PHYS 111 Lecture Notes PDF

NJALA UNIVERSITY School of Basic Sciences Physics Department FIRST SEMESTER LECTURE NOTES SESSION: 2023/2024 GENERAL PHYSICS I – PHYS 111 November 2023 Course Name: Forces, Motion and Introduction to Thermal Physics Course Code: PHYS 111 Credit Hou...

NJALA UNIVERSITY School of Basic Sciences Physics Department FIRST SEMESTER LECTURE NOTES SESSION: 2023/2024 GENERAL PHYSICS I – PHYS 111 November 2023 Course Name: Forces, Motion and Introduction to Thermal Physics Course Code: PHYS 111 Credit Hour: 3 Credit hours Name of Instructor: Course Objectives This is an introductory level course in physics for students from all disciplines. This course introduces a broad range of physical phenomena and gives learners the opportunity to gain an understanding of fundamental physical principles in Newtonian mechanics and thermodynamics, use these principles to describe the world around us, and develop problem solving and mathematical skills. Course Learning Outcomes Upon successful completion of the course, students will be able to 1. obtain the prerequisite body of knowledge and skills that will provide a basis for further academic training 2. appreciate and apply the scientific method to investigations of all phenomena 3. appreciate and apply the physics of everyday life 4. Communicate effectively, particularly to the scientific community using the language of physics and mathematics. 5. Handle equipment in a safe and effective manner with regard to their own safety and the safety of others. Detailed Content UNIT I - Introduction to Physics: Concepts of Physics, relationship between physics and other subjects, importance of studying physics, Applications of Physics in Real Life UNT II - Measurements and Quantities: Concepts of measurement, importance of measurement in real life, Fundamental and Derived Quantities, Systems of measurements and Units, Basic SI units and measurement of length, time and mass. Principle of venire, dimensional analysis, uncertainty in measurement and significant figures, conversion of units, 1 estimates and order-of-magnitude calculations, vectors and scalars – Addition, resolution, and multiplication of vectors. UNT III - Particle Kinematics: Definition, Motion an types, Rectilinear motion and parameters, graphical representation of motion, equations of uniformly accelerated motion, vertical motion under gravity, relative motion, kinematics in two dimensions, UNT IV - Dynamics- Forces and Motion: Definition of force, categories, and types, Newton's Laws of Motion. Principle of Linear Impulse and Momentum - Principle of Linear Impulse and Momentum for a System of Particles, Conservation of Linear Momentum for a System of Particles. Applications of Newton’s Laws. UNT V - Force and Equilibrium: Concepts of equilibrium and types, general conditions of equilibrium, moment and parallel forces, Applications of parallel forces, center of gravity and center of mass, non-parallel forces and applications, equilibrium and limb movement, stability and equilibrium. UNT VI – Work, Energy and Power: work of a force, principle of work and energy, principle of work and energy for a system of particles, energy resources and conversion, power and efficiency, conservative forces and potential energy, conservation of energy. UNT VII – Curvilinear Motion: Definition and motion parameters of Curvilinear Motion, Tangential Components of Curvilinear Motion, equations of uniform, Example of Curvilinear Motion - conical pendulum, motion in a vertical circle, motion of a cyclist round a horizontal curve, motion of a vehicle round a horizontal curve, banking of curves, Components of Curvilinear UNT VIII - Oscillatory Motion: Basic concepts of oscillatory motion, Simple Harmonic Motion (SHM)- period and Frequency, Linear restoring forces and simple harmonic motion, Simple harmonic motion graphs, simple pendulum – determination of gravitational intensity (g). UNT IX - Introduction to Thermal Physics: Temperature and its measurement including thermocouple, Thermal expansion of solids, liquids, specific heat and latent heat, Ideal gas laws. Absolute zero of temperature. Transfer of heat- conduction, convection and Radiation. 2 Assessment methods Tests = 20% Assignments = 10% Practical/Field work = 20% Final Examination = 50 % Recommended Text Books/References: 1. Cutnell, John, Johnson, Kenneth, Physics, 5th ed. 2001 2. Yukon College, Physics 100 Laboratory Manual, 2017 3  UNIT I – INTRODUCTION TO PHYSICS 1.0 What is Physics? Physics is the study of the world around us. In a sense, we are more qualified to do physics than any other science. From the day we are born we study the things around us in an effort to understand how they work and relate to each other. Learning how to catch or throw a ball is Physics undertaking for example. Physics is the branch of science that investigates the fundamental concepts of matter, energy, and space and the relationships between them. 1.1 Structure of Physics Physics, like other sciences, starts with observation in the world around us or from laboratory experiments designed to obtain facts. The investigation of electricity, for example began when it was noticed that amber (a glass-like fossil) attracts small lightweight objects when it is rubbed with cloth. A. Concepts: To help make sense of the facts of physics and explain the behaviour of the physical world, physicists invent terms called concepts. These concerned quantities that can be measured. Some such as length are very basic and easily measured, while others like potential difference in electricity are less ‘concrete’ and require more sophisticated measuring instruments. Four of the most useful concepts are those of Atoms, Energy, Fields and Waves. B. Hypothesis: A Hypothesis is a tentative explanation of some regularity of nature. If a hypothesis is to be useful, it should suggest new experiment that becomes test of the hypothesis. If a hypothesis passes many tests, it becomes known as a Theory. C. Theory: Frequently in physics, what we are dealing with is not directly accessible to our senses and in such cases, we sometimes use theories or ‘thought models’ to help us explain things. A theory is a tested explanation of basic natural phenomena. Examples include, Wave theory, Field theory Kinetic theory etc. D. Laws: Experiments show that in many cases, relationships called laws or principles exists between concepts. They summarize a large number of facts in an economical way. A law is a concise statement or mathematical equation about a fundamental relationship or 1  regularity of nature. Examples include, Hooke’s law, Ohm’s law, Boyle’s law, Newton’s laws of motion etc. 1.2 Branches of Physics For the purpose of study, it is often convenient to divide physics into different branches such as Mechanics, Heat, Waves, Electricity, Magnetism, Atomic and Nuclear etc. However, many concepts and laws cut across these artificial boundaries and are useful in more than one branch. i) Mechanics: Is concerned with the study of position (statics) and motion (dynamics) of matter in space. ii) Statics: represent the branch of mechanics associated with bodies in equilibrium. There are two kinds of equilibrium. The first is static equilibrium, in which the body remains at rest under the influence of opposing forces whose magnitudes are equal. For example, a book resting on the study table, for instance, exerts a force (i.e., its weight) on the table, this force being opposed by an equal and opposite reaction force exerted by the table on the book. In the second type of equilibrium, called dynamic equilibrium, the body is in motion with constant velocity. Examples of bodies in dynamic equilibrium in mechanics include the following: An object falling at constant velocity through a viscous liquid: the weight of the object is balanced by the forces of friction in the fluid and the force exerted by the fluid, referred to as the upthrust of the liquid. The net downward force on the object is therefore zero, and the body moves with constant velocity in a straight line, in accordance with Newton’s first law of motion. A hot-air balloon coming to land at constant speed in a straight line: The forces acting are the weight of the balloon, the upward force due to the hot air in the balloon, and air resistance. Again, the net force on the balloon is zero, and the balloon moves with a constant speed in a straight line. Since the net force on the body in these situations is zero, the body does not accelerate. 