General Physics PDF - Undergraduate Level

Ministry of Science and Higher Education Prepared By:  Wondimagegn Anjulo (Assistant Professor, Arbaminch University)  Birhanu Mengistu (PhD Candidate, Haramaya University)  Wondimeneh Leul (MSc, Hawassa University)  Tamirat Bekele (MSc, Addis Ababa Science...

Ministry of Science and Higher Education Prepared By:  Wondimagegn Anjulo (Assistant Professor, Arbaminch University)  Birhanu Mengistu (PhD Candidate, Haramaya University)  Wondimeneh Leul (MSc, Hawassa University)  Tamirat Bekele (MSc, Addis Ababa Science and Technology University) October – 2019 Addis Ababa – Ethiopia  MODULE OVERVIEW The module provides students with a firm foundation in the role of physics in different science, technology and engineering fields together with mathematics and problem-solving skills. And also it prepares students to know the applications of physics in multidisciplinary areas that are at the forefront of technology in 21 st Century, such as agricultural and archeological sciences, health and medical sciences, earth and space sciences, electronics and electromagnetism, communication technology, energy systems, and other related engineering and science fields that require a very solid background in physics. This module will be taught in an introductory undergraduate level and is primarily designed for a broader audience of science students. The goal of the course is to give an overview of the various physics based analysis and dating techniques used in science and technology. High school mathematics and physics concepts are enough as prerequisite for this course. Laws, principles, and methods of physics will be taught in a more descriptive manner using simple mathematics. The course covers preliminaries, mechanics, fluid mechanics, electromagnetism and electronics, thermodynamics, oscillations and waves, and cross-cutting applications of physics in different areas of science and technology. For this course a total of 10 experiments relevant to Mechanics, Electricity and Magnetism, and Electronics will be carried out. I. List of Experiments from Mechanics  Measurements of basic constants, length,.mass and time  Free fall  Hook‘s law  Density of liquids  Simple pendulum II. List of Experiments from Electricity and Magnetism  Calibration of voltmeter and ammeter from galvanometer  Ohm‘s law, parallel and series combination of resistors III. List of Experiments from electronics  V-I characteristics of diode ii   Rectification  Logic gate From these recommended experiments, at least six experiments to be performed. Simulation experiments from the Internet can be used to supplement laboratory activities whenever possible. Manuals for the experiments will be prepared at the respective Universities. It is recommended that the number of students per laboratory session to be between 25 and 30. Module Objectives: Upon completion of this module students should be able to:  Discuss basic physics by refreshing and summarizing the previous preparatory physics concepts before tackling the advanced physics courses.  Explain the kinematics and dynamics of particles in one and two dimensions.  State principles of fluids in equilibrium and solve problems applying Pascal‘s principle, Archimedes‘s, principles and Bernoulli‘s equation in various situations.  Explain the basic concepts of charges, fields and potentials.  Analyze direct and alternating current circuits containing different electric elements and solve circuit problems.  Demonstrate the use and the working system of cells (batteries), resistors, generators, motors and transformers.  Explain the first law of thermodynamics for a closed system and apply it to solve problems.  Discuss systems that oscillate with simple harmonic motion.  Explain the application of physics in different sciences and technology fields.  Apply and describe a variety of experimental techniques and grasp the general guidelines of laboratory.  Develop the skill of laboratory work. iii  ACKNOWLEDGEMENT In performing our module writing task, we had to take the help and guideline of some respected persons, who deserve our greatest gratitude. The completion of this assignment gives us much Pleasure. We would like to show our deepest gratitude to Dr. Eba Mijena, Director General for Academic and Research of Ministry of Science and Higher Education (MoSHE), for giving us a good guideline for write up and completion of the module throughout numerous consultations. We would also like to expand our deepest gratitude to all those who have directly and indirectly participated in writing this module. The Module Team October 26, 2019 iv  TABLE OF CONTENTS MODULE OVERVIEW....................................................................................................................... ii ACKNOWLEDGEMENT................................................................................................................... iv CHAPTER ONE..................................................................................................................................... 1 PRELIMINARIES.................................................................................................................................. 1 1.1. Physical Quantities and Measurement...................................................................................... 1 1.1.1. Physical quantities............................................................................................................ 2 1.1.2. SI Units: Basic and Derived Units..................................................................................... 3 1.1.3. Conversion of Units.......................................................................................................... 3 1.2. Uncertainty in Measurement and Significant Digits.................................................................. 4 1.2.1. Significant digits.............................................................................................................. 6 1.3. Vectors: composition and resolution......................................................................................... 8 1.3.1. Vector Representation...................................................................................................... 8 B. Geometric Method........................................................................................................................... 8 1.3.2. Vector Addition................................................................................................................ 9 B. Parallelogram law of vector addition................................................................................................ 9 1.3.3. Components of Vector...................................................................................................... 9 1.4. Unit Vector............................................................................................................................ 11 1.4.2. Finding a Unit Vector..................................................................................................... 12 CHAPTER TWO.................................................................................................................................. 15 KINEMATICS AND DYNAMICS OF PARTICLES............................................................................ 15 2.1. Kinematics in One and Two Dimensions................................................................................ 16 2.1.1. Displacement, velocity and Acceleration in 1D and 2D................................................... 16 2.1.2. Motion with Constant Acceleration................................................................................. 18 2.1.3. Free Fall Motion............................................................................................................. 21 2.1.4. Projectile Motion............................................................................................................ 22 2.2. Particle Dynamics and Planetary Motion................................................................................ 26 2.2.1. The Concept of Force as A Measure of Interaction.......................................................... 27 2.2.2. Type of Forces................................................................................................................ 27 v  2.2.3. Newton‘s Laws of Motion and Applications................................................................... 29 2.2.4. Uniform Circular Motion................................................................................................ 37 2.2.5. Newton‘s Law of Universal Gravitation.......................................................................... 38 2.2.6. Kepler‘s Laws, Satellites Motion and Weightlessness......................................................... 40 2.3. Work, Energy and Linear Momentum.................................................................................... 43 2.3.1. Work and Energy........................................................................................................... 44 2.3.2. Power............................................................................................................................. 50 2.3.3. Linear Momentum.......................................................................................................... 51 2.3.4. Collisions....................................................................................................................... 