cos ( 90 ) = 0 We can take dot product easily between any two vectors. 𝐴⃗ = 𝐴𝑥 𝑖̂ + 𝐴𝑦 𝑗̂ ⃗⃗ = 𝐵𝑥 𝑖̂ + 𝐵𝑦 𝑗̂ 𝐵 𝐴⃗. 𝐵 ⃗⃗ = (𝐴𝑥 𝑖̂ + 𝐴𝑦 𝑗̂). (𝐵𝑥 𝑖̂ + 𝐵𝑦 𝑗̂) = 𝐴𝑥 𝐵𝑥 𝑖̂. 𝑖̂ + 𝐴𝑥 𝐵𝑦 𝑖̂. 𝑗̂ + 𝐴𝑡 𝐵𝑥 𝑗̂. 𝑖̂ + 𝐴𝑦 𝐵𝑦 𝑗̂. 𝑗̂ = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 Now you can do all these things in the same way in 3 dimensions. There would be only one more unit vector needed which is 𝑘̂. Generalization in 3 dimensions 𝐴⃗ = 𝐴𝑥 𝑖̂ + 𝐴𝑦 𝑗̂ + 𝐴𝑧 𝑘̂ ⃗⃗ = 𝐵𝑥 𝑖̂ + 𝐵𝑦 𝑗̂ + 𝐵𝑧 𝑘̂ 𝐵 𝐴⃗. 𝐵 ⃗⃗ = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧 Application: Projectile Motion: Consider the example of a ball being thrown. When the ball is thrown, it follows a trajectory until it reaches its destination. The ball's velocity is composed of two distinct components: one in the y- direction and another in the x-direction. Notably, the x and y components of velocity are independent of each other. Acceleration is a factor affecting the ball's motion. Upon release, the ball descends in the negative y-direction, experiencing acceleration in the y-direction. However, there is no acceleration acting in the x-direction. For the sake of clarity, let's denote the acceleration of the ball in the y-direction as "ay." Acceleration along y-axis is 𝑎𝑦 = −𝑔 Velocity along x is constant Acceleration along x-axis is 𝑎𝑥 = 0 Figure 3.9. Shows the projectile motion of an object at various instants.  V0 x = V0 cos  V0 y = V0 sin  As presented in Figure 3.4 the velocity V of the ball is represented by two components: horizontal Vx and vertical Vy. As the ball traverses its trajectory, its velocity undergoes changes. Notably, the value of Vx remains constant because the acceleration ax in the horizontal x direction is equal to zero, indicating no acceleration along the x-direction. In contrast, the value of Vy changes as the ball moves. At its highest point in the trajectory, Vy becomes zero. As discussed earlier, the gradient dy/dx is also zero at this point. As x increases beyond this point, dy/dx continues to be zero. Subsequently, the ball descends, causing its vertical component Vy to become negative. Vy eventually reaches zero again at a lower point in the trajectory, and further descent results in negative values for Vy. A more detailed examination using mathematical formulas will provide further insight into this phenomenon. Along x-axis As the variable “x” undergoes change, it is important to note that when “x” is constant, it equals the product of velocity (v) and time (t). The initial velocity “v” is the same as the velocity with which the ball was initially thrown. When the ball is thrown, the component along the horizontal direction remains constant when time is equal to zero. At this point, the value of “x” is zero, and this choice of coordinates is convenient, as it aligns with the starting point of the ball. While alternative coordinate choices are valid, setting “x” to zero at “t” not equal to zero proves to be a practical choice. Examining this concept in mathematical terms, the relationship is described through the following formulas. X direction  𝑉𝑥 = 𝑉0𝑥 𝑥 = 𝑥0 + 𝑉0𝑥 𝑡 𝑎𝑥 = 0 Y-direction 𝑎𝑦 = −𝑔 𝑉𝑦 = 𝑉0𝑦 − 𝑔𝑡 1 𝑦 = 𝑦0 + 𝑉0𝑦 𝑡 − 𝑔𝑡 2 2 How much its velocity changes depend on the value of “t”. In the equation, 𝑉𝑦 = 𝑉0𝑦 − 𝑔𝑡, if “t” is equal to zero, then the values of Vy will be equal to the initial vertical velocity, but with the increase of “t”, they are decreasing and finally when they reach their maximum, where Vy will be zero. Consider the following scenario: an inquiry into the maximum height a ball can attain when thrown lightly versus when thrown with greater force. The analysis suggests that a faster throw results in a higher ascent. To quantify this, calculations are performed with the objective of determining the optimal height. Subsequently, attention is turned to calculating the maximum distance the ball can cover in the horizontal (x) direction—the maximum range. To address these questions, algebraic methods are employed to derive the necessary equations and relationships that govern the ball's trajectory. 