2 iii) Dynamics: represent the branch of mechanics that describe motion and its courses. iv) Heat Energy: Is the form of energy that can flow from one point to another because of difference in temperature. It flows from points of higher to lower temperatures. v) Waves: Wave is a disturbance, which transfers energy from one point to another through an elastic medium with the average positions of the particle in the medium remaining constant. vi) Electricity: This deals with the study of electric charges that are not moving (electrostatics) or in motion (current electricity). vii) Magnetism: Is concerned with the study of field and force of attraction (or repulsion) exerted by a magnet. A magnet is a device, which exerts magnetic force. When electricity and magnetism are combined together a special phenomenon known as electromagnetism is observed. viii) Atomic and nuclear physics: Is concerned with the study of atom and its constituents. 1.3 Relationship Between Physics and Other Subjects, Physics is the most fundamental and all-inclusive of the sciences, and has had a profound effect on all scientific development. In fact, physics is the present- day equivalent of what used to be called natural philosophy, from which most of our modern sciences arose. Students of many fields find themselves studying physics because of the basic role it plays in all phenomena. 1. Physics and Mathematics: Physics relates strongly with mathematics. Many physics concepts and formulae are expressed mathematically. 2. Physics and Biology: Knowledge of lenses in physics is used in making microscope used in study of cells in biology. Physics formulae are used in calculation of magnification by microscopes. The knowledge of levers helps to explain locomotion in Biology. 3. Physics and Chemistry: Physics has helped in explaining forces within atoms and therefore atomic structure. It is this structure of the atom that then determines the reactivity of the atom as explained in chemistry. 3  4. Physics and Religion: Orderliness of systems in the universe can be traced back to the Creator. Many wonders of creation include the anomalous expansion of water, the rainbow etc. 5. Physics and Geography: Accurate use of instruments and physics concepts can establish weather patterns and explain formation of rainfall, pressure variations etc. Use of magnetic properties of lodestone and other materials help navigators to determine direction. 6. Physics and Technology: The applications of physics in everyday life are numerous. We use physics in our everyday life activities such as walking, playing, watching, listening, cutting, cooking, and opening and closing things. Let’s have a look at the main application areas of physics! 1. 4 Why Study Physics? There are hundreds of possible college majors and minors or subjects. So why should you study physics? i) Physics helps us to understand how the world around us works, from can openers, light bulbs and cell phones to muscles, lungs and brains; from earthquakes, tsunamis and hurricanes to quarks, DNA and black holes, etc. ii) Physics helps us to organize the universe. It deals with fundamentals, and helps us to see the connections between seemly disparate phenomena. iii) Physics gives us powerful tools to help us to express our creativity, to see the world in new ways and then to change it. iv) Studying physics strengthens quantitative reasoning and problem-solving skills that are valuable in areas beyond physics. Students who study physics or engineering physics are prepared to work on forefront ideas in science and technology, medicine, in academia, the government, or the private sector, as well as in economics, finance, management, law and public policy. v) Majoring in physics provides excellent preparation for graduate study not just in physics, but in all engineering and information/computer science disciplines; in the life sciences including molecular biology, genetics and neurobiology; in earth, atmospheric and ocean science; in finance and economics; and in public policy and journalism. 4  vi) Physics is the basis for most modern technology, and for the tools and instruments used in scientific, engineering and medical research and development. Manufacturing is dominated by physics-based technology. 1.5 Applications of Physics in Real Life 1. Alarm Clock: Physics gets involved in your daily life right after you wake up in the morning. The buzzing sound of an alarm clock helps you wake up in the morning as per your schedule. The sound is something you can’t see, but hear or experience. Physics studies the origin, propagation, and properties of sound. It works on the concept of Quantum Mechanics. 2. Steam Iron: Right after you wake up in the morning and start preparing for your school/office, you need an ironed cloth, and that’s where Physics comes into play. The steam iron is such a machine that uses a lot of Physics to make it work. The foremost principle of Physics used in the steam iron is “Heat.” Heat is a type of energy transfer from a warmer substance to a colder one. Ironing works by having a heated metal base- the soleplate. 3. Walking: Now, when you get ready for your office/school, whatever medium of commutation is, you certainly have to walk up to a certain distance. You can easily walk is just because of Physics. While walking in a park or on a tar road, you have a good grip without slipping because of roughness or resistance between the soles of your shoes and the surface of the road. This resistance, which is responsible for the grip, is called “Friction” or “Traction.” However, when a banana peel comes under your foot, you suddenly fall. Now, what makes you fall? Well, it’s due to the reduced friction between your shoes and the surface of the road due to the slippery banana peel. 4. Ball Point Pen: Whether you are at your workplace or in your school, a Ball Point Pen is your weapon. Had Physics not been there, you would not have been able to write with a Ball Point Pen on paper. In this case, the concept of gravity comes into play. As your pen moves across the paper, the ball turns and gravity forces the ink down onto the top of the ball where it is transferred onto the paper. 5. Headphones/Earphones: When you get tired of work or studies, listening to music comes in handy. Have you ever thought about how your headphones/earphones work? Well, it’s again 5  because of Physics. The concept of magnetism and sound waves are involved in the science of your headphones/earphones. When you plug your headphones/earphones into an electrical source, the magnet in your headphones/earphones creates an electromagnetic field, which ultimately results in sound waves. 6. Car Seat-Belts: Have you ever noticed on which principle your car seat-belt works? Well, it’s again Physics. When you tighten your car seat-belt, it works on the concept of “Inertia.” Inertia is the unwillingness or laziness of a body to change its state of rest or motion. In case of a car collision, your seat-belt helps prevent your body from moving in a forward direction as your body resists being stopped because of inertia of motion. 7. Camera Lens: The phenomenon of “Selfie” has engulfed people of every age group. You entertain yourself by clicking photos. The Lens used in a camera works on the principle of Optics. The set of convex lenses provides the camera with an image outside of the camera. 8. Cell Phones: Cellphones have become like Oxygen gas in modern social life. Hardly, anyone would have been untouched by the effects of a cell phone. Whether conveying an urgent message or doing incessant gossip, cell phones are everywhere. But do you know how a cell phone works? It works on the principle of electricity and the electromagnetic spectrum, undulating patterns of electricity and magnetism. 9. Batteries: Whether in cellphones, cars, torches, toys, or any other appliance, batteries act as saviours of electricity. Batteries work on the principle of capacitance. Since the late 18th century, capacitors have been used to store electrical energy. Benjamin Franklin was the first to coin the phrase “battery” for a series of capacitors in an energy store application. 10. Doppler Radar: To check over speeding vehicles, police often use Doppler Radars. Doppler Radars work on the principle of the Doppler Effect. The Doppler Effect is nothing but a change in the pitch of a sound when the source of the sound is moving relative to the listener. It is because the frequency of the sound wave changes as the source of sound moves closer to or farther from the listener. 