53 2.3.5. Center of Mass............................................................................................................... 56 CHAPTER THREE............................................................................................................................... 62 FLUID MECHNICS.............................................................................................................................. 62 3.1. Properties of Bulk Matter....................................................................................................... 62 3.2. Density and Pressure in Static Fluids...................................................................................... 68 3.3. Buoyant Force and Archimedes‘ Principles............................................................................ 70 3.3.1. Archimedes‘ principle.................................................................................................... 72 3.4. Moving Fluids and Bernoulli Equations (Fluid Dynamics)...................................................... 73 3.4.1. Bernoulli‘s Equation....................................................................................................... 74 CHAPTER FOUR................................................................................................................................. 80 HEAT AND THERMODYNAMICS..................................................................................................... 80 4.1. The concept of Temperature and the Zeroth law of Thermodynamics..................................... 80 4.2. Thermal Expansion................................................................................................................ 83 4.3. The Concept of Heat, Work and Internal Energy.................................................................... 85 4.4. Specific Heat and Latent Heat................................................................................................ 86 4.5. Heat Transfer Mechanisms..................................................................................................... 89 4.6. The First Law of Thermodynamics......................................................................................... 92 CHAPTER FIVE................................................................................................................................... 99 OSCILLATIONS, WAVES AND OPTICS........................................................................................... 99 5.1. Simple Harmonic Motion....................................................................................................... 99 5.1.1. Periodic and Oscillatory Motion................................................................................... 100 5.1.2. Displacement, Velocity and Acceleration in a SHM...................................................... 103 5.2. The simple Pendulum........................................................................................................... 104 vi  5.3. Wave and Its Characteristics................................................................................................ 106 5.4. Resonance............................................................................................................................ 108 5.5. The Doppler Effect............................................................................................................... 108 5.6. Image Formation by Thin Lenses and Mirrors...................................................................... 111 CHAPTER SIX................................................................................................................................... 119 ELECTROMAGNETISM AND ELECTRONICS............................................................................... 119 6.1. Coulomb‘s Law and Electric Fields...................................................................................... 120 6.2. Electric Potential.................................................................................................................. 125 6.3. Current, Resistance and Ohm‘s Law..................................................................................... 130 6.4. Electrical Energy and Power................................................................................................ 134 6.5. Equivalent Resistance and Kirchhoff‘s Rule......................................................................... 134 6.6. Magnetic Field and Magnetic Flux....................................................................................... 140 6.7. Electromagnetic Induction.................................................................................................... 144 6.8. Insulators, Conductors and Semiconductors.......................................................................... 146 6.9. Diodes................................................................................................................................. 154 6.10. Transistors....................................................................................................................... 158 CHAPTER SEVEN............................................................................................................................. 166 CROSS CUTTING APPLICATIONS OF PHYSICS........................................................................... 166 7.1. Physics in Agriculture and Environment............................................................................... 168 7.2. Physics in Industries............................................................................................................. 172 7.3. Physics in Health Sciences and Medical Imaging.................................................................. 173 7.3. Physics and Archeology....................................................................................................... 177 7.4. Application in Earth and Space Sciences.............................................................................. 181 7.5. Applications in Power Generation........................................................................................ 184 REFERENCES.................................................................................................................................... 193 vii viii  CHAPTER ONE PRELIMINARIES The word physics comes from the Greek word meaning ―nature‖. Today physics is treated as the base for science and have various applications for the ease of life. Physics deals with matter in relation to energy and the accurate measurement of natural phenomenon. Thus physics is inherently a science of measurement. The fundamentals of physics form the basis for the study and the development of engineering and technology. Measurement consists of the comparison of an unknown quantity with a known fixed quantity. The quantity used as the standard of measurement is called ‗unit‘. For example, a vegetable vendor weighs the vegetables in terms of units like kilogram. Learning Objectives: At the end of this chapter, you will be able to:  Explain physics.  Describe how SI base units are defined.  Describe how derived units are created from base units.  Express quantities given in SI units using metric prefixes.  Describe the relationships among models, theories, and laws.  Know the units used to describe various physical quantities.  Become familiar with the prefixes used for larger and smaller quantities.  Master the use of unit conversion (dimensional analysis) in solving problems.  Understand the relationship between uncertainty and the number of significant figures in a number. 1.1. Physical Quantities and Measurement Self Diagnostic Test:  Why do we need measurement in physics and our day-to-day lives?  Give the names and abbreviations for the basic physical quantities and their corresponding SI units.  What do you mean by a unit? 1 Definitions: Physical quantity is a quantifiable or assignable property ascribed to a particular phenomenon or body, for instance the length of a rod or the mass of a body. Measurement is the act of comparing a physical quantity with a certain standard. Scientists can even make up a completely new physical quantity that has not been known if necessary. However, there is a set of limited number of physical quantities of fundamental importance from which all other possible quantities can be derived. Those quantities are called Basic Physical Quantities, and obviously the other derivatives are called Derived Physical Quantities. 1.1.1. Physical quantities A. Basic Physical Quantities: Basic quantities are the quantities which cannot be expressed in terms of any other physical quantity. Example: length, mass and time. B. Derived Physical Quantities: Derived quantities are quantities that can be expressed in terms of fundamental quantities. Examples: area, volume, density. Giving numerical values for physical quantities and equations for physical principles allows us to understand nature much more deeply than qualitative descriptions alone. To comprehend these vast ranges, we must also have accepted units in which to express them. We shall find that even in the potentially mundane discussion of meters, kilograms, and seconds, a profound simplicity of nature appears: all physical quantities can be expressed as combinations of only seven basic physical quantities. We define a physical quantity either by specifying how it is measured or by stating how it is calculated from other measurements. For example, we might define distance and time by specifying methods for measuring them, such as using a meter stick and a stopwatch. Then, we could define average speed by stating that it is calculated as the total distance traveled divided by time of travel. 