General Physics 1 (Mechanics) Part 1 PDF

General Physics 1 (Mechanics) Prof. M.E. Emetere Part 1 Table of Content Space and time Units and dimension vectors and scalars differentiation of vectors Kinematics Space and time  Introduction Native knowledge have always taken the concept of ‘space and time’ for granted...

General Physics 1 (Mechanics) Prof. M.E. Emetere Part 1 Table of Content Space and time Units and dimension vectors and scalars differentiation of vectors Kinematics Space and time  Introduction Native knowledge have always taken the concept of ‘space and time’ for granted. Space is regarded as just emptiness while time is a term that simply ticks on incessantly. For physicists, ‘space and time’ form a system of staggering complexity that defies ardent efforts to understand some concepts in Physics. In other words ‘space and time’ are vital concept that makes it complex to unify Physics theories. The complexity of ‘space and time’ in physics is like asking a stranger to determine the household that may likely be having the most delicious meal without asking him/her to visit each household. In other words, ‘space and time’ means different thing to most branches of Physics. In the slides below, the different types of ‘space and time’ will be discussed.  Introduction ………… Contd In this lecture, the learning objectives are to: Identify the types of ‘space and time’ available in Science; Elaborate on the classical of ‘space and time’; Elaborate on the relativity view of ‘space and time’; distinguish between the classical, relativity and quantum view of ‘space and time’; and solve some problems related to classical view of ‘space and time’. Types of Time Absolute Time Relative Time Proper Time Coordinate Time Psychological Time Biological Time Universal Time Atomic Time Sidereal Time Types of Time Solar Time Planck Time Historical Time Cosmological Time Clock Time Dynamical Time Event Time Absolute Time In classical physics, especially as proposed by Isaac Newton, absolute time is the idea that time flows uniformly, regardless of any external factors. It is considered the same for all observers, independent of their motion or position. Example: The ticking of a clock is assumed to be the same everywhere in the universe, irrespective of motion or gravity. Relative Time In Albert Einstein’s theory of relativity, time is not absolute but relative. It depends on the observer's speed and position in a gravitational field. Time can slow down or speed up depending on how fast an object is moving relative to the speed of light or its proximity to a massive object. Example: Time passes more slowly for astronauts traveling at high speeds compared to people on Earth (time dilation), and clocks near a strong gravitational field tick more slowly (gravitational time dilation). Proper Time In the context of relativity, proper time is the time interval measured by an observer moving along a specific trajectory. It's the time experienced by an object or person in their own reference frame. Example: An astronaut traveling in space would experience proper time differently from someone remaining stationary on Earth, but to the astronaut, time would seem to pass normally. Coordinate Time Coordinate time is the time measured by an observer using a fixed reference frame. It is used in relativity to describe events as they occur across different spatial locations. Example: Time recorded by distant observers in the International Celestial Reference System (ICRS) as opposed to the local clock on Earth. Psychological Time This refers to the subjective experience of time as perceived by individuals. It’s how time feels to people, which can vary depending on circumstances, emotions, and mental states. Example: Time may feel like it’s passing more quickly when you’re engaged in an exciting activity and more slowly when you’re bored or anxious. Biological Time Biological time refers to the natural cycles that govern living organisms, often referred to as biological rhythms or circadian rhythms. It’s the internal clock that regulates physiological processes in animals and plants. Example: The human sleep-wake cycle, which typically follows a 24- hour pattern, is an example of biological time. Universal Time Universal Time is a time standard based on the Earth's rotation. It is essentially the same as Greenwich Mean Time (GMT) and is used in astronomy, navigation, and broadcasting as a reference time. Example: UT is used in global communications and satellite positioning systems. Atomic Time Atomic time is measured using the vibrations of atoms, specifically cesium atoms, which oscillate at a very precise frequency. The International System of Units (SI) defines a second based on atomic time. Example: The International Atomic Time (TAI) is a time standard that ensures extremely accurate timekeeping, used in global positioning systems (GPS). Sidereal Time Sidereal time is a timekeeping system used by astronomers to track the rotation of the Earth relative to distant stars, rather