6 11. Some other areas of technology that requires knowledge of physics are: In medicine: X-rays, lasers, scanners which are applications of physics are used in diagnosis and treatment of diseases. Communication: Satellite communication, internet, fibre optics are applications of internet which requires strong foundation in physics. In industrial applications: In the area of defense, physics has many applications e.g., war planes, laser-guided bombs which have high level accuracy. Entertainment Industry: In entertainment industry, knowledge of physics is used in mixing various colours to bring out the desirable stage effects 7  UNT II - Measurements and Quantities 2.0 Introduction Physics is often referred to as the science of measurement. The practice of physics relies on making measurements of quantities. These measurements are then used to arrive at relationships among physical entities, based on which certain physical laws are formulated. Making measurements is therefore central to the understanding of the behaviour of the physical world. Measurements provide a deeper perception of physical phenomena. Without measurements therefore, our knowledge of any subject is of a meager and insufficient kind. For instance, an automobile mechanic might measure the diameter of an engine cylinder; Refrigeration technicians might be concerned with volume, pressure and temperature measurements; electrician’s measure electric resistance and current, civil and mechanical engineers are concerned with dimensions and effects of forces whose magnitudes must be accurately determined. 2.1 Nature of measurement All measurements in physics involve; A knowledge and recognition of the physical quantity involved A procedure for comparing magnitudes. This comparison involves counting and the result is a numerical expression of magnitude A generally accepted unit. It follows therefore that any statement of magnitude in physics is incomplete unless it includes both a number and a unit. 2.2 Units and Systems of Units It is possible to qualitatively describe the nature and behaviour of many physical systems. However, to do so more precisely, we must go beyond qualitative to quantitative statements for which we need standards. If you want to measure something, you have to use a unit. For example, it is useless to say that a person’s mass is 10, 20, 30 or 50 if we do not know whether it is measured in kilograms or some other units such as stones or pounds. To measure a quantity, we always compare it with some reference standard. To say that a rope is 10 metres/meters long is to say that it is 10 times as long as an object whose length is defined as 1 metre/meter. Such a standard is called a unit of the quantity. 8 Therefore, unit of a physical quantity is defined as the established standard used for comparison of the given physical quantity. Systems of Units In earlier days, many systems of units were followed to measure physical quantities. The three systems commonly followed were: 1. The British System: The British system of foot−pound−second or fps system. As the name implies, the British system of units was developed in Britain, and the units were in use in many countries up to mid-1960s. 2. The Gaussian system of centimetre − gram − second or cgs system, 3. Metric system of metre − kilogram − second or the mks system. The metric system uses decimal conversions to obtain multiples of standard units. The multiplying factor is always a power of ten. The base units of the metric system are abbreviated ‘m’ for metre, ‘kg’ for kilogram, and ‘s’ for second. Multiples and sub-multiples are obtained by combining prefixes with these units. Thus, one centimetre is one hundredth of a metre. 2.3 Systeme Internationale d’Unites To bring uniformity, the General Conference on Weights and Measures in the year 1960, accepted the SI system of unit (The internationally accepted system of units for scientific work). This system is essentially a modification over mks system and is, therefore rationalized mksA (metre- kilogram- second- ampere) system. This rationalization was essential to obtain the units of all the physical quantities in physics. 2.3.1 Uniqueness of SI system The SI system is logically far superior to all other systems. The SI units have certain special features which make them more convenient in practice. Permanence and reproducibility are the two important characteristics of any unit standard. The SI standards do not vary with time as they are based on the properties of atoms. Further SI system of units are coherent system of units, in which the units of derived quantities are obtained as multiples or submultiples of certain basic units. 9 2.3.2 Rules Governing the use of S.I Units Units should be written in full using internationally agreed symbols The letter ‘s’ is never used to show a plural form. We write 3kg or 5cm and not 3kgs or 5cms because ‘s’ is the symbol for time in seconds. Capital letters are used in symbols only for certain units named after famous scientists such as Newton (N), or Joule (J) etc. When symbols are combined as quotient, e.g., meter per second, they are written as m/s or ms-1 2.4 Quantities and Units There are two types of physical quantities and units: Base (or fundamental) quantities and units and Derived quantities and units. 2.4.1 Fundamental Quantities and Units Basic or Fundamental Quantities: Fundamental quantities are quantities which cannot be expressed in terms of any other physical quantity. They are quantities on which the measurements of other quantities depend. Base Units: The units in which the fundamental quantities are measured are called fundamental units. These are units based on arbitrary value for some physical quantities from which others could then be derived. They are units from which all derived units can be obtained. In the SI system of units there are Seven fundamental quantities and Units and two supplementary quantities and Units as below. Fundamental Quantities Unit Symbol Length Meter m Time Second s Mass Kilogram kg Temperature Kelvin K Electric current Ampere A Luminous intensity Candela cd Amount of substance Mole mol 10  Supplementary quantities Unit Symbol Plane angle Radian rad Solid angle Steradian sr Length: Length is defined as the distance between two points. The SI unit of length is metre. One standard metre is equal to 1 650 763.73 wavelengths of the orange − red light emitted by the individual atoms of krypton − 86 in a krypton discharge lamp. Mass: Mass is the quantity of matter contained in a body. It is independent of temperature and pressure. It does not vary from place to place. The SI unit of mass is kilogram. The kilogram is equal to the mass of the international prototype of the kilogram (a plantinum − iridium alloy cylinder) kept at the International Bureau of Weights and Measures at Sevres, near Paris, France. An atomic standard of mass has not yet been adopted because it is not yet possible to measure masses on an atomic scale with as much precision as on a macroscopic scale. Time: Until 1960 the standard of time was based on the mean solar day, the time interval between successive passages of the sun at its highest point across the meridian. It is averaged over a year. In 1967, an atomic standard was adopted for second, the SI unit of time. One standard second is defined as the time taken for 9 192 631 770 periods of the radiation corresponding to unperturbed transition between hyperfine levels of the ground state of cesium − 133 atom. Atomic clocks are based on this. In atomic clocks, an error of one second occurs only in 5000 years. Ampere: The ampere is the constant current which, flowing through two straight parallel infinitely long conductors of negligible cross-section, and placed in vacuum 1 m apart, would produce between the conductors a force of 2 × 10 -7 Newton per unit length of the conductors. Kelvin: The Kelvin is the fraction of 1∕273.16 of the thermodynamic temperature of the triple point of water. 