2 Measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of meters (for sprinters) or kilometers (for distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way. 1.1.2. SI Units: Basic and Derived Units SI unit is the abbreviation for International System of Units and is the modern form of metric systemfinallyagreeduponattheeleventhInternationalconferenceofweightsandmeasures,1960. This system of units is now being adopted throughout the world and will remain the primary system of units of measurement. SI system possesses features that make it logically superior to any other system and it is built upon 7 basic quantities and their associated units (see Table 1.1). Table 1.1: Basic quantities and their SI units Table 1.2: Derived quantities, their SI units and dimensions 1.1.3. Conversion of Units Measurements of physical quantities are expressed in terms of units, which are standardized values. To convert a quantity from one unit to another, multiply by conversions factors in such a 3 way that you cancel the units you want to get rid of and introduce the units you want to end up with. Below is the table for commonly used unit conversions (see Table 1.3). Table 1.3: Unit conversion of basic quantities Examples: 1. Length 0.02in can be converted into SI unit in meters using table 1.3 as follow: Solution: 0.02in= 0.02 x0.0254m = 0.000508m = 5.08 x10-4 m = 0.503 mm or 508µm. 2. Honda Fit weighs about 2,500 lb. It is equivalent to 2500 x0.4536kg = 1134.0kg. Activities: 1. A common Ethiopian cities speed limit is 30km/hr. What is this speed in miles per hours? 2. How many cubic meters are in 250,000 cubic centimeters? 3. The average body temperature of a house cat is 101.5oF. What is this temperature in Celsius? 1.2. Uncertainty in Measurement and Significant Digits Measurements are always uncertain, but it was always hoped that by designing a better and better experiment we can improve the uncertainty without limits. It turned out not to be the case. No measurement of a physical quantity can be entirely accurate. It is important to know, therefore, just how much the measured value is likely to deviate from the unknown, true, value of the quantity. The art of estimating these deviations should probably be called uncertainty analysis, 4 but for historical reasons is referred to as error analysis. This document contains brief discussions about how errors are reported, the kinds of errors that can occur, how to estimate random errors, and how to carry error estimates into calculated results. Uncertainty gives the range of possible values of the measure and, which covers the true value of the measure and. Thus uncertainty characterizes the spread of measurement results. The interval of possible values of measure and is commonly accompanied with the confidence level. Therefore, the uncertainty also indicates a doubt about how well the result of the measurement presents the value of the quantity being measured. All measurements always have some uncertainty. We refer to the uncertainty as the error in the measurement. Errors fall into two categories: 1. Systematic Error - errors resulting from measuring devices being out of calibration. Such measurements will be consistently too small or too large. These errors can be eliminated by pre-calibrating against a known, trusted standard. 2. Random Errors - errors resulting in the fluctuation of measurements of the same quantity about the average. The measurements are equally probable of being too large or too small. These errors generally result from the fineness of scale division of a measuring device. Physics is an empirical science associated with a lot of measurements and calculations. These calculations involve measurements with uncertainties and thus it is essential for science students to learn how to analyze these uncertainties (errors) in any calculation. Systematic errors are generally ―simple‖ to analyze but random errors require a more careful analysis and thus it will be our focus. There is a statistical method for calculating random uncertainties in measurements. The following general rules of thumb are often used to determine the uncertainty in a single measurement when using a scale or digital measuring device. 1. Uncertainty in a scale measuring device is equal to the smallest increment divided by 2. Example: Meter Stick (scale device) 5  2. Uncertainty in a digital measuring device is equal to the smallest increment. Example: A reading from digital Balance (digital device) is 5.7513 kg, therefore When stating a measurement, the uncertainty should be stated explicitly so that there is no question about it. However, if it is not stated explicitly, an uncertainty is still implied. For example, if we measure a length of 5.7cm with a meter stick, this implies that the length can be anywhere in the range 5.65 cm ≤ L ≤ 5.75 cm. Thus, L =5.7cm measured with a meter stick implies an uncertainty of 0.05 cm. A common rule of thumb is to take one-half the unit of the last decimal place in a measurement to obtain the uncertainty. In general, any measurement can be stated in the following preferred form: Measurement = xbest± Where, xbest= best estimate of measurement, = uncertainty (error) in measurement. 1.2.1. Significant digits Whenever you make a measurement, the number of meaningful digits that you write down implies the error in the measurement. For example if you say that the length of an object is 0.428 m, you imply an uncertainty of about 0.001 m. To record this measurement as either 0.4 or 0.42819667 would imply that you only know it to 0.1m in the first case or to 0.00000001 m in the second. You should only report as many significant figures as are consistent with the estimated error. The quantity 0.428m is said to have three significant digits, that is, three digits that make sense in terms of the measurement. Notice that this has nothing to do with the "number of decimal places". The same measurement in centimeters would be 42.8cm and still be a three significant figure. The accepted convention is that only one uncertain digit is to be reported for a measurement. In the example if the estimated error is 0.02m you would report a result of 0.43 ± 0.02 m, not 0.428 ± 0.02 m. Students frequently are confused about when to count a zero as a significant figure. The rule is: If the zero has a non-zero digit anywhere to its left, then the zero is significant, otherwise it is not. For example 5.00 has 3 significant figures; the number 0.0005 has only one significant 6 figure, and 1.0005 has 5 significant figures. A number like 300 is not well defined. Rather one should write 3 x 102, one significant figure, or 3.00 x 102, 3 significant figures. When writing numbers, zeros used ONLY to help in locating the decimal point are NOT significant others are. See the following examples: 1) 0.0062 cm has 2 significant figures 2) 4.0500 cm has 5 significant figures Rules for significant digits: Rule 1: When approximate numbers are multiplied or divided, the number of significant digits in the final answer is the same as the number of significant digits in the least accurate of the factors. Example: ( ) (. ) Least significant factor (45) has only two (2) digits so only two are justified in the answer. The appropriate way to write the answer is P = 7.0 N/m2. Rule 2: When approximate numbers are added or subtracted, the number of significant digits should equal the smallest number of decimal places of any term in the sum or difference. Example: 9.65 cm + 8.4 cm - 2.89 cm = 15.16 cm Note that the least precise measure is 8.4cm. Thus, answer must be to nearest tenth of cm even though it requires 3 significant digits. The appropriate way to write the answer is 15.2cm. Example: Find the area of a metal plate that is 8.71 cm by 3.2 cm. A = LW = (8.71 cm) (3.2 cm) = 27.872 cm2 In general to determine significant digits in a given number 1. All non-zero numbers are significant. 2. Zeros within a number are always significant. 3. Zeros that do nothing but set the decimal point are not significant. Both 0.000098 and 0.98 contain two significant figures. 4. Zeros that aren‘t needed to hold the decimal point are significant. For example, 4.00 has three significant figures. 5. Zeros that follow a number may be significant. 7 1.3. Vectors: composition and resolution A scalar is a quantity that is completely specified by a number and unit. It has magnitude but no direction. Scalars obey the rules of ordinary algebra. Examples: mass, time, volume, speed, etc. A vector is a quantity that is specified by both a magnitude and direction in space. Vectors obey the laws of vector algebra. Examples are: displacement, velocity, acceleration, momentum, etc. 1.3.1. Vector Representation A. Algebraic Method Vectors are represented algebraically by a letter (or symbol) with an arrow over its head (Example: velocity by ⃗, momentum by ⃗) and the magnitude of a vector is a positive scalar and is written as either by |A| or A. B. Geometric Method When dealing with vectors it is often useful to draw a picture (line with an arrow). Here is how it is done:  Vectors are nothing but straight arrows drawn from one point to another.  