𝑉0𝑥 = 𝑉0 𝑐𝑜𝑠 𝜃 𝑉0𝑦 = 𝑉0 𝑠𝑖𝑛 𝜃 𝑉𝑦 = 𝑉0𝑦 − 𝑔𝑡 𝑣𝑦 = 0 𝑣0 𝑠𝑖𝑛 𝜃 − 𝑔𝑡 = 0 𝑣0 𝑠𝑖𝑛 𝜃 𝑡= 𝑔 1 𝑦 = (𝑣0 𝑠𝑖𝑛 𝜃)𝑡 − 𝑔𝑡 2 2 𝑣0 𝑠𝑖𝑛 𝜃 1 𝑣0 𝑠𝑖𝑛 𝜃 2 𝐻 = (𝑣0 𝑠𝑖𝑛 𝜃) ( )− 𝑔( ) 𝑔 2 𝑔  (𝒗𝟎 𝒔𝒊𝒏 𝜽)𝟐 𝑯= 𝟐𝒈 In the quest to achieve the maximum height for a thrown ball, it is essential to set the angle 90, as the sine function attains its maximum value at 90 degrees. Thus, to propel the ball to its highest point, aligning it straight upward is optimal. On the other hand, if the objective is to maximize the horizontal distance covered by the ball, the strategy involves achieving the maximum range “R”. Given that the speed of the ball remains constant in the horizontal direction, the distance covered “x” in time “t” is crucial. At two distinct points in time, “y” attains the value of zero: firstly, at the release point (initial point) and secondly, at the landing point (final point). Consequently, there are two unique values of “x” where “y” equals zero. The first is evident at “x = 0”, and the second, denoted as x = R, represents the distance from the starting to the ending point. Notably, the choice of θ is pivotal for achieving the maximum range. To optimize the horizontal distance covered, it is recommended to throw the ball at an angle of θ = 45o. 𝑥 = (𝑣0 𝑐𝑜𝑠 𝜃)𝑡 𝑥 𝑡= (𝑣0 𝑐𝑜𝑠 𝜃) 1 𝑦 = (𝑣0 𝑠𝑖𝑛 𝜃)𝑡 − 𝑔𝑡 2 2 𝑥 1 𝑥 2 = (𝑣0 𝑠𝑖𝑛 𝜃) − 𝑔( ) (𝑣0 𝑐𝑜𝑠 𝜃) 2 (𝑣0 𝑐𝑜𝑠 𝜃) 𝑔 𝑠𝑒𝑐 2 𝜃 = 𝑥 𝑡𝑎𝑛 𝜃 − 𝑥 2 ( ) 2𝑣0 2 𝑔 𝑦 = 𝑥 [𝑡𝑎𝑛 𝜃 − 𝑥 ( 2 𝑐𝑜𝑠 2 𝜃 )] = 0 2𝑣0 This equation has two solutions for x, 𝑥 = 0, 𝑎𝑛𝑑 𝑥 = 𝑅 (𝑟𝑎𝑛𝑔𝑒) 𝑔 [𝑡𝑎𝑛 𝜃 − 𝑅 ( 2 𝑐𝑜𝑠 2 )] = 0 2𝑣0 𝜃 𝑔 𝑅( 2 𝑐𝑜𝑠 2 𝜃 ) = 𝑡𝑎𝑛 𝜃 2𝑣0 2𝑣0 2 𝑐𝑜𝑠 2 𝜃 𝑅=. 𝑡𝑎𝑛 𝜃 𝑔  2𝑣0 2 𝑐𝑜𝑠 2 𝜃 𝑠𝑖𝑛 𝜃 𝑅=. 𝑔 𝑐𝑜𝑠 𝜃 2𝑣0 2 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 𝑅= 𝑔 𝒗𝟎 𝟐 𝒔𝒊𝒏 𝟐 𝜽 𝑹= 𝒈 Since − 1  sin 2  1 therefore (sin 2 ) max = 1 v0 2 v0 2  Rmax = (sin 2 ) max = g g  Physics-PHY101-Lecture#04 FORCE AND NEWTON’S LAWS Till now in this course of physics, we have looked at kinematics which includes basic concepts such as displacement, velocity, and acceleration. We talked about one dimension that if the body can move only on one dimension i.e., on a straight line then how do we define velocity and acceleration? We talked about derivatives, then we generalized it into two and three dimensions. Then some interesting problems were also solved. But this question remains to be raised why do bodies/objects move like this, where do they get this acceleration from? We will look at these issues in today’s lecture. The theme of today’s lecture is dynamics. By dynamics, we mean the force which acts on the body and gives it acceleration. In this context, we will talk about Newton’s three laws of Motion. So, the real question arises where does acceleration come from? For this, we will need a new concept which we will call Force. Dynamics means it is the study of forces and the resulting motion. So how does a body get motion and what is the effect of force on it will be discussed in this lecture? Before Newton’s time, it was believed that the natural state of everything is to come to a standstill, which means that everything wants to come to a standstill or stop, so for example: if we roll a ball then the ball stops after some time or if you throw any other thing then it moves forward for a while and then stops as if it is a natural state that is the of rest and if something moves then there must be some force acting on it, hence this idea was common before Newton’s time that some force behind moving body. For this reason, it was believed that the sun, the moon and all the other planets are moving then there must be some force behind them. For example: if mars rotates in its orbit like this then there must be something pushing it. This was an old idea but after Newton a great revolution took place. Newton said that the natural state of everything is that it wants to continue its movement. The modern view is that objects tend to remain in their initial state, that is they want to remain in the same state in which they were and unless some force acts on it will maintain their condition. Isaac Newton was probably the world’s greatest scientist and thinker. He wrote a book called Principia Mathematica about 350 years ago in which he proposed three laws of motion. Newton’s First Law of Motion: It states that everything either remains at rest or moves with constant speed unless some force acts on it. This means that the body moving will keep moving on its own without the application of external force and can be considered as a free body. Frame of reference: If we measure the movement of a body and someone else measures the motion of the same body then there can be differences between the two, so here we have to talk to you about reference frames as shown in figure 4.1. These are two frames of reference, one we can call S, and the other can be called S'. Imagine that you are at rest and standing on land