11 Candela: The candela is the luminous intensity in a given direction due to a source, which emits monochromatic radiation of frequency 540 × 1012 Hz and of which the radiant intensity in that direction is 1∕683 watt per steradian. Mole: The mole is the amount of substance which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. Radian: The radian is the angle subtended by an arc of length r at the center of circle of radius r. it is approximately equal to 57° Solid Angle – Steradian: The steradian is the solid angle at the centre of a sphere subtended by an area of r2on the surface of the sphere of radius r. The total solid angle about a point is therefore 4π steradians 2.4.2 Derived Quantities and Units Derived Quantities: Quantities that can be expressed in terms of fundamental quantities are called derived quantities. They are quantities that depend on the Base quantities for their measurements and units Derived Units: The units used to measure derived quantities are called derived units. In the S.I system derived units are expressed as combinations of two or more fundamental units. Examples of derived quantities and units Quantity Derivation Unit Symbol Area Length× Length meter× meter = (meter)2 m2 Volume Length× Length × Length meter× meter × meter = (meter)3 m3 mass ki log ram kg m3 ( meter ) 3 Density volume Speed/ Dis tan ce / length meter ms Velocity time sec ond 2 Acceleration velocity length meter m s = ( time ) ( sec ond ) 2 2 time 12  Force mass × acceleration ki log ram  meter kgm s 2 = N ( sec ond ) 2 2.5 Conversion of Units We convert a quantity q from one unit of measurement [Q] to another [Q]*, given a conversion factor c such that Q  = c Q   For a value (q) expressed in units [Q], the equivalent value (q)* in units [Q]* is c Q    q  = ( q ) Q  = c ( q ) Q  = ( q ) Q      Q  Hence ( q)* = c ( q) Example 1 The speed of light is 3.0 10ms −1 , and a light year is the distance that light travels in one year. Calculate the length of the light year in (i) kilometres, (ii) Earth radius (RE = 6,400 km), (iii) Astronomical units AU, the average distance between the Earth and the Sun, where 1 AU = 1.50 108 km. Solution  km  Speed of light = 3 108 m / s =  3 108 m   / s = 3 10 km / s 8  1000  This means that light travels 3 108 km every second. In one year (= 365  24  60  60) s, it will travel a distance D given by i. D = 3 108 km / s  365days  24hours  60 min 60s = 9.46 1012 km RE ii. 9.46 1012 km = 9.46  1012 km  = 1.478  1012 RE 6400km AU iii. 9.46  1012 km = 9.46  1012 km  = 6.3  104 AU 150,000,000 km 13 2.6 Orders of Magnitude We refer to an order of magnitude of a quantity as the power of ten of the number that describes that quantity. In order of magnitude calculations, the results are reliable to within a factor of 10. For example, the superscript 8 is more important than the number 3 in 3.0 108 Answers to some physical problems can be approximated through calculation of order of magnitude. These approximations are reasonable guesswork based on certain assumptions, which must be modified if higher precision is needed. 2.7 Some Basic Measurements and Instruments Much of the theories in physics have gained acceptance due to the fact that they have been verified experimentally through measurements. Basic experimental work in physics covers measurements of mass, length, and time. 2.7.1 Measurement of mass The main instrument for the measurement of mass is the balance (or chemical balance). Laboratory balances are calibrated to give mass readings in kg, g, mg (milligram), or g (microgram). Mechanical balances read down to grams or tenths of a gram, while mg and g measurements require sensitive electronic balances. 2.7.2 Measurement of Weight A spring balance measures the weight of an object by using the weight (force) to extend the spring. The extension of the spring is proportional to the weight of the object (Hooke’s law) Differences between Mass and Weight No Mass Weight 1 It is the quantity of matter a substance It is the force of gravity exerted on an object contains 2 It is measured by means of a beam balance, It is measured by means of a spring balance lever balance, top pan balance or electronic balance. 3 It has only magnitude and is therefore a It has both magnitude and direction and is scalar quantity therefore a vector quantity. 14  4 It is constant (it does not change) It varies over the surface of the earth and from one planet to another. 5 It is measured in kilograms (kg) It is measured in Newton (N) 2.7.3 Measurement of Length This category includes all distance measurements. Examples are thickness, diameter, length, and radius of curvature. Common types of instruments are the meter rule, the Vernier caliper, and the micrometer screw gauge. Metre rule -measured to be exactly 1 m from end to end; with graduations 1 mm apart. Vernier caliper - The vernier caliper enables fractions of a main scale subdivision (such as mm) to be determined. The vernier scale is such that 10 divisions on it are equivalent to 9 divisions on the main scale. Hence 1 vernier scale division is equivalent to 0.9 main scale divisions. The instrument is capable of reading to 0.01 cm The difference between a main scale division and the vernier scale division is known as the vernier constant. Thus, on a main scale calibrated in mm, Vernier constant = 1 mm – 0.9 mm = 0.1 mm = 0.01 cm, making the instrument capable of reading to 0.01 cm. To get the measurement, the vernier constant is multiplied by the reading on the vernier scale that coincides with the main scale. Micrometre screw gauge The micrometer can give readings to 0.01 mm (or 0.001 cm), or less in some cases. The scale has two parts – a main scale on the hub, graduated in mm, and a circular scale on the sleeve, with 50 marks equally spaced. Each complete revolution on the sleeve scale moves the jaws by 0.50 mm on the main scale. Hence the minimum reading on this instrument is 0.01 mm. The instrument is therefore capable of giving readings to 0.01 mm. The micrometer screw gauge is more suitable for measuring small thickness (e.g. diameter of a wire). The main parts of the screw gauge are shown below 15 2.7.4 Measurement of time Time is measured in seconds, for which the standard stopwatch is the main instrument used to measure. For example, some stopwatches have one revolution of the second-hand equivalent to 30 seconds instead of 60 seconds in others. 2.8 Errors of Observation Technology today depends on instruments used to measure and control events. However, no measurement can be perfectly accurate. The uncertainty in the measurement of a physical quantity is called error. It is the difference between the true value and the measured value of the physical quantity. Errors occur in every measurement and may be classified into many categories. 1. Constant Errors It is the same error repeated every time in a series of observations. Constant error is due to faulty calibration of the scale in the measuring instrument. In order to minimize constant error, measurements are made by different possible methods and the mean value so obtained is regarded as the true value. 2. Gross Errors Gross errors arise due to one or more of the following reasons. Improper setting of the instrument. Wrong recordings of the observation. Not taking into account sources of error and precautions. 16  Usage of wrong values in the calculation. Gross errors can be minimized only if the observer is very careful in his observations and sincere in his approach. 3. Systematic errors (or instrumental) Systematic errors (or instrumental) are errors which occur due to a certain pattern or system. These errors can be minimized by identifying the source of error. Instrumental errors due to calibration, zero errors, and sometimes if the pointer is not pivoted at the centre of an angular scale, personal errors due to individual traits and errors due to external sources are some of the systematic errors. 4. Random errors It is very common that repeated measurements of a quantity give values which are slightly different from each other. These errors have no set pattern and occur in a random manner. Hence, they are called random errors. They can cause an increase or decrease in a reading at random. They can be minimized by repeating the measurements many times and taking the arithmetic mean of all the values as the correct reading. A typical example of random error is parallax error, which often occurs if the measuring pointer is viewed at an angle. The most common way of expressing an error is percentage error. If the accuracy in measuring a x quantity x is x , then the percentage error in x is given by 100% x Errors are estimated in one of two ways. In the first method, a large number of independent measurements are made, and the mean or average of the readings is calculated. This mean is assumed to be the value closest to the true value, while the spread of the measurements (i.e. the difference between the lowest and highest recorded readings) is used as an estimate of the error. In the second method of error estimation, only one reading is taken; the error is taken as the smallest reading on the scale. If a meter rule is used, for instance, the error will be of the order of 1 mm, this being the smallest graduation on the meter rule. 17 2.8.1 GRAPHS In experimental work in science, we often come across quantities connected through equations. We call such quantities variables - the independent variable, chosen arbitrarily, and the dependent variable, whose value depends on that of the independent variable. Following result tabulation, we plot the independent variable along the x-axis (called the abscissa), and the dependent variable along the y-axis (or ordinate). The shape of the graph depends on the relationship that connects the two quantities being plotted. At this initial level in the study of science, most graphs are of the linear type, for which the slope and intercepts on the axes are the most important properties. Other important simple forms of relationships commonly used for experiments are as follows: The parabola, for which the equation y = kx 2 holds, where k is a constant, The rectangular hyperbola, yx = cons tan t. This graph shows inverse proportion. Independent quantities: If there is no relationship between two quantities, they are said to be independent of each other. On a graph of two independent quantities, one of them assumes the same value irrespective of the value of the other. 