Zero vector is just a vector of zero length - a point.  Length of vectors is the magnitude of vectors. The longer the arrow the bigger the magnitude.  It is assumed that vectors can be parallel transported around. If you attach beginning of vector ⃗to end of another vector ⃗⃗then the vector ⃗+ ⃗⃗is a straight arrow from begging of vector ⃗to end of vector ⃗⃗. A vector changes if its magnitude or direction or if both magnitude and direction change. We add, subtract or equate physical quantities of same units and same characters (all the terms on both sides of an equation must be either scalar or vector). A vector may be multiplied by a pure number or by a scalar. Multiplication by a pure number merely changes the magnitude of the vector. If the number is negative, the direction is reversed. When a vector is multiplied by a scalar, the new vector also becomes a different physical quantity. For example, when velocity, a vector, is multiplied by time, a scalar, we obtain a displacement. 8 1.3.2. Vector Addition A single vector that is obtained by adding two or more vectors is called resultant vector and it is obtained using the following two methods A. Graphical method of vector addition Graphically vectors can be added by joining their head to tail and in any order their resultant vector is the vector drawn from the tail of the first vector to the head of the last vector. In Figure 1 graphical technique of vector addition is applied to add three vectors. The resultant vector R = A + B + C is the vector that completes the polygon. In other words, R is the vector drawn from the tail of the first vector to the tip of the last vector B. Parallelogram law of vector addition The parallelogram law states that the resultant R of two vectors A and B is the diagonal of the parallelogram for which the two vectors A and B becomes adjacent sides. All three vectors A, B and R are concurrent as shown in Figure 2. A and B are also called the components of R. The magnitude of the diagonal (resultant vector) is obtained using cosine law and direction (i.e. the angle that the diagonal vector makes with the sides) is obtained using the sine law. Applying cosine and sine laws for the triangle formed by the two vectors: Cosine law: √ Sine law: 1.3.3. Components of Vector Considering Figure 3 below, components of the given vector A are obtained by applying the trigonometric functions of sine and cosine. 9  Figure 3: Components of vector A ………………….. is the x component of A …………………. Is the y component of A The components Ax and Aycan be added to give back A as their resultant. A = A x + Ay Because Ax and Ay are perpendicular to each other, the magnitude of their resultant vector is obtained using Pythagoras theorem. √ Similarly, any three dimensional vector A can be written as the sum of its x, y and z components. A = A x + A y + Az And its magnitude becomes √ The direction angles that this vector makes with the three axes, is given by the direction of cosines. 10  Figure 4: Vector in three dimensional space ( * ( * ( * 1.4. Unit Vector A unit vector is a vector that has magnitude of one and it is dimensionless and its sole purpose is to point a given vector in specified direction. It is usually denoted with a ―hat‖. ̂ There is a special set of three unit vectors that are exceptionally useful for problems involving vectors, namely the Cartesian coordinate axis unit vectors. There is one of them for each positive coordinate axis direction. These unit vectors are so prevalent that we give them special names. For a two-dimensional x-y coordinate system we have the unit vector ̂ pointing in the +x direction, and, the unit vector ̂ pointing in the +y direction. For a three-dimensional x-, y- and z- coordinate system, we have those two, and one more, namely the unit vector ̂ pointing in the +z direction. Any vector can be expressed in terms of unit vectors. Consider, for instance, a vector A with components Ax, Ay, and Az. The vector formed by the product Ax ̂ has magnitude │Ax│ in the +x direction. This means that Ax ̂ isthe x-component of vector A. Similarly, Ay ̂is the y-component of vector A and, Az ̂ is the z-component vector of A. Thus A can be expressed as: ⃗ ̂ ̂ ̂ The vector ⃗ ̂ ̂ ̂ is depicted in the figure 4above, along with the vectors ̂, ̂, and ̂ drawn so that is clear that the three of them add up to ⃗ 1.4.1. Vector addition in Unit Vector Notation Adding vectors that are expressed in unit vector notation is easy in that individual unit vectors appearing in each of two or more terms can be factored out. The concept is best illustrated by means of an example. 11 Let ⃗ ̂ ̂ ̂ and ⃗⃗ ̂ ̂ ̂ ⃗ ⃗⃗ ( )̂ ( )̂ ( )̂ We see that the sum of vectors that are expressed in unit vector notation is simply the sum of the x components times ̂, plus the sum of the y components times ̂, plus the sum of the z component times ̂. 1.4.2. Finding a Unit Vector Consider the vector ⃗ ̂ ̂ ̂ The unit vector rˆ in the same direction as the vector ⃗is simply the vector ⃗divided by itsmagnitude r. Or ⃗ ̂ ̂ ̂ ⃗ ̂ ̂ ̂ The result makes it clear that each component of the unit vector is simply the corresponding component, of the original vector, divided by the magnitude √ of the original vector. 12 Chapter Summery Physical quantity is the property of an object that can be quantified. Measurement is the act of comparing a physical quantity with its unit. Basic quantities are the quantities which cannot be expressed in terms of any other physical quantity. Example: length, mass and time. Derived quantities are quantities that can be expressed in terms of fundamental quantities. Example: area, volume, density. Uncertainty gives the range of possible values of the measure and, which covers the true value of the measure and thus uncertainty characterizes the spread of measurement results. A scalar is a quantity that is completely specified by a number and its unit. It has magnitude but no direction. Scalars obey the rules of ordinary algebra. Examples: mass, time, volume, A vector is a quantity that is specified by both a magnitude and direction in space. Vector can be represented either by Algebraic method or Geometric method. A single vector that is obtained by adding two or more vectors is called resultant vector and it is obtained using the following two methods Vectors can be added using the ways Graphical method of vector addition or Parallelogram law of vector addition. A unit vector is a vector that has magnitude of one and it is dimensionless and a sole purpose of unit vector is to point-that is, to specify a direction. It is usually denoted with a ―hat‖. 13 Chapter Review Questions and Problems 1. Vector ⃗ has magnitude of 8units and makes an angle of 450 with the positive x-axis. Vector ⃗⃗ also has the same magnitude of 8units and directed along the negative x-axis. Find a. The magnitude and direction of ⃗ ⃗⃗ b. The magnitude and direction of ⃗ ⃗⃗ 2. Given the displacement vectors ⃗ ̂ ̂ ̂ , ⃗⃗ ̂ ̂ ̂. Find the magnitudes of the vectors a) ⃗+ ⃗⃗ b) 2 ⃗- ⃗⃗ 3. If ⃗ ̂ ̂ ⃗⃗ ̂ ̂ ⃗ ̂.̂ Find a and b Such that ⃗ b ⃗⃗ ⃗ 4. Find a unit vector in the direction of the resultant of vectors ⃗ ̂ ̂ ̂ , ⃗⃗ ̂ ̂ ̂ and ⃗ ̂ ̂ ̂ 14  CHAPTER TWO KINEMATICS AND DYNAMICS OF PARTICLES Mechanics is the study of the physics of motions and how it relates to the physical factors that affect them, like force, mass, momentum and energy. Mechanics may be divided into two branches: Dynamics, which deals with the motion of objects with its cause – force; and kinematics describes the possible motions of a body or system of bodies without considering the cause. Alternatively, mechanics may be divided according to the kind of system studied. The simplest mechanical system is the particle, defined as a body so small that its shape and internal structure are of no consequence in the given problem. More complicated is the motion of a system of two or more particles that exert forces on one another and possibly undergo forces exerted by bodies outside of the system. The principles of mechanics have been applied to three general realms of phenomena. The motions of such celestial bodies as stars, planets, and satellites can be predicted with great accuracy thousands of years before they occur. As the second realm, ordinary objects on Earth down to microscopic size (moving at speeds much lower than that of light) are properly described by mechanics without significant corrections. The engineer who designs bridges or aircraft may use the Newtonian laws of mechanics with confidence, even though the forces may be very complicated, and the calculations lack the beautiful simplicity of celestial mechanics. The third realm of phenomena comprises the behavior of matter and electromagnetic radiation on the atomic and subatomic scale. Learning Objectives: After going through this unit students will be able to:  Understand the general feature of motion of a particle.  Know how particles interact with the action of force.  Explain the relationship between force and work done. 15 2.1. Kinematics in One and Two Dimensions Self Diagnostic Test  What does kinematics deals about?  Can you state the kinematical quantities that describe the motion of objects?  Can you distinguish instantaneous and average velocities? And accelerations? A formal study of physics begins with kinematics. The word ―kinematics‖ comes from a Greek word ―kinesis‖ meaning motion, and is related to other English words such as ―cinema‖ (movies) and ―kinesiology‖ (the study of human motion). Kinematics is the branch of mechanics that describes the motion of objects without reference to the causes of motion (i.e., forces). Kinematics is concerned on analyzing kinematical quantities used to describe motion such as velocity, acceleration, displacement, time, and trajectory. Objects are in motion all around us. Planets moving around the sun, car moving along a road, blood flowing through veins, etc, are some examples of motion. Objectives At the end of this section you will be able to:  Define kinematic terms such as position, displacement, velocity and acceleration  Identify the difference between average and instantaneous velocity  Identify the difference between average and instantaneous acceleration  Drive kinematic equations for motions with constant acceleration  Explain projectile motion and solve problems related to it  Solve problems related to the concepts discussed in this chapter 2.1.1. Displacement, velocity and Acceleration in 1D and 2D Definition: Kinematical Quantities Position: - The location of an object with respect to a chosen reference point. 16 Displacement: - The change in position of an object with respect to a given reference frame. For 1D (for one-dimensional motion) = For 2D Y ⃗ ⃗ ⃗ ⃗ -------------------------------------------------- (2.1.1) ⃗⃗ S X Distance (S):- The length of the path followed by the object. Average and Instantaneous Velocities: Average Velocity (⃗⃗ ):-is the total displacement divided by the total time. ⃗ ⃗ ⃗ ⃗ ------------------------------------------------------------- (2.1.2) Average Speed: - is the total distance traveled by the object divided by the total elapsed time. ( ) ------------------------------------------------ (2.1.3) ( ) Average speed and average velocity of an object do not provide the detail information of the entire motion. We may need to know the velocity or speed of the particle at a certain instant of time. ⃗ Instantaneous Velocity (⃗⃗): - is the limiting value of the ratio as approaches zero. ⃗ ⃗= ------------------------------------------------ (2.1.4) The instantaneous speed:-It is the magnitude of the instantaneous velocity Average and Instantaneous Accelerations: If the velocity of a particle changes with time, then the particle is said to be accelerating. Average acceleration: is the change in velocity ( ⃗) of an object divided by the time interval during which that change occurs. ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ------------------------------------------------ (2.1.5) 17 Instantaneous acceleration: -The limit of average acceleration as approaches zero. ⃗⃗ ⃗= ------------------------------------------------ (2.1.6) Example A person walks first at a constant speed of 5m/s along the straight line from point A to point B, and then back along the same line from B to A at a constant speed of 3m/s a) What is his average speed over the entire trip b) What is his average velocity over the entire trip Solution 5m/s a) By definition, average speed is ( ) A x B ( ) 3m/s The total distance covered in the entire trip is Therefore, ( ) ( ) b) Average velocity over the entire trip is zero, because for the entire trip ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ 2.1.2. Motion with Constant Acceleration For motion with constant acceleration,  The velocity changes at the same rate throughout the motion. 18   Average acceleration over any time interval is equal to the instantaneous acceleration at any instant of time. ⃗⃗ ⃗⃗ ⃗⃗ ⃗= = , assuming ti= 0. Rearranging this equation gives ⃗ ⃗ ⃗ ------------------------------------------------ (2.1.7) For motion with constant acceleration, average velocity can be written as: ⃗⃗ ⃗⃗ ⃗ ------------------------------------------------ (2.1.8) ⃗ By definition ⃗ for ti=0 ⃗= ⃗ ( ⃗⃗ ⃗⃗ ) ⃗ ⃗ but ⃗ ⃗ ⃗ ⃗ ⃗ ⃗⃗⃗⃗ ⃗ ------------------------------------------------ (2.1.9) ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ Again, ⃗= ⃗ but ⃗ and t= (⃗⃗⃗⃗⃗ ⃗ ) (⃗⃗⃗⃗⃗ ⃗) ⃗ +2 ------------------------------------------------ (2.1.10) For 2D motion ⃗ ̂+ ̂, ⃗ ̂+ ̂, ⃗ ̂+ ̂ ⃗ ⃗ ⃗ { } ⃗ ⃗ ⃗⃗⃗ ⃗ { } Example 1 A track covers 40m in 8.5s while smoothly slowing down to a final speed of 2.8m/s. Find a) Its original speed b) b) its acceleration 19 Solution We are given that a) ⃗= ⃗ the magnitude of ⃗ is the magnitude of ⃗ is ( ) ( ) We are asked to find a) b) ⃗ ⃗⃗ ⃗⃗ ⃗⃗ b) ⃗ = The magnitude of the acceleration is Example 2 A jet plane lands with a speed of 100m/s and slows down at a rate of 5m/s2 as it comes to rest. a) What is the time interval needed by the jet to come to rest? b) Can this jet land on an airport where the runway is 0.8km long? Solution We are given that a) the magnitude of the acceleration is We are asked to find a) b) b. To determine whether the jet can land on the 0.8km runway, we need to calculate the distance through which the jet moves as it comes to rest ( * ( * ( ) ( ) 20  Activity: 1. At t=0s, a particle moving in the x-y plane with constant acceleration has a velocity of ⃗ ( ̂-2 ̂) m/s, and is at the origin. At t=3s, the particle‘s velocity is ⃗ ( ̂+7 ̂) m/s. Find (a)the acceleration of the particle (b) Its coordinates at t=3s 2.1.3. Free Fall Motion The motion of an object near the surface of the Earth under the only control of the force of gravity is called free fall. In the absence of air resistance, all objects fall with constant acceleration, g, toward the surface of the Earth. On the surface of the Earth, the generally accepted value is g = 9.8 m/s2. The acceleration due to gravity varies with latitude, longitude and altitude on Earth‘s surface. And it is greater at the poles than at the equator and greater at sea level than a top mountain. There are also local variations that depend upon geophysics. The value of 9.8 m/s2, with only two significant digits, is true for most places on the surface of the Earth up to altitudes of about 16 km. Example A girl throws a ball upwards, giving it an initial speed u = 15 m/s. Neglect air resistance. (a) How long does the ball take to return to the boy‘s hand? (b) What will be its velocity then? Solution: (a) We choose the positive y upward with its origin at the girl‘s hand, i.e. yi =0, see the Fig. below. Then, the ball‘s acceleration is negative (downward) during the ascending and descending motions, i.e. a=−g=−9.8m/s2. When the ball returns to the girl‘s hand its position y is zero. Since u=15m/s, yi=0, y=0, and a=−g, then we can find t from y – yi=ut− gt2 it follows: 0= (15m/s) t− (9.8 m/s2)×t2 ⇒ t = 2×(15m/s) 9.8 m/s2 =3.1s 21  (b) We are given u =15m/s, yi =0, y =0, and a =−g=− 9.8 m/s2. To find v, we use (c) v2 = u2−2 g(y−yi) it follows that: v2 = u2−0 ⇒ v =±u =±u =±15 m/s We should select the negative sign, because the ball is moving downward just before returning to the boy‘s hand, i.e. v =−15 m/s. 2.1.4. Projectile Motion Projectile is any object thrown obliquely into the space. The object which is given an initial velocity and afterwards follows a path determined by the gravitational force acting on it is called projectile and the motion is called projectile motion. A stone projected at an angle, a bomb released from an aero plane, a shot fired from a gun, a shot put or javelin thrown by the athlete are examples for the projectile. Consider a body projected from a point 'O' with velocity 'u'. The point 'o' is called point of projection and 'u' is called velocity of projection. Figure 2.1: Motion of a projectile Velocity of Projection (u): the velocity with which the body projected. Angle of Projection (α): The angle between the direction of projection and the horizontal plane passing through the point of projection is called angle of projection. Trajectory (OAB): The path described by the projectile from the point of projection to the point where the projectile reaches the horizontal plane passing through the point of projection is called trajectory. The trajectory of the projectile is a parabola. Basic assumptions in projectile motion 22   The free fall acceleration (g) is constant over the range of motion and it is directed downward.  The effect of air resistance is negligible. With the above two basic assumption the path of the projectile will be a down ward parabola. For projectile motion =0 (Because there is no force acting horizontally) The horizontal position of the projectile after some time t is: ( ) = (0, 0) if the projectile is initially at the origin ( ) ------------------------------------------------ (2.1.11) The vertical position of the projectile after some time t = ------------------------------------------------ (2.1.12) The horizontal components of the velocity But =0 = ------------------------------------------------ (2.1.13) 23 The vertical components of the velocity ------------------------------------------------ (2.1.14) Horizontal Range and Maximum Height - When the projectile reaches the maximum height (the peak), (time to reach maximum height) At t= ( )- ( )2 ------------------------------------------------ (2.1.15) The Range(R) is the horizontal displacement of the projectile covered in a total time of flight. Where ------------------------------------------------ (2.1.16) When ( )( ) But, ------------------------------------------------ (2.1.17) The range (R) is maximum, when Example 1 A rocket is fired with an initial velocity of 100m/s at an angle of 55 0 above the horizontal. It explodes on the mountain side 12s after its firing. What is the x-and y- coordinates of the rocket relative to its firing point? Solution: We are given that ( ) ( ) ( ) We are asked to find the horizontal position(x) x 24 and the vertical position (y) of the rocket ( ) ( )( )( ) ( )( ) y Example 2 A plane drops a package to a party of explorer. If the plane is travelling horizontally at 40m/s and is 100m above the ground, where does the package strike the ground relative to the point at which it is released? Solution - We are given that Since the vertical displacement is below the reference point - We are asked to find the horizontal displacement (x) ( ) ( ) (1) We can find t from equation for the vertical displacement ( ) But ( ) ( ) Substituting this value of t in (1) ( ) ( )( ) Activities 1. A ball is thrown with an initial velocity of ⃗⃗ ( ̂+ ̂) m/s. When it reaches the top of its trajectory, neglecting air resistance, what is its a) velocity? b) Acceleration? 