and exist in frame S. There is another person who exists in frame S' and it is moving away from us at velocity V. Let’s assume a point P having distance x from our frame of reference. Now the person who is moving will say that I measured this distance as x'. Figure 4.1: Frames of references S and S'. The relation between these two distances is given by x = x − V t and at t = 0 x = x So initially ( t = 0 ) both distances are measured to be the same, but as the second observer moved forward, there was a difference between x and x’ which is called a frame of reference. According to this, two different observers will have different sets of coordinates for the same point and whatever measurements we make will depend on these coordinates. Now the question arises of whose measurements are more accurate. Inertial and non- Inertial Frame of Reference: Newton's first law applies in every frame which is moving with constant velocity and such frames are called inertial frames. Newton's law is valid in all inertial frames and all non-accelerating frames are called inertial frames. For example: Imagine that you are sitting in the car and there is a hydrogen or helium balloon in that car, and it is suspended inside the car. Now when the car accelerates, we will observe that the balloon starts moving backwards. It seems like some force is acting on the balloon. But in reality there is no force acting on it, but we feel it as if it’s moving backwards. And that is because you are not in the same inertial frame. The frame which accelerates is a non-inertial frame and apparently, a force acts in it sometimes called a fictitious force. It appears to be force but in reality, it is only observable because we have chosen the wrong frame of reference. So, Newton’s first law is true only in inertial frames. Difference between inertial and non-inertial frames of reference: Inertial frame The frame of reference which is moving with uniform velocity and does not accelerate (a=0). Obeys Newton’s law of motion. Example, a train moving with uniform velocity is an inertial frame of reference. Non-inertial The frame of reference which is accelerating (a≠0) is called non-inertial frame frame of reference. Does not obey Newton’s law of motion. Example, a freely falling elevator is taken as non-inertial frame of reference. Law of inertia: Now the question arises when you want to change the state of motion of a body, you feel some resistance. How much does a body resist when you try to change its state of motion? It depends on its mass. The heavier the body, the more it will resist change in its state of motion. There are many examples of this. For example: If you push a light body (like a shopping cart), it moves easily. If you push a heavy body (like a car or bus) it is possible that you may not even be able to move it. We define this resistance as “inertia”. So, inertia is the resistance to change in motion in other words resistance to acceleration and mass quantifies it. The greater the mass, inertia will be more accordingly. Now the question arises whether mass means size? to which the answer is no. Sometimes smaller bodies offer more resistance. It is related to density. To which the answer is also no. Is this related to weight? Is mass the same as weight? This is also not the answer. Mass and weight are not equal. Figure 4.2: Force F acting on a body having Mass m and causing acceleration a. Now look at Figure 4.2. Force F is acting on a mass whose value is m. Now the more force applied to it the acceleration ‘a’ will be greater accordingly. So mathematically it is written as aF More force leads to more acceleration. But on the other hand, if we consider a body having more mass then the acceleration will be less. 1 a m Hence F a m  F = kma (k=1) Increasing the force increases the acceleration but increasing mass reduces acceleration. Newton’s second law of motion: Newton's second law of motion states that the product of mass and acceleration is equal to the total external force acting upon mass. Now there can be many forces (shown by subscripts) applied simultaneously in different directions. The total net force will produce acceleration. F F = ma → a = m Where F = F1 + F + 2 + F3 +..... By this, we can also define mass as: F m= a So, how to rightly write this equation? There is an equation and there are three variables. Here you can measure the acceleration separately because acceleration is the rate of change of