2.8.1.1 Slope and intercepts of linear graphs The biggest advantage of a graph is that it gives at a glance, a picture of how the measured quantities are related to each other. The simplest graph is the linear graph, in which the values of the variables when plotted lie on a straight line. Mathematically, a linear relationship between variables x and y is expressed by the general equation y = mx + c , where m is the slope of the graph, and c is the intercept on the ordinate axis. If a straight-line graph is obtained, the slope and intercept of the graph can be used to determine the numerical values of physical quantities 2.8.2 Possible Error for a Calculated Answer Suppose w is calculated from an equation of the form w = x + y , and then the errors in x and y are  x and  y. The error in the calculated w is  w given by ( w +  w) = ( x +  x ) + ( y +  y )  w +  w = x +  x + y +  y = ( x + y ) + ( x +  y ) w +  w = w + ( x +  y )  w =  x +  y..................................(1) 18  If w is calculated from an equation of the form w = x − y , then ( w +  w) = ( x +  x ) − ( y +  y )  w +  w = x +  x − y −  y = ( x − y ) + ( x −  y ) w +  w = w + ( x −  y )  w =  x −  y..................................(2) If w is calculated from an equation of the form w = xy , then ( w +  w) = ( x +  x )( y +  y )  ( w +  w ) = xy + x y + y x +  x y w +  w = w + x y + y x +  x y If the errors  x and  y are both small, then  x y is negligible  w +  w = w + x y + y x  w = + x y + y x Dividing through by xy , we get w w x y y x = + = xy w xy xy w x  y = + w x y Multiplying through by 100%, gives w x y 100% = 100% +  100% w x y This shows that the percentage error in w equals the sum of the percentage errors in x and y 2.9-Dimensional Analysis The word dimension denotes the physical nature of the quantity. The three fundamental properties are length, mass and time. They are specified by the symbols L, M, and T. The dimension of a physical quantity is the way we express the quantity in terms of the fundamental units of mass M, length L, and time T. The usual symbol for expressing dimensions is to enclose the quantity in square brackets, […]. The tables below give the dimensions and units of a few physical quantities. 19  Dimensions of Fundamental Quantities Dimensional Formulae of some Derived Quantities 2.9.1 Dimensional Quantities 1. Dimensional Constants: Constants which possess dimensions are called dimensional constants. Planck’s constant, universal gravitational constant, are dimensional constants. 20  2. Dimensional variables: Dimensional variables are those physical quantities which possess dimensions but do not have a fixed value. Example − velocity, force, etc. 3. Dimensionless quantities: There are certain quantities which do not possess dimensions. They are called dimensionless quantities. Examples are strain, angle, specific gravity, etc. They are dimensionless as they are the ratio of two quantities having the same dimensional formula. 2.9.2 Principle of Homogeneity of Dimensions An equation is dimensionally correct if the dimensions of the various terms on either side of the equation are the same. This is called the principle of homogeneity of dimensions. This principle is based on the fact that two quantities of the same dimension only can be added up, the resulting quantity also possessing the same dimension. The equation A + B = C is valid only if the dimensions of A, B and C are the same. 2.9.3 Uses of dimensional analysis The method of dimensional analysis is used to 1. Convert a physical quantity from one system of units to another. 2. Check the dimensional correctness of a given equation. 3. Establish a relationship between different physical quantities in an equation. 1. To Convert a Physical quantity from one system of units to another Given the value of G in cgs system is 6.67 × 10−8 dyne cm2 g−2. Calculate its value in SI units. In cgs system In SI system Gcgs = 6.67 10−8 G =? M1 = 1g M 2 = 1kg L1 = 1cm L2 = 1m T1 = 1s T2 = 1s The dimensional formula for gravitational constant is  M −1 L3T −2  In cgs system, dimensional formula for G is  M 1x L1yT1z  21 Step 1: Find the SI unit of universal gravitation constant The universal gravitational constant is the gravitational force acting between two bodies of unit mass, kept at a unit distance from each other. Gm1m2 Ie F = r2 Fr 2 Force  (length)r 2 G = = m1m2 mass  mass In terms of units, we have x = 6.67 10−11 Nm2 kg −2 Nm 2 Hence, the S.I unit of G is kg 2 Step 2: Convert the S.I unit of G into c.g.s Unit.  N  m2  kg ( ms )  m −2 2 m3 i.e 1  = 1  = 1  putting into the c.g.s system of unit gives  kg  2 kg 2 kg  s 2 (100cm ) 3  N  m2  106 cm3 3 cm 3 1  = 1 = = 10  kg  2 (1000 g )  s 2 103 g  s 2 gs 2 So the value of 1 S.I unit of G in c.g.s unit is 103 cm3 g −1s −2 Step 3: Calculate the value of 6.67 10−8 c.g.s unit of G to S.I unit  N  m2  i.e 103 cm3 g −1s −2 = 1 2   kg  6.67 10−8 dyne cm2 g−2 = 6.67 10−8 cm3 g −1s −2 = x ( In S.I Unit )  N  m2   ( 6.67 10−8 cm3 g −1s −2 ) 1 2  = x (10 cm g s ) 3 3 −1 −2  kg   6.67 10−8 cm3 g −1s −2   N  m 2   6.67 10 −8   N  m 2  −11 −2 x =  3 3 −1 −2  2 = 3  2  = 6.67  10 Nm kg 2  10 cm g s   kg   10   kg  x = 6.67 10−11 Nm2 kg −2 22  2. To check the Dimensional Correctness of a given Equation This method works on the principle that if an equation is valid, the dimensions of both sides of the equation must be identical. If the dimensions are not identical, then the equation is adjudged as incorrect even without further analysis. Let us take the equation of motion s = ut + (½ ) at 2 Applying dimensions on both sides,  L  =  LT −1  T  +  LT −2  T 2  (½ is a constant having no dimension)  L =  L +  L As the dimensions on both sides are the same, the equation is dimensionally correct. 3. Deriving Empirical Relationships - To Establish a Relationship between the Physical Quantities in an Equation By this, we can work out a general equation connecting physical variables. Let us find an expression for the time period T of a simple pendulum. The time period T may depend upon (i) mass m of the bob (ii) length l of the pendulum and (iii) acceleration due to gravity g at the place where the pendulum is suspended. i.e T  m xl y g z  T = km xl y g z...........................(1) Where k is a dimensionless constant of proportionality. Rewriting equation (1) with dimensions, T 1  =  M x   Ly   LT −2  =  M x L( y + z )T −2 z  z   Comparing the powers of M, L and T on both sides x = 0, (y + z) = 0 and −2z = 1 Solving for x, y and z, x = 0, y = ½ and z = –½ From equation (1), T = km0l ½ g −½ ½ l  l T =k  =k g g 23  l Experimentally the value of k is determined to be 2π. T = 2 g 2.9.4 Limitations of Dimensional Analysis The value of dimensionless constants cannot be determined by this method. This method cannot be applied to equations involving exponential and trigonometric functions. It cannot be applied to an equation involving more than three physical quantities. It can check only whether a physical relation is dimensionally correct or not. It cannot tell whether the relation is absolutely correct or not. For example, applying this technique s = ut + (¼ ) at 2 is dimensionally correct whereas the correct relation is s = ut + (½ ) at 2 2.10 Scalars and Vectors There are two classes of physical quantities – scalars and vectors. Scalars: A scalar quantity is identified completely by quoting only its magnitude. Examples of scalars include mass, volume, density, time, and temperature. Scalar quantities can be positive or negative. With only magnitude, it means that mathematical operations like addition, subtraction, multiplication, and division of scalars follow the normal rules of the algebra. Vectors: Vectors, on the other hand, are quantities that can be completely, described using both magnitude and direction. Examples of vectors are displacement, velocity, acceleration, torque, and momentum. In the combination of two or more vectors (by addition, subtraction, or multiplication), both the magnitudes and the directions of the vectors must be taken into account. Working with vectors is, therefore, not as straightforward as working with scalars For example, a car travels at 120kmh-1. This is the speed and is a scalar quantity. If the car travels from Freetown to Bo at 120kmh-1, then it is called its velocity and is a vector because it tells us the direction. Clearly, traveling from Freetown to Bo is not the same as traveling from Bo to Freetown, as direction is crucially important. 