2. An astronaut on a strange planet can jump a maximum horizontal distance of 15m if his initial speed is 3m/s. What is the free fall acceleration on the planet? 25 2.2. Particle Dynamics and Planetary Motion Self Diagnostic Test  What do you think about the cause for the change in the state of motion of an object?  What makes planets to revolve around the sun keeping their trajectory? In the previous section, we have described motion in terms of displacement, velocity, and acceleration without considering what might cause that motion. Here we investigate what causes changes in the state of motion. What cause particles to remain at rest or accelerate? It is because of the mass of the object and forces acting on it. Knowledge of Newton's laws and the ability to apply them to various situations will allow us to explain much of the motion we observe in the world around us. They are also very important for analyzing things (like bridges) that don't move much (a subject called Statics that's important in some Engineering programs). Newtonian dynamics was initially developed in order to account for the motion of the Planets around the Sun, which we discuss the problem in this part of the unit while discussing Kepler‘s laws of planetary motion. Objectives At the end of this section, you will be able to:  State the three Newton‘s laws of motion  Explain the behavior of action-reaction forces  Describe the nature and types of friction forces  Apply Newton‘s laws of motion in solving some problems  Discuss how an object accelerates in uniform circular motion.  State Kepler‘s laws of planetary motion Force: any interaction that changes the motion an object. A force moves or tends to move, stops or tends to stop the motion of the object. The force can also change the direction of motion of an object. It can also change the shape or size of a body on which it acts. Net force: is defined as the vector sum of all the forces acting on the object. The object accelerates only if the net force ( ⃗ ) acting on it is not equal to zero. 26 2.2.1. The Concept of Force as A Measure of Interaction In physics, any of the four basic forces gravitational, electromagnetic, strong nuclear and weak forces govern how particles interact. All other forces of nature can be traced to these fundamental interactions. The fundamental interactions are characterized on the basis of the following four criteria:  the types of particles that experience the force,  the relative strength of the force,  the range over which the force is effective, and  the nature of the particles that mediate the force. 2.2.2. Type of Forces Self diagnostic test List the types of forces you know and try to classify them as contact forces and field forces. Forces are usually categorized as contact and non-contact. Contact Force It is a type of force that requires bodily contact with another object. And it is further divided into the following. A. Muscular Forces Muscles function to produce a resulting force which is known as ‗muscular force‘. Muscular force exists only when it is in contact with an object. We apply muscular force during the basic day to day work of our life such as breathing, digestion, lifting a bucket, pulling or pushing some object. Muscular force comes in handy to simply our work. B. Frictional Forces When an object changes its state of motion, ‗frictional force‖ acts upon. It can be defined as the resisting force that exists when an object is moved or tries to move on a surface. The frictional force acts as a point of contact between two surfaces that is it arises due to contact between two surfaces. Examples; lighting a matchstick or stopping a moving ball come under frictional force. 27 C. Normal Force When a book is lying on the table, even though it seems that it‘s stationary, it‘s not. An opposing force is still acting on the book wherein the force from gravity is pulling it towards the Earth. This force is the ‗normal force‘. They always act perpendicular to the surface. 1. Applied Force When you push a table across the room, you apply a force that acts when it comes in contact with another object. This is ‗applied force‘; i.e. a force that is applied to a person or object. 2. Tension Force Tension is the force applied by a fully stretched cable or wire anchored on to an object. This causes a ‗tension force’ that pulls equally in both directions and exerts equal pressure. 3. Spring Force Force exerted by a compressed or stretched spring is ‗spring force‘. The force created could be a push or pull depending on how the spring is attached. 4. Air Resisting Force Air resisting forces are types of forces wherein objects experience a frictional force when moving through the air. These forces are resistive in nature. Non-Contact Force 28 It is a type of force that does not require a physical contact with the other object. It is further divided into the following types of forces: 5. Gravitational Force Gravitational force is an attractive force that can be defined by Newton‘s law of gravity which states that ‗gravitational forces between two bodies are directly proportional to the product of their masses and inversely proportional to the square of the distance between them‘ (more on this later). It is a force exerted by large bodies such as planets and stars. Example: water droplets falling down 6. Magnetic Force The types of forces exerted by a magnet on magnetic objects are ‗magnetic forces‘. They exist without any contact between two objects. 7. Electrostatic Force The types of forces exerted by all electrically charged bodies on another charged bodies in the universe are ‗electrostatic forces‘. These forces can be both attractive and repulsive in nature based on the type of charge carried by the bodies. 2.2.3. Newton’s Laws of Motion and Applications Laws of motions are formulated for the first time by English physicist Sir Isaac Newton in 1687. Newton developed the three laws of motion in order to explain why the orbits of the planets are ellipses rather than circles, at which he succeeded. Newton‘s laws continue to give an accurate account of nature, except for very small bodies such as electrons or for bodies moving close to the speed of light. Newton’s First law of Motion: “Everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.” This is sometimes called the Law of Inertia. Essentially, it makes the following two points:  An object that is not in motion will not move until a force acting upon it.  An object in constant motion will not change its velocity until a force acts upon it. 29 Another way of stating Newton's First Law of motion: A body that is acted on by no net force moves at a constant velocity (which may be zero) and zero acceleration. So with no net force, the object just keeps doing what it is doing. It is important to note the words net force. This means the vector sum of total forces acting upon the object must add up to zero. An object sitting on a floor has a gravitational force pulling it downward, but there is also a normal force pushing upward from the floor, so the net force is zero. Therefore, it doesn‘t move. Newton's Second law of Motion: The acceleration acquired by a point particle is directly proportional to the net force acting on the particle and inversely proportional to its mass and the acceleration is always in the direction of the net force. Mathematically, ∑ F = ma …………………………………………(2.2.1) Where ∑F is the net force acting on the particle, m is the mass of the particle and ais the acceleration of the particle. You'll note that when the net forces on an object sum up to zero, we achieve the state defined in Newton's First Law: the net acceleration must be zero. We know this because all objects have mass (in classical mechanics, at least). If the object is already moving, it will continue to move at a constant velocity, but that velocity will not change until a net force is introduced. Obviously, an object at rest will not move at all without a net force. Example 1 A box with a mass of 40 kg sits at rest on a frictionless tile floor. With your foot, you apply a 20 N force in a horizontal direction. What is the acceleration of the box? Solution: The object is at rest, so there is no net force except for the force your foot is applying. Friction is eliminated. Also, there's only one direction of force to worry about. So this problem is very straightforward. We begin the problem by defining the coordinate system. The mathematics is similarly straightforward: 30  F = ma F/m=a 20 N / 40 kg = a = 0.5 m / s2 The problems based on this law are literally endless, using the formula to determine any of the three values when you are given the other two. As systems become more complex, you will learn to apply frictional forces, gravity, electromagnetic forces, and other applicable forces to the same basic formulas. Example 2 A 3kg object undergoes an acceleration given by ⃗ ( ̂ )̂. Find the magnitude of the resultant force. Solution We are given that ⃗ ⃗ ⃗⃗⃗⃗⃗ ( ̂ ̂) ( )( ̂ ̂) We are asked to find the magnitude of ⃗ ⃗ ( ̂ ̂) The magnitude of the force is | ⃗| √ Newton's Third law of Motion States that ―For every action there is always an equal and opposite reaction.