velocity and velocity is the rate of change of position. As position can be measured as a function of time so, velocity and consequently time-dependent acceleration can be measured. There is a need to measure force and how do we do this? We can give you many simple examples of this. Example: We can consider a spring balance. In spring balance, we can place some mass and there is a spring. The more mass we place, the more spring will extend. Similarly, if we pull something with a spring balance, there is a scale on the spring balance showing the magnitude pulling force. If we pull a rubber band, this rubber band exerts a force on our hand and the more it is pulled, the greater the force. So, a rubber band can measure force. Introduction to force: Force is the vector and hence it has a magnitude and also a direction. The direction of force is also the direction of acceleration. There are many types of forces. Let us consider contact forces. When two bodies come in contact with each other, one body exerts a force on another body that is a force acts on it, i.e., when we push a box, a force acts on the box, air exerts pressure on a moving car which is called air resistance. Consider a rope with a weight tied at one end that results in tension which is also a type of force. Just like other physical quantities force also has dimensions given by, L  ML   Force =  Mass   Acceleration =  M    2  =  2  =  MLT −2  T  T  Units for force in the MKS system is Newton. Newton is defined as, if we apply 1 Newton force on 1 kilogram of mass then its acceleration becomes 1 m/s2. m 1N = 1kg.1 s2 Force F means external forces only. Consider a body made up of atoms (as all matter is made up of atoms), every atom attracts or repels the other atom. These interactions are internal forces so they cancel out each other. When we say that the acceleration of a body is F/m, then that F refers to external force only. External force actually is a total force so if various forces are acting such as F1, F2 and F3 till FN then we have to add all these, given by: N F = F1 + F2 + F3 +....... + FN =  Fi i =1 here we see a new symbol which is called the Greek symbol “Sigma ∑” or symbol of summation. So, this summation means we add F1, and F2 till FN and this makes the total force. As force is a vector quantity it can be added by two methods. One method is already discussed (the Parallelogram method) earlier. This can also be done and the other way is through vectors addition by components. The following sample problem (figure 4.3) can help us better understand the addition of two forces by components method. We are given two forces (F1 whose value is 4 N along the negative y-axis) and F2 whose value is 5 N at an angle of 36.9° with the x-axis (angle not mentioned on PPT slide). Figure 4.3: Sample problem. Components of F1: F1x = 0 (as no component along x-axis) F1 y = −4 N (as it is along negative y-axis) Components of F2: F2 x = F2 cos(36.9) 5*0.8 = 4 N F2 y = F2 sin(36.9) 5*0.6 = 3N The sum of components along the x-axis, F x = F1x + F2 x = 0 + 4 = 4 N F y = F1 y + F2 y = −4 + 3 = −1N So, the magnitude of resultant force F can be calculated by, F +F 2 2 F = x y = (4)2 + (−1) 2 = 17 N = 4.12 N Exercise for student: Determine the direction of resultant force F. Mass and Weight: Now we will discuss the difference between weight W and mass m. Weight actually is a force, the force due to gravity. If I have a kilogram of matter it will have a specific weight on earth’s surface, but if I take it to the moon’s surface, its weight will be different. The mass will be the same but its weight is different on both (earth and moon). Let’s look at it in the formula, the weight is the force of gravity and if you apply the following equation. F = ma Where F becomes the weight W, mass m remains tha same, and acceleration a is replaced by acceleration due to gravity g. W = mg Like forces, weight is also a vector and is measured in Newton. If we measure the weight of the same body on earths and moon’s surface, we will observe that the weight on earth will be seven times as compared to weight on moon although mass is same. If we go out into space where there is no celestial body, then our weight will be zero there as there will be no acceleration due to gravity, although mass will be non-zero and remain the same. Difference between mass and weight: Mass Weight Mass is a property of matter. The mass of an Weight depends on the effect of gravity. object is the same everywhere. Weight increases or decreases with higher or lower gravity. Mass can never be zero. Weight can be zero if no gravity acts upon an object, as in space. Mass does not change according to location. Weight varies according to location.  