24  2.10.1 Representations of Vectors 1. Graphical representation Vectors are represented graphically by drawing a line with an arrowhead. In this representation, the length of the line is proportional to the magnitude of the vector (by choice of a convenient scale), and the arrow points in the direction of the vector. The following notions become part of this mode of representation. 2. Analytical representation In this representation, the vector is identified by a letter, which may be either in boldface (e.g., a, or → →  A), a letter with an underline (e.g., a, or A), or a letter with an arrow on the top  a or A . Whichever   method is used, the magnitude and direction of the vector are stated separately as part of the representation. 2.10.2 Some Vector Type Identical vectors: Identical vectors have the same magnitude and point in the same direction – i.e., they are in fact parallel vectors. In vector algebra, we can replace a vector by its identical vector. Collinear vectors: Collinear vectors lie along the same straight line, and may be either in the same, or opposite direction. These vectors may or may not have the same magnitude. Non-collinear vectors: Non-collinear vectors do not lie along the same straight line. In general, their directions are at an angle to each other. Negative of a vector: The negative of a given vector is a vector having the same magnitude as the given vector, but points in the opposite direction. Thus, if the vector has a magnitude of 5 units, and points east, the negative vector will have a magnitude of 5 units in a direction West. 2.10.3 Vector Addition and Subtraction Using any of the above representations and some ideas of vector algebra, we can combine vectors in addition, subtraction as well as multiplication. In the addition of two vectors using scale drawing, the two vectors are added by placing the ‘tail’ of one vector to the ‘head’ of the other. 25  1. Addition of two collinear vectors: The simplest case of vector addition is that for collinear vectors. The sum of two collinear vectors pointing in the same direction is a vector with a magnitude equal to the sum of the magnitudes of the two vectors in the same direction as the two vectors. If two collinear vectors are in opposite directions, their sum has a magnitude equal to the difference in magnitudes of the two vectors, pointing in a direction the same as that of the larger vector in magnitude A B C=A+B (a) + = (b) A D E=A+D + = a) in the same direction and (b) opposite direction. E.g., 1: The General Addition and (Subtraction) for two Non-collinear Vectors Addition uses the equivalent vectors, and subtraction results from adding one vector to the negative of the other. The resultant of two or more vectors is the single vector that produces the same effect as the given vectors put together. Subtraction is treated as the addition of one vector to the negative of the other vector (i.e., A-B = A+(-B). The resultant of two non-collinear vectors can be found by calculation using the parallelogram law of vector addition. 26  2. Parallelogram law of vector addition: This law states that If two vectors, say, P and Q, are represented in magnitude and direction by the adjacent sides of a parallelogram, their resultant could be represented in magnitude and direction by the diagonal of the parallelogram drawn from their point of intersection or action. Diagram If OA and OB represent the two given vectors, P and Q, in magnitude and direction, the diagonal OC represents their resultant. We use the cosine and sine rules in trigonometry to find the magnitude and direction of the resultant vector R. The magnitude R of the resultant is given by: R 2 = P2 + Q2 + 2PQcos It can be shown that the angle  which the resultant R makes with the vector P is tan = Q sin  P + Q cos  Q R = Alternatively, sin  sin (180 −  ) This last formula is commonly referred to as the sine rule. Example Two ropes are attached to a heavy box to pull it along a smooth floor. One rope applies a force of 2.0 kN in the direction due east; the other rope applies a force of 1.50 kN due south-east. Calculate the magnitude and direction of the force that would be applied by a single rope to have the same effect as the two forces put together. 27 Solution We can represent the force vectors as below (diagram not drawn to scale): The vectors labeled P and Q represent the 2.0 kN and the 1.5 kN forces, respectively. Their resultant is labeled R. The magnitude of R is R 2 = P2 + Q2 + 2PQ cos45o = 20002 + 15002 + 2  2000  1500  0.7071 R = 3239 N (or 3.239 kN). 2000 3239 Also = giving  = 25.9o sin  sin (180 − 45 ) 2.10.3.1 Components of a vector Or Vector Resolution From the parallelogram law, we see that two vectors can be represented by a single vector that will have the same effect as the two given vectors. Conversely, it is possible to split a single vector into two vectors which, added together (using the parallelogram law), results in the given vector, both in magnitude and direction. We take the given vector as the diagonal of some parallelogram. The pairs of vectors that give this diagonal are the components of the vector. Since any given line can be the diagonal of a nearly infinite number of parallelograms, it follows that a given vector can have many components. However, for practical purposes, the most useful pair of components are those that are at right angles to each other – e.g., the horizontal and vertical components, the radial and tangential components, and components along and perpendicular to an inclined plane. These are sometimes called the resolved components to distinguish them from all other components. In most cases, we drop the word ‘resolved’ and simply refer to them as the components, with the understanding that we are talking about components at right angles to one another. 28 The vector A is resolved into components, Ax , and Ay , along the horizontal and vertical, respectively, and their vector sum is A. We define the component of a vector as follows: The component of a vector in any direction is the effective value of the vector in that direction. From the diagram, Ax = A cos  and Ay = A sin  Applying Pythagoras theorem to the diagram gives A2 = Ax2 + Ay2 = ( A cos  ) + ( A sin  ) 2 2 ( A cos  ) + ( A sin  ) 2 2  A = Ax2 + Ay2 = =A Thus A = Ax + Ay (vector sum), For more than one vector, the x-component of the sum is found by adding the individual X-components; similarly, the y-component of the sum is found by adding the Y-components. If Rx and Ry represent the sums of the x- and y- components, then Rx =  Axi and Ry =  Ayi The magnitude of the resultant vector R can then be calculated using the relationship R = Rx2 + Ry2 The direction  of the resultant vector relative to the positive x-axis is given by Rx tan  = Ry Example A tension force of 2000 N is applied in a rope to pull a heavy box along a horizontal floor. If the rope makes an angle of 25 to the vertical, calculate the horizontal and vertical components of the applied tension. Which of the components is effective in pulling the box along the floor? 