than the Sun. A sidereal day is slightly shorter than a solar day. Example: Sidereal time is used in celestial navigation and for aligning telescopes with stars. Solar Time Solar time is based on the position of the Sun in the sky. It measures time according to the Earth’s rotation relative to the Sun. Apparent solar time measures the exact position of the Sun, while mean solar time averages out the variations in the Sun’s position due to Earth’s elliptical orbit. Example: Sundials measure apparent solar time. Modern clocks are based on mean solar time. Planck Time Planck time is the smallest meaningful unit of time in quantum mechanics. It represents the time it takes for light to travel one Planck length (the smallest distance measurable in the universe) in a vacuum. It is on the order of 10−43 seconds. Example: Planck time is used in theoretical physics, especially in discussions about the Big Bang and quantum gravity. Historical Time This refers to how time is understood in terms of human history, divided into eras, epochs, centuries, etc. Example: The Geological Time Scale or the Anthropocene Epoch represent different periods in Earth’s history. Cosmological Time Cosmological time refers to time on the largest scales of the universe. It is used to measure the age and evolution of the universe, from the Big Bang to the present. Example: The universe is approximately 13.8 billion years old, according to cosmological time. Clock Time Clock time is the everyday, practical time measured by clocks and used in daily life, business, and industry. It is broken down into hours, minutes, and seconds. Example: Standard 24-hour clock systems used worldwide to coordinate events. Dynamical Time Dynamical time is used in the precise measurement of planetary motion. Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT) are used for solar system dynamics and celestial mechanics. Example: Astronomers use dynamical time to predict the positions of celestial bodies like planets. Event Time Event time is the duration taken for an event to occur. It varies from one geographical location to another. Like the clock time, it is broken down into hours, minutes, and seconds. Example: Unlike the standard 24-hour clock systems used worldwide to coordinate events, it is dependent on the adopted time system that is used. Types of Space Physical Space (Real Space) Euclidean Space Non-Euclidean Space Curved Space (Riemannian Space) Minkowski Space Metric Space Topological Space Hilbert Space Phase Space Personal Construct Space (Psychology) Types of Space Vector Space Space-Time Personal Space (Psychological Space) Cognitive Space Outer Space Social Space Imaginary Space Digital Space (Virtual Space) Configurational Space Hyperspace Physical Space (Real Space) The three-dimensional continuum in which physical objects exist and events occur. It is what we experience in our everyday life, and it consists of height, width, and depth. Example: The space around us where we move, such as a room, the atmosphere, or outer space. Euclidean Space The space described by Euclidean geometry, where the familiar rules of geometry apply (e.g., parallel lines never meet, the sum of angles in a triangle is 180°). It is the standard space used in classical geometry. Example: A flat, two-dimensional surface like a sheet of paper or a three-dimensional box where traditional geometric principles hold. Non-Euclidean Space A type of space where Euclidean geometry does not apply. It includes spaces with curvature, such as spherical geometry (on a curved surface like Earth) or hyperbolic geometry (used in models of negative curvature). Example: The surface of a globe is an example of spherical geometry, where the sum of angles in a triangle is greater than 180°. Curved Space (Riemannian Space) A type of non-Euclidean space used in Einstein’s theory of general relativity, where space is curved by the presence of mass and energy. Space is not flat but can be warped by gravitational fields. Example: The bending of light around a star due to gravitational lensing, as predicted by general relativity, occurs in curved space. Minkowski Space The four-dimensional spacetime framework used in special relativity, where time is treated as a dimension alongside the three spatial dimensions. Events in the universe are located in this four- dimensional continuum. Example: The path of a moving object in space and time can be plotted in Minkowski space, where the speed of light is constant, and time and space are interconnected. Hilbert Space A type of non-Euclidean space used in Einstein’s theory of general relativity, where space is curved by the presence of mass and energy. Space is not flat but can be warped by gravitational fields. Example: The bending of light around a star due to gravitational lensing, as predicted by general relativity, occurs in curved space. Vector Space A mathematical structure made up of vectors, where vectors can be added together and scaled by numbers (scalars). Vector spaces are fundamental in linear algebra. Example: The three-dimensional space where vectors like displacement, velocity, and force are represented is a vector space. Outer Space The vast expanse beyond Earth’s atmosphere, commonly referred to as space or the cosmos. It is the region where stars, planets, and galaxies exist. Example: The region where astronauts travel during space missions, far beyond the gravitational influence of Earth.  