Mass is a scalar quantity. It has magnitude. Weight is a vector quantity. It has magnitude and is directed toward the center of the Earth or other gravity well. Mass may be measured using an ordinary Weight is measured using a spring balance. balance. Mass usually is measured in grams and Weight often is measured in newtons, a unit kilograms. of force. Newton’s third law of motion: Newtown’s third law says that the action of every force produces a negative force. The magnitude of these two forces is the same and direction is opposites (antiparallel). Example: Two boxing gloves A and B collide and come in contact with each other while punching. A Pushes on B and B pushes on A. So, these are action and reaction forces are in opposite direction but have same magnitude (negative sign on the right-hand side of equation in figure 4.4 shows opposite direction) Figure 4.4: Action reaction forces. Note that FA on B is equal to negative FB on A, so the effect of A is on B & B is on A. A is not affecting C, D or anything else, nor is B on anything else. It is so that A is on B & B is on A. Example: An apple is lying on the table, so this is the force of the earth on the apple. We will call it action, and this is due to gravity. Its reaction is not the force of the table on the earth, but the force of the apple on the earth. Earth pulls this apple down, and on the other hand this apple pulls the earth to itself. We assume earth is not pulled to apple which is wrong. Apple is light and very small in comparison to the earth, but the apple also pulls the earth toward itself. Now because the earth is so heavy its acceleration is very less. But apple definitely exerts a force on earth. Table also exerts as a force on apple and its reaction is that apple also exerts a force on table. Now you can ask how the table exerts a force on the apple. How does the force exerted by the table affect it? It’s obvious that if there was no apple there would be no force. But if you keep the apple on the table there is a slight bend produced in the table as a result of which a force is produced in upward direction. To make this discussion interesting we will narrate a story of a very educated horse. He studied Newton’s law in his spare time and especially Newton’s third law. One day, while he was studying, his owner got angry. His owner attached a cart to him and ordered horse to pull it. The horse said that there is no use of it because he has just read Newton’s third law and according to that if he pulls the cart then the cart will also pull him back with same force. He will not move forward hence it is useless to pull the cart. The owner got angry, he gave a pat at the back and what happened? The cart started moving forward. Why did it happen? If the horse is right then the cart would not have moved but there is a mistake in it. So, let’s see where the mistake is. If we look at this student in figure 4.5 walking forward on the ground. Similar to the student depicted in figure we also push ground backwards with our feet. The ground pushes us forward as a reaction. Newton's third law says that if we push something back, it pushes us forward. In this way we move forward. So whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first.  Figure 4.5: Person ‘P’ walking on ground ‘G’. FGP is a backward force exerted on ground by a person. FPG is forward force exerted on a person by ground. Now look at Figure 4.6 of assistant boy ‘A’ boy is moving forward on ground ‘G’ and at the same time he is also pulling a sledge ‘S’ forward. When he pushes ground backward (FGA), the ground pushes him forward (FAG). The tension in this rope (FAS) also pulls this boy backwards. But if the total force produced in the forward direction is more than the backward pull, then this sledge will move forward. Figure 4.6. Assistant boy ‘A’ moving forward on ground ‘G’ and also pulling sledge ‘S’ forward. Now one more example is depicted in figure 4.7. Your hand is on the table in front of you, and with your hand, you are applying pressure on the table and pushing it downwards but table is pushing your hand upward in opposite direction.  Figure 4.7: Action-reaction pair for hand and table. Let’s consider another example shown in figure 4.8. The person is standing and pushing against the wall and the wall pushes the person back with same force. If the action-reaction forces are equal and opposite in direction, why they do not cancel out each other? Figure 4.8: Action-reaction pair for person pushing wall. Figure 4.9 elaborates forces involved with more details. A person is leaning against a wall. Two forces are acting on this person. One is his weight and written it as Fm on f meaning the force of mass ‘m’ on the floor ‘f’. This force act in downward direction. Reaction to this force i.e., the force of floor on the man Ff on m acts upward and cancel out each other. The force exerted on the left side is force of man on the wall Fm on w and the force of wall on the man Fm on w is on opposite direction. Both these forces are equal in magnitude and in opposite direction. Figure 4.9: Forces involved for physical scenario of man ‘m’ leaning against wall ‘w’. Now if we push a body with much greater force then there is possibility that it will start moving in the direction of applied force. Such an example is shown in figure 4.10. A block/box is on frictionless (ice) surface. Man pushes block/box to the left (negative x-axis) by a force Fm on b (force of man on block). Reaction to this is force of block on man (Fb on m) is towards right side (positive x-axis). Although both forces (Fm on b and Fb on m) are equal and opposite in direction but block moves to left side. So how is it possible? Why are these forces not cancelling each other? Figure 4.10: A block/box on frictionless (ice) surface being pushed by man. The answer to why these forces is not cancelling each other out is that a body moves depending upon the total external force acting on that body (having mass m). The force that will act on a body will create acceleration in it. By considering ONLY the block/box as the whole system we can answer above mentioned questions. The force on block/box (Fon box) is equal to force of man on block/box (Fm on b) which by newton’s second law is equal to product of mass m and acceleration of block/box (abox). Mathematically it can be written as Fon box = Fm on b = mabox Acceleration of block/box (abox) will be, Fm on b abox = m Misconception: Table 4.1: This table enlist few misconceptions related newton’s laws Sr. No. Claim True or False. Explanation 1. If something is moving, This is a false claim. A body moving at constant there must be a net velocity has no net force on it. An accelerating body force on it. must have a net force on it. 2. All equal and opposite This claim is false. The weight of a book sitting on a forces are action- tabletop and the normal force of the table acting on the reaction pairs. book are equal and opposite, but they are not an action- reaction pair! 3. If there is a force on an This claim is false. Only a net force on the object leads object, it must be to acceleration. accelerating. In our daily lives, we observe the validation of Newton's law on many occasions, especially the first and second laws. There is a concept about these which we call inertia that a body tends to maintain its state of motion or rest and its speed doesn’t change. This resistance to change of state is proportional to its mass. An example involving glass and paper is discussed in the video lecture followed by the stretching of rubber band as a function of applied force (amount of water). We will pay close attention to the applications of Newton’s law during the next few lectures. The application of Newton’s laws is not limited or localized. These laws are applicable in the whole universe (also termed as universal set of laws). Interestingly concepts at the base of Newton’s laws do not exist in the world. They only exist in our minds. For example, if we talk about free particles (as mentioned earlier that free particle is the one on which no force acts on it) but is there any free particle in reality? The answer is NO. There is no such thing as a free particle. If we even take a particle/body million and millions of miles away from Earth, there will be a minute force acting on it due to gravity. We have also talked about this earlier there is such a thing as a point mass or point body. Despite these facts, we are related to the abstract concept that Newton's laws related to things that do not exist yet this is the achievement of human thinking. We will discuss the application of these laws