29  Solution The force F of 2000 N pulls the box along the floor, with the rope inclined at 65 to the horizontal (the same as 25 to the vertical). The horizontal component Fx is Fx = F cos 65 = 2000cos 65 = 845 N The vertical component Fy is Fy = F sin 65 = 2000sin 65 = 1812.6 N The component that is effective in pulling the box along the floor is Fx 2.10.4 Cartesian Unit vectors a a A unit vector is a vector with unit magnitude. We represent the unit vector a by a = = a a where a = a is the notation representing the magnitude of the vector a. a = aâ Thus, any vector can be represented by the Product of its magnitude and the unit vector in the direction of that vector. We can also represent components of vectors in terms of unit vectors. The letters i and j are commonly used for unit vectors in the horizontal and vertical directions, respectively, in the Cartesian coordinate system. In three dimensions, the letters i, j, and k are used. We, therefore, write a general vector r in the form r = xi + yj where x and y are the horizontal and vertical components, respectively. In three dimensions, we write r = xi + yj + zk 30 Operations on Cartesian Vectors i) Addition and Subtraction To add or subtract vectors given in the Cartesian ( i, j, k ) form, the coefficients of i, j and k are collected separately. If r1 = x1i + y1 j + z1k and r2 = x2i + y2 j + z2 k , then r1 − r2 = ( x1i + y1 j + z1k ) − ( x2i + y2 j + z2k ) = ( x1 − x2 ) i + ( y1 − y2 ) j + ( z1 − z2 ) k ii) Modulus Suppose a vector r in the cartesian system is given by r = xi + yj + zk , then the modulus of r , using Pythagoras theorem twice will be r = x2 + y 2 + z 2 iii) Equal Vectors If two vectors r1 = x1i + y1 j + z1k and r2 = x2i + y2 j + z2 k are equal, then r1 = r2  x1 = x2 and y1 = y2 , and z1 = z2 Example Vectors P and Q are represented as P = 6i + 5j, and Q = 3i – 2j. Determine i. the magnitude and direction of each vector, ii. the magnitude and direction of the resultant of the two vectors, iii. the magnitude and direction of the difference between the two vectors Solution i. The magnitude of P is given by P = x 2 + y 2 = 62 + 52 = 61 = 7.8units The angle P makes with the horizontal is θ, y  y 5 Where tan  =   = tan −1   = tan −1   = 39.8 x x 6 The magnitude of Q is given by Q = x 2 + y 2 = 32 + ( −2 ) = 13 = 3.6units 2 31 The angle Q makes with the horizontal is θ, y  y  −2  Where tan  =   = tan −1   = tan −1   = −33.7 = 326.3 x x  3  ii. The resultant R of the two vectors is R = P + Q = ( 6i + 5 j ) + ( 3i − 2 j ) = ( 6 + 3) i + (5 − 2 ) j = 9i + 3 j This resultant has magnitude R = 92 + 33 = 90 = 9.5units at an angle y  y 3 tan  =   = tan −1   = tan −1   = 18.4 to the horizontal x x 9 iii. The difference S of the two vectors is S = P − Q = ( 6i + 5 j ) − ( 3i − 2 j ) = ( 6 − 3) i + ( 5 + 2 ) j = 3i + 7 j This has magnitude S = 32 + 73 = 58 = 7.6units at an angle y  y 7 tan  =   = tan −1   = tan −1   = 66.8 to the horizontal x x 3 2.10.4.1 Multiplication of vectors There are three kinds of vector multiplication, as outlined below. 1. Multiplication by a scalar: the vector is multiplied by a scalar quantity. An example is the quantity momentum, defined as the product of mass (a scalar) and velocity (a vector). The result of multiplying a vector by a scalar is to produce another vector, with a magnitude equal to a multiple (or sub-multiple) of the given vector, pointing in the same direction as the vector. Multiplying a vector by –1 gives the negative of the vector, which, as we saw earlier results in a vector of the same magnitude but opposite in direction 2. Scalar (or dot) product: The scalar (or dot) product of two vectors, A and B (written as A.B), is defined by A.B = ABcos where  is the angle between the directions of the two vectors. The result of the dot product of two vectors is a scalar quantity – hence the description scalar product. An example of the scalar product of two vectors in physics is work, defined as the (dot) product of Force (a vector) and displacement (also a vector). 32  Properties of the Scalar Product i) Parallel Vectors If the vectors are parallel, then for like parallel vectors (parallel vectors in the same direction)  = 0  A.B = ABcos = ABcos0=AB For unlike parallel vectors (parallel vectors in opposite directions)  = 180  A.B = ABcos = ABcos180=-AB Therefore, for the Cartesian unit vectors i, j and k i.i = j. j = k.k = 1 ii) Perpendicular Vectors If the vectors are perpendicular (at right angles)  = 90  A.B = ABcos = ABcos90=0 Hence, for the Cartesian unit vectors i, j and k i. j = j.k = k.i. = 0 iii) Commutative The scalar product is commutative. This means that A.B = B.A From definitions A.B = A.B cos  and B.A = BA cos . Now AB cos  = BA cos  , therefore, A.B = B.A iv) Distributive The scalar product is distributive over addition. This means that A. ( B + C) = A.B + A.C Calculating Scalar (or dot) product in Cartesian Form If r1 = x1i + y1 j + z1k and r2 = x2i + y2 j + z2 k , then we can find r1.r2 by using the properties given above. r1.r2 = ( x1i + y1 j + z1k ). ( x2i + y2 j + z2k ) 33 = ( x1 x2i.i + y1 y2 j. j + z1 z2k.k ) + ( x1 y2i. j + y1z2 j.k + z1x2k.i ) + ( y1x2 j.i + z1 y2k. j + x1z2i.k ) But i.i = j. j = k.k = 1 and i. j = j.k = k.i. = 0  r1.r2 = x1 x2 + y1 y2 + z1 z2 Example Calculate the scalar products of the following pairs of vectors: i. vector A (20 units, direction N), and vector B (5 units, direction W) ii. vector P (3 units, direction E), and vector Q (5 units, direction N) iii. vector X (5 units, direction S), and vector Y (2 units, direction SW) iv. vector U (20 units, direction N), and vector V (3 units, direction S) Solution i. Angle between A and B =  = 90o A.B = ABcos = 20  5 cos 90 = 0 ii. Angle between P and Q =  = 90o. P.Q = PQcos = 3  5 cos 90 = 0 iii. Angle between X and Y =  = 45o. X.Y = XYcos = 5  2 cos 45 = 7.07units iv. Angle between U and V =  = 180o. U.V = UVcos = 20  3 cos180 = −60units Classwork Find the scalar product of A = 2i − 3 j + 5k and B = i − 3 j + k. Hence find the cosine of the angle between A and B 3. Vector (or cross) product: The vector (or cross) product of two vectors is the multiplication of two vectors in such a manner that the product is a vector quantity. The  vector product of two vectors A and B is defined as AxB = −BxA = ABsin u where  is the angle between the directions of the two vectors, and û is a unit vector perpendicular to both vectors A and B, in a right-hand rotation from A to B. If the vectors are parallel, i.e.,  = 0 or  = 180 then, AxB = −BxA = ABsin0=ABsin180=0 34 Therefore, for the Cartesian unit vectors i, j and k ixi = j xj = k xk = 0 If the vectors are perpendicular, i.e.  = 90 , then AxB = −BxA = ABsin90=AB Hence, for the Cartesian unit vectors i, j and k i xj = k and j xi = −k j xk = i and k xj = −i k xi = j and i xk = − j An example of the vector product is moment, a vector quantity obtained by multiplying a force vector and a displacement vector. The direction of the product is perpendicular to the two given vectors. 4. The triple product We can multiply three vectors in two ways The first of these is called the scalar triple product. Also called the mixed product, the scalar product of three vectors A, B, and C is the product ( AxB ).C, or A.(BxC). This product gives a scalar quantity, the reason why it is called the scalar triple product. In this computation, the vector product is always done first, and the product of this vector multiplication is “dotted” with the other vector. With this, the brackets are often omitted. The other product is the vector triple product, written as ( AxB ) xC, or Ax ( BxC). This product gives a vector quantity, and the order of the multiplication is important. The brackets are hence necessary to indicate this order. 35  UNT III – PARTICLE KINEMATICS 3.0 Introduction The world around us is made up of things that are constantly moving. Some of these are living (or animate) bodies (e.g., insects, animals), whilst others are inanimate. In the study of motion in physics, we are concerned with only the physical motion. Thus, the study of motion concerns mainly the motion of inanimate objects. Examples of such objects are the motor car, the train, the aircraft, the bicycle, the ceiling fan, and the to-and-fro movement of a mass hanging in the vertical plane at the end of a string – the simple pendulum. 3.1 Rest and Motion Rest: When a body does not change its position with respect to time, then it is said to be at rest. Motion: Motion is the change of position of an object with respect to time. This means that the body is in one position at a given time, and at another after an interval of time. A moving body has a particular frame of reference in which its motion can be measured. The frame of reference shows the point from which the motion started, the region in which the motion took place, the rate of change of position and the direction of motion. 