Classical View of Space and Time In classical mechanics, developed by Isaac Newton, space and time were considered absolute and independent entities: Absolute Space: Space is a fixed, unchanging stage in which objects move. It exists independently of any objects within it and provides a reference frame for motion. Absolute Time: Time flows uniformly and is independent of any events or observers. All observers would measure the same intervals of time regardless of their motion. In this view: Objects move through space, and time progresses the same for all observers. Space and time are distinct and separate entities.  Relativity View of Space and Time Albert Einstein’s theory of relativity fundamentally altered our understanding of space and time. He showed that they are not absolute or independent but are interwoven into a single entity known as spacetime. Spacetime: In special relativity, space and time are unified into a four- dimensional continuum called spacetime. Events are described by four coordinates: three for space (x, y, z) and one for time (t). Observers moving relative to each other will have different perceptions of time and space, but they all agree on spacetime intervals. Spacetime Curvature: In general relativity, gravity is not a force in the traditional sense but a result of the curvature of spacetime caused by mass and energy. Massive objects, such as planets or stars, bend the fabric of spacetime around them. Objects moving through spacetime follow these curved paths, which we perceive as gravitational attraction.  Relativity View of Space and Time Einstein’s Special Relativity introduced the idea that space and time are relative to the observer's motion, with the following key ideas: Relativity of Simultaneity: Events that appear simultaneous to one observer may not be simultaneous to another observer in motion relative to the first. In other words, time is not the same for all observers. Time Dilation: Moving clocks run slower compared to those at rest, meaning time "dilates" or stretches as the speed of the observer increases. Length Contraction: Objects in motion relative to an observer appear shorter in the direction of motion. Speed of Light: The speed of light is constant and is the same for all observers, regardless of their motion.  Classical Relativity Quantum Aspect Physics (Special/General) Mechanics Part of spacetime; Background for Fixed, absolute not fixed, can be quantum fields; Space framework. curved by uncertain at small mass/energy. scales. Relative, changes Uncertain at small Absolute, flows with velocity and scales; time and Time uniformly for gravity (time space may be linked all. dilation). in novel ways.  We live in a three-dimensional Ways of writing vector notation world, and for the purpose of this course we can consider space and F = ma time to be a fixed framework against ! ! which we can make measurements F = ma of moving bodies. F = ma Each point P in space can be labeled with a distance and direction from z axis some arbitrarily chosen origin O. Expressed in terms of unit vectors xˆ , yˆ , zˆ r r = x xˆ + y yˆ + z zˆ z y axis It is equivalent to write the vector as x y an ordered triplet of values r = ( x, y , z ) We can also write components of x axis vectors using subscripts v = (v x , v y , v z ) a = (a x , a y , a z ) Classwork (space identification) The following diagram shows a variety of displacement vectors in space. Express each vector in component (i,j) notation and (x,y) coordinate. Classwork: Solution A as shown by the arrow, is located on coordinates (6,1) and (13,4). The subtraction gives the (i,j) notation i.e. "⃑ = 7&̂ + 3)̂ B as shown by the arrow, is located on coordinates (3,7) and (3,1). The subtraction gives the (i,j) notation i.e. * = −6)̂ C as shown by the arrow, is located on coordinates (-14,4) and (-6,6). The subtraction gives the (i,j) notation i.e. "⃑ = 8&̂ + 2)̂ Exercise 1 The following diagram shows a variety of displacement vectors in space. Find the vector in component (i,j) notation and (x,y) coordinate of points D-I.  