‖ To understand this law, consider two bodies A and B that are interacting and let FBA is the force applied on body A by body B, and FAB is the force applied on body B by body A. These forces will be equal in magnitude and opposite in direction. In mathematical terms, it is expressed as: FBA = - FAB or FAB + FBA = 0 This is not the same thing as having a net force is zero, however. Action and reaction forces are not treated the same as the forces acting on stationary object, normal force and weight of the object. 31  Note that:  Action and reaction forces are always exist in pair  A single isolated force cannot exist  Action and reaction forces act on different objects Activities 1. Find the force needed to accelerate a mass of 40kg from velocity ⃗ ( ̂ ̂ ̂) to ⃗ ( ̂ ̂ ̂) in 10s 2. If a man weighs 900N on earth, what is his weight on Jupiter where the acceleration due to gravity is 25.9m/s2? Forces of Friction Self Diagnostic Test  What makes you walk on the surface of the Earth without slipping?  How can you express your interaction with the surrounding air while you are walking or running? Frictional force refers to the force generated by two surfaces that are in contact and either at rest or slide against each other. These forces are mainly affected by the surface texture and amount of force impelling them together. The angle and position of the object affect the amount of frictional force.  If an object is placed on a horizontal surface against another object, then the frictional force will be equal to the weight of the object.  If an object is pushed against the surface, then the frictional force will be increased and becomes more than the weight of the object. Generally friction force is always proportional to the normal force between the two interacting surfaces. Mathematically Ffrict  Fnorm 32  Ff= µFN……………………………………………….(2.2.2) Where the proportionality constant µ is the coefficient of friction Forces of friction are very important in our everyday lives. They allow us to walk or run and are necessary for the motion of wheeled vehicles. Therefore, friction forces are categorized in the following manner: a) Static friction: exists between two stationary objects in contact to each other. Mathematically static friction is written as = -------------------------------------------------------- (2.2.3) b) Kinetic friction: arises when the object is in motion on the surface. The magnitude of the force of kinetic friction acting between two surfaces is ------------------------------------------------------- (2.2.4) Where is called the coefficient of kinetic friction.  The values of and depend on the nature of the surfaces, but is generally less than ( ). Typical values range from around 0.03 to 1.0.  The direction of the friction force on an object is parallel to the surface with which the object is in contact and opposite to the actual motion (kinetic friction) or the impending motion (static friction) of the object relative to the surface.  The coefficients of friction are nearly independent of the area of contact between the surfaces. Example A 25.0-kg block is initially at rest on a horizontal surface. A horizontal force of 75.0 N is required to set the block in motion. After it is in motion, a horizontal force of 60.0 N is required to keep the block moving with constant speed. Find the coefficients of static and kinetic friction from this information. Solution 33 We are given that = but , )( ( ) We are asked to find = but ( )( ) Application of Newton’s Laws of Motion In this section we apply Newton‘s laws to objects that are either in equilibrium ( ⃗= 0) or accelerating along a straight line under the action of constant external forces. Remember that when we apply Newton‘s laws to an object, we are interested only in external forces that act on the object. We assume that the objects can be modeled as particles so that we need not worry about rotational motion. We usually neglect the mass of any ropes, strings, or cables involved. The following procedure is recommended when dealing with problems involving Newton‘s laws: 1. Draw a sketch of the situation. 2. Consider only one object (at a time), and draw a free-body diagram for that body, showing all the forces acting on that body. Do not show any forces that the body exerts on other bodies. If several bodies are involved, draw a free-body diagram for each body separately, showing all the forces acting on that body. 3. Newton's second law involves vectors, and it is usually important to resolve vectors into components. Choose an x and y axis in a way that simplifies the calculation. 4. For each body, Newton's second law can be applied to the x and y components separately. That is the x component of the net force on that body will be related to the x component of that body's acceleration: Fx=max, and similarly for the y direction. 5. Solve the equation or equations for the unknown(s). Example 1 34 A bag of cement of weight 300 N hangs from three ropes as shown in the figure below. Two of the ropes make angles of 53.0° and 37.0° with the horizontal. If the system is in equilibrium, find the tensions , , and in the ropes. Solution We can draw two free body diagrams for the problem as follows a) b) y x ∑ Since the system is in equilibrium, ∑ ⃗ {∑ F ( )∑ (a) ∑ (b) F ( )∑ (c) Substituting(c) in (b) (d) But Substituting for in (d) Gives ( ) 35  ( ) Example 2 A block of mass m slides down an inclined plane as shown in the figure below. Find the expression for the acceleration of the block. (a)If the inclined plane is frictionless (b)If the inclined plane has coefficient of kinetic friction Solution The free body diagram for the problem is x a) y N x b) y N fk a θ mgcosθ aθ mgcosθ mg mgsinθ mg mgsinθ θθ a) If the inclined plane is frictionless, the only forces acting on the block are the gravitational force (mg) downward and the normal force (N). Resolving mg into parallel and perpendicular to the direction of motion of motion ∑ b) If the inclined plane has coefficient of kinetic friction , the forces acting on the block are shown in free body diagram (b) ∑ 36  ( ) 2.2.4. Uniform Circular Motion Self Diagnostic Test Do you know that objects moving with constant speed can have acceleration? When does this occur? Uniform Circular Motion is motion of objects in a circular path with a constant speed. Objects moving in a circular path with a constant speed can have acceleration. ⃗⃗ ⃗= There are two ways in which the acceleration can occur due to:  change in magnitude of the velocity  change in direction of the velocity For objects moving in a circular path with a constant speed, acceleration arises because of the change in direction of the velocity. Hence, in case of uniform circular motion:  Velocity is always tangent to the circular path and perpendicular to the radius of the circular path.  Acceleration is always perpendicular to the circular path, and points towards the center of the circle. Such acceleration is called the centripetal acceleration ⃗ ⃗ The angle Δ in figure (a) and (b) are the same ⃗⃗ ( ⃗) By SAS similarity ⃗ ⃗ ⃗⃗ ⃗ ⃗= ( )( ) 37  (Centripetal acceleration) ------------------------------------------------ (2.3.1) Period (T):- Time required for one complete revolution For a particle moving in a circle of radius r with a constant speed ------------------------------------------------ (2.3.2) Activity An athlete rotates a discus along a circular path of radius 1.06m.If the maximum speed of the disc us is 20m/s, determine the magnitude of the maximum centripetal acceleration. 2.2.5. Newton’s Law of Universal Gravitation Self Diagnostic Test What do you think when the Earth is always revolving around the Sun without slipping and leaving its line of revolution? Gravity is the weakest of the four basic forces found in nature, and in some ways the least understood. Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions. But Newton was not the first person to suspect that the same force caused both our weight and the motion of planets. His forerunner Galileo Galilei had contended that falling bodies and planetary motions had the same cause. Some of Newton‘s contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections; circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph, it had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose a mechanism that caused them to follow these paths and not others. The gravitational force is always attractive, and it depends only on the masses involved and the distance between them. Newton’s universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them. The force is directly 38 proportional to the product of their masses and inversely proportional to the square of the distance between them. Gravitational attraction is along a line joining the centers of mass of these two bodies. The magnitude of the force is the same on each, consistent with Newton‘s third law. CM F F CM m M r Figure 2.2: Interaction of two objects with gravitational force ⃗ ̂ ………………………………………………………….(2.3.4) Here, r is the distance between the centers of mass of the bodies, G is the gravitational constant, whose value found by experiment is G= 6.674×10−11 Nm2/kg2in SI units. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that even large objects like mountains exert gravitational forces on us. In fact, our body weight is the force of attraction of the entire Earth on us with a mass of 6 x 1024kg. Recall that the acceleration due to gravity g is about 9.8m/s2 on Earth. We can now determine why this is so. The weight of an object mg is the gravitational force between it and Earth. Substituting mg for F in Newton‘s universal law of gravitation gives Where, m is the mass of the object, M is the mass of Earth, and r is the distance to the center of Earth (the distance between the centers of mass of the object and Earth). The mass m of the object cancels, leaving an equation for g: ……………………………………………… (2.3.5) Substituting known values for Earth‘s mass and radius (to three significant figures), ( )( ) 39 and we obtain a value for the acceleration of a falling body: The distance between the centers of mass of Earth and an object on its surface is very nearly the same as the radius of Earth, because Earth is so much larger than the object. This is the expected value and is independent of the body’s mass. Newton‘s law of gravitation takes Galileo‘s observation that all masses fall with the same acceleration a step further, explaining the observation in terms of a force that causes objects to fall, in fact, in terms of a universally existing force of attraction between masses. 2.2.6. Kepler’s Laws, Satellites Motion and Weightlessness The basic laws of planetary motion were established by Johannes Kepler (1571-1630) based on the analysis of astronomical observations of Tycho Brahe (1546−1601). In 1609, Kepler formulated the first two laws. The third law was discovered in 1619. Later, in the late 17th century, Isaac Newton proved mathematically that all three laws of Kepler are a consequence of the law of universal gravitation. Kepler’s First Law (Law of Orbits) States that the orbit of each planet in the solar system is an ellipse, the Sun will be on one focus. The points F1 and F2represented in figure are known as the foci of the ellipse. The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse. Kepler’s Second Law (The Law of Areas) States that ―the radius vector connecting the centers of the Sun and the Planet sweepsout equal areas in equal intervals of time.‖The Figure below shows the two sectors of the ellipse having equal areas corresponding to the same time intervals. The second 40 law describes the speed (which is constantly changing) at which any given planet will move while orbiting the sun. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. Kepler’s Third Law (The Law of Harmony) States that “the square of the orbital period of a planet is proportional to the cube of the average distance between the centers of the planet and the sun.” Unlike Kepler's first and second laws that describe the motion characteristics of a single planet, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. As an illustration, consider the orbital period and average distance from sun (orbital radius) for Earth and mars as given in the table below. Period Average T2/R3 Planet (s) Distance (m) (s2/m3) Earth 3.156 x 107 s 1.4957 x 1011 2.977 x 10-19 Mars 5.93 x 107 s 2.278 x 1011 2.975 x 10-19 Observe that the T2/R3 ratio is the same for Earth as it is for mars. In fact, if the same T2/R3 ratio is computed for the other planets, it can be found that this ratio is nearly the same value for all the planets (see table below). Amazingly, every planet has the same T2/R3 ratio. Planet Period Average T2/R3 41  (yr) Distance (au) (yr2/au3) Mercury 0.241 0.39 0.98 Venus.615 0.72 1.01 Earth 1.00 1.00 1.00 Mars 1.88 1.52 1.01 Jupiter 11.8 5.20 0.99 Saturn 29.5 9.54 1.00 Uranus 84.0 19.18 1.00 Neptune 165 30.06 1.00 Pluto 248 39.44 1.00 (NOTE: The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the earth to the sun - 1.4957 x 1011 m. The orbital period is given in units of earth- years where 1 earth year is the time required for the earth to orbit the sun - 3.156 x 107 seconds. ) Activity:  In what frame(s) of reference are Kepler’s laws valid? Are Kepler’s laws purely descriptive, or do they contain causal information? Satellite motion and Weightlessness Self Diagnostic Test Astronauts on the orbiting space station are weightless because:  There is no gravity in space and they do not weigh anything.  Space is a vacuum and there is no gravity in a vacuum.  Space is a vacuum and there is no air resistance in a vacuum.  The astronauts are far from Earth's surface at a location where gravitation has a minimal effect. Astronauts who are orbiting the Earth often experience sensations of weightlessness. These sensations experienced by orbiting astronauts are the same sensations experienced by anyone who has been temporarily suspended above the seat on an amusement park ride. Not only are the 42 sensations the same (for astronauts and roller coaster riders), but the causes of those sensations of weightlessness are also the same. Unfortunately however, many people have difficulty understanding the causes of weightlessness. The cause of weightlessness is quite simple to understand. However, the stubbornness of one's preconceptions on the topic often stands in the way of one's ability to understand. What was your answer for the self-diagnostic test given above? Read the statements given below and use your knowledge of contact and non-contact force to understand well and find the answer. (MIND YOU: none of them are answers!) Weightlessness is simply a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or pull upon it. Weightless sensations exist when all contact forces are removed. These sensations are common to any situation in which you are momentarily (or perpetually) in a state of free fall. When in free fall, the only force acting upon your body is the force of gravity, a non- contact force. Since the force of gravity cannot be felt without any other opposing forces, you would have no sensation of it. You would feel weightless when in a state of free fall. 2.3. Work, Energy and Linear Momentum On a typical day, you probably wake up, get dressed, eat breakfast, and head off to work. After you spend all day at your job, you go home, eat dinner, walk the dog, maybe watch some TV, and then go to bed. In this sense, work can be just about anything - construction, typing on a keyboard, driving a bus, teaching a class, cooking food, treating patients, and so much more. But in physics, work is more specific. This is the displacement of an object due to force. How much work is done depends on the distance the object is moved. Work can be defined as transfer of energy due to an applied force. In physics we say that work is done on an object when energy is transferred to that object. If one object transfers (gives) energy to a second object, then the first object does work on the second object. The energy of a moving object is called kinetic energy. 43 The work done on an object by conservative force over any displacement is a function only of the difference in the positions of the end-points of the displacement. This property allows us to define a different kind of energy for the system than its kinetic energy, which is called potential energy. Potential energy is a state of the system, a way of storing energy as of virtue of its configuration or motion, while work done in most cases is a way of changing this energy from one body to another. When only conservative forces act within an isolated system, the kinetic energy gained (or lost) by the system as its members change their relative positions is balanced by an equal loss (or gain) in potential energy. This balancing of the two forms of energy is known as the principle of conservation of mechanical energy. Objectives At the end of this section, you will be able to:  Define work, kinetic energy and potential energy  Calculate the work done by a constant force  Derive work-kinetic energy theorem and apply in solving related problems  State the principle of conservation of mechanical energy  Solve problems related to the topics discussed in this section. 2.3.1. Work and Energy Self Diagnostic Test Can we use work and energy interchangeably? Can work be expressed in terms of the kinetic energy of an object and vice versa? The terms work and energy are quite familiar to us and we use them in various contexts. In physics, work is done when a force acts on an object that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. If no displacement takes place, no work is said to be done. Therefore for work to be done on an object, three essential conditions should be satisfied:  Force must be exerted on the object  The force must cause a motion or displacement 44   The force should have a component along the line of displacement If a particle subjected to a constant force ⃗ undergoes a certain displacement, ⃗, the work done W by the force is given by: ⃗ ⃗ ⃗ ⃗  -------------------------------------- (2.3.1) Where θ is the angle between ⃗ and ⃗. Work is a scalar quantity and its SI unit is Joule (J). Where, The sign of work depends on the direction of ⃗⃗⃗⃗ ⃗. Hence, the work done by the applied force is positive when the projection of ⃗⃗⃗⃗onto ⃗ is in the same direction