3.1.1 Branches of Motion The study of motion is divided into two branches Dynamics: which is the study of motion and the forces responsible for the motion, and Kinematics: which is the study of motion alone, making no reference to the forces producing the motion. We assume that the moving object can be represented by a geometrical point, or particle – hence the description Particle Kinematics. Particle: A particle is ideally just a piece or a quantity of matter, having practically no linear dimensions but only a position. 3.1.2 Categories of Motion To study the motion of the object, one has to study the change in position (x, y, z coordinates) of the object with respect to the surroundings. It may be noted that the position of the object changes even due to the change in one, two or all the three coordinates of the position of the objects with respect to time. Thus, motion can be classified into three categories: 36  i) Motion in one dimension: Motion of an object is said to be one dimensional, if only one of the three coordinates specifying the position of the object changes with respect to time. Example: An ant moving in a straight line, running athlete, etc. ii) Motion in two dimensions: In this type, the motion is represented by any two of the three coordinates. Example: a body moving in a plane. iii) Motion in three dimensions: Motion of a body is said to be three dimensional, if all the three coordinates of the position of the body change with respect to time. Examples: motion of a flying bird, motion of a kite in the sky, motion of a molecule, etc. 3.1.3 Types of Motion Motion comes in many forms, some of which include: i) Translation (or Displacement) motion: This is the motion in which the body moves from one geometric point in space to another, over an interval of time. Examples: A car moving between two points on the road performs this kind of motion. Motion of woman pushing her baby’s pram/stroller from one point to another along a straight line. ii) Oscillatory (or vibratory) Motion: Oscillatory (or vibratory) motion is the motion of a body about a fixed point. I.e., it is the motion that repeats itself in equal intervals of time along the same path. Example: motion of a plucked violin or guitar the to and fro movement of the wings of a bird in flight or the swinging of the pendulum bob. iii) Random Motion: This is a type of motion with no specific direction. Example is the zigzag motion often exhibited by gas molecules or that of pollen particles in water. The direction describing this type of motion is observed to change continuously leaving no definite path. iv) Circular Motion: This is the motion of a body in a circle at a constant speed. Example: the motion of the earth round the sun. 37 3.2 Motion in one Dimension (Rectilinear Motion) The motion along a straight line is known as rectilinear motion. The important parameters required to study the motion along a straight line are position, distance/displacement, speed/velocity, and acceleration. 3.2.1 Rectilinear Motion parameters 1. Position: The position of an object is the location of that object with respect to a particular reference frame or coordinate system. P Q O X1 X2 Displacement = ( X2 − X1 ) , ( X2 X1 ) Generally, the position of a body is conveniently specified by the projection on two or three axis of a rectangular coordinate system. These two axes describe a plane in which the object might be located. (i.e., the horizontal axis commonly X – axis and the vertical axis, commonly Y – axis). To locate the position of a body in a plane, move X – steps parallel to the horizontal axis and Y – steps parallel to the vertical axis corresponding to where the body is placed, the point of intersection of the two movements gives the position of the body. Y P ( X, Y ) O X ( X, Y ) , are the coordinates locating the position of the object P. The first coordinates is always the horizontal followed by the vertical coordinate. In space, three axes may also be used ( X, Y, Z ) to locate the position of an object. In this case, move X – steps parallel to X – axis, Y - steps parallel to the Y – axis and Z – steps parallel to the Z – axis all corresponding to where the body is found. The point of intersection of the three movements gives the position of the object described by the coordinates ( X, Y, Z ). 38  Z Z1 P ( X1 , Y1 , Z1 ) O X Y1 X1 2. Distance: Is defined as the magnitude of the length between two points or magnitude of length moved by an object. It is a scalar quantity and its S.I unit is the meter (m). 3. Displacement: Is defined as the distance moved is a specified direction. It is a vector quantity and its S.I unit is the meter (m). Example: suppose you walk 3 m to the east and then 4 m towards the north. The distance you have travelled is 7m but your displacement, the distance between where you started and where you ended up is only 5m. Because displacement is a vector, we also need to say that the 5m had been moved in a certain direction north of east. 4. Speed: Is the rate of change of distance with no regard to direction. Thus, if a car covers a Dis tan ce ( X ) distance, X in a time interval, t, then, speed =. It simply tells you how fast a body Time ( t ) is moving. The SI unit of speed is meter per second (ms-1) and is a scalar quantity. Uniform Speed: A body is said to move with uniform speed if its rate of change of distance is constant. 39  Average Speed: In most cases, bodies do not maintain steady speed, possibly due to traffic or road condition. Under such situation, the total distance covered over the time period is referred to as the average speed. i.e. Average speed = Total distance Total time taken Similarly, if the initial and final speeds of a body in motion are ‘u’ and ‘v’ respectively, its average speed is given by: u+v Average speed = 2 5. Velocity: Is the rate of change of displacement. It is a vector quantity and its S.I unit is meter per second (ms-1). Uniform Velocity: A body is said to move with uniform velocity if its rate of change of displacement is constant. Mathematically, Velocity = change in displacement ( X ) Time (t) 6. Acceleration: Is the rate of change of velocity or the measure of the rate at which velocity changes. Acceleration = change in velocity Time taken The S.I unit of acceleration is meter per second square (ms-2). It is a vector quantity. If the velocity of an object decreases with time, it is said to be decelerating or retarding. Hence, deceleration can be defined as negative acceleration. Thus, if the initial and final velocity of a body are u and v respectively, then  v −u Acceleration, a =    t  Where t = time taken, If v is greater than u the body is said to be accelerating. If v is less than u, the body is said to be decelerating. Uniform Acceleration: A body is said to have a uniformly acceleration if its rate of change of velocity is constant. 40 Example 1: A motorcycle stating from rest moves with uniform acceleration until it attains a speed of 108 km/h after 15 seconds. Find its acceleration. Solution Time taken t = 15 s Initial velocity, u = 0 ms-1 Final velocity v = 108 km/h. V in m/s = 108 x 1000 = 30ms-1 60 x 60  v − u   30 − 0  −2 Using the relation a=  =  = 2ms  t   15  Example 2: A cat runs 20 m in 35 seconds. Find (i) Its speed (ii) how long it takes to run 75 m? Solution Distance x = 20 m Time taken, t = 35 s Speed = ? i) Using the equation below, Dis tan ce ( X ) 20 Speed = = = 0.57ms −1 Time ( t ) 35 ii) Distance x = 75m Speed = 0.57ms-1 Time taken = ? Dis tan ce ( X ) 75 time = = = 132s speed 0.57 3.2.2 Graphical Presentations of Motion a) Distance/Displacement – time graph: This is the graph of distance/displacement plotted against time. It is used to determine the speed (distance – time graph) or velocity (displacement – time graph) of a body in motion. In situations, where the speed/velocity of the motion is uniform, a straight-line graph is obtained and its gradient gives the speed (distance – time graph) or velocity (displacement – time graph) of the motion. 41  Distance/displacement x (m) X X1 X2 O t (s ) t1 t2 Gradient or slope = Speed/velocity = ( x 2 − x1 ) ( t2 − t1 ) In situations where the body is stationary, the speed or velocity is zero since there is no change in distance/displacement. b) Velocity – time graph: This is the graph of velocity plotted against time. It is used to determine the acceleration of an object in motion. In situations where the motion is uniform, a straight-line graph is obtained. The gradient/slope of the graph gives the acceleration. Velocity (ms-1) v u O t (s ) t1 t2 Gradient/slope = acceleration, a (Acceleration), a = (v − u ) = (v − u ). In the case where the velocity is not changing, the ( t2 − t1 ) t acceleration is zero. In the other case where the final velocity is less than the initial velocity (v