Exercise 2 The following diagram shows a variety of displacement vectors in space. In space and time, find the (x,y) coordinate and vector in component (i,j) notation if the time taken to cover each displacement are A=2s, B=3s, C=5s, D=3s, E=4s, F=2s, G=5s, H=4s, and I=7s Units and dimension Introduction Before the 17th century, the systems of weights and measures around the world were very confusing. Units of length, land area, and weight varied from region to region SI unit system : Internationally accepted system of units is Système Internationale d’ Unites (French for International system of Units) or SI. It was developed in 1960 by an international committee and recommended by General Conference on Weights and Measures in 1971. SI lists 7 base units referred to as fundamental unit as presented the displayed in the next slide. Along with it, there are two units - radian or rad (unit for plane angle) and steradian or sr (unit for solid angle). They both are dimensionless. Fundamental Units Quantity SI Unit Length meter Mass kilogram Time second Temperature Kelvin Electric Current Ampere Luminous Intensity Candela Amount of Substance mole  Types of Unit System The base units for length, mass and time in these systems are as follows : In Centimetre–gram–second (CGS) system of units are centimeter, gram and second respectively. In Foot-pound-second (FPS) system of units are foot, pound and second respectively. In Meter-kilogram-second (MKS) system of units are meter, kilogram and second respectively. MKS unit of force is Newton ads CGS unit is dyne. MKS unit of work/energy is Joule and CGS unit is erg. Some important physical quantities unit conversion: 1 Foot = 30.48 cm 1 hour = 3600 seconds 1 pound= 454 gram 1 Newton = 105 dyne 1 joule =107 erg 9.8 m/s2= 980 cm/s2  Advantages of SI Unit System SI is a coherent system of units i.e system based on a certain set of fundamental units, from which all derived units are obtained by multiplication or division without introducing numerical factors. SI is a rational system of units as it assigns only one unit to be a particular physical quantity. For example, joule is the unit for all types of energy. This is not so in other systems of units. SI is an absolute system of units. There are no gravitational units on the system. This use factor ‘g’ is thus eliminated. SI is a metric system i.e the multiples and sub multiples of units are expressed as power of 10.  Definition of Fundamental Units 1 meter : The meter is the length of the path travelled by light in vacuum during a time interval of 1/299792 458 of a second. 1 kilogram: The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. The international prototype of the kilogram, an artifact made of platinum-iridium, is kept at the BIPM under the conditions specified by the 1st CGPM in 1889 (CR, 34-38) when it sanctioned the prototype and declared: 1 second : The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. 1 Ampere: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 Newton per meter of length.  Definition of Fundamental Units…..Contd 1 Kelvin: The Kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. 1 Candela: The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. 1 mole: The mole is the amount of substance as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol.” 1 Radian & 1 steradian :  Physical Quantities and their types Physical Quantities: Those quantities which can be measured and, are necessary to describe any physical phenomenon. There are two types of physical quantities: Fundamental physical quantities: Those quantities which are independent and do not depend on other quantities are called as fundamental quantities. Mass ,Length and Time are called Fundamental physical quantities Derived Physical quantities: Those physical quantities which can be expressed in terms of fundamental physical quantities are area, volume, force, work, pressure etc  Length as a Fundamental Unit Distance Length (m) Radius of visible universe 1 x 1026 To Andromeda Galaxy 2 x 1022 To nearest star 4 x 1016 Earth to Sun 1.5 x 1011 Radius of Earth 6.4 x 106 Sears Tower 4.5 x 102 Football field 1.0 x 102 Tall person 2 x 100 Thickness of paper 1 x 10-4 Wavelength of blue light 4 x 10-7 Diameter of hydrogen atom 1 x 10-10 Diameter of proton 1 x 10-15 Time as a Fundamental Unit Interval Time (s) Age of universe 5 x 1017 Age of Grand Canyon 3 x 1014 32 years 1 x 109 One year 3.2 x 107 One hour 3.6 x 103 Light travel from Earth to Moon 1.3 x 100 One cycle of guitar A string 2 x 10-3 One cycle of FM radio wave 6 x 10-8 Life time of neutral pi meson 1 x 10-16  Mass as a Fundamental Unit Object Mass (kg) Milky Way Galaxy 4 x 1041 Sun 2 x 1030 Earth 6 x 1024 Boeing 747 4 x 105 Car 1 x 103 Student 7 x 101 Dust particle 1 x 10-9 Top quark 3 x 10-25 Proton 2 x 10-27 Electron 9 x 10-31 Neutrino 1 x 10-38 Prefixes of Units Unit Conversion Converting between different systems of units Useful Conversion factors: 15.0 in = ? cm 1 inch = 2.54 cm 1m = 3.28 ft æ 2.54 cm ö 15.0 in ç ÷ = 38.1cm 1 mile = 5280 ft è 1in ø 1 mile = 1.61 km Example: convert miles per hour to meters per second: mi mi ft 1 m 1 hr m 1 = 1 ´ 5280 ´ ´ = 0.447 hr hr mi 3.28 ft 3600 s s Unit Conversion........ Contd Dimensions: An Introduction Dimension: The dimensions of a physical quantity are the powers to which the fundamental quantities are raised to represent that physical quantity. They are represented by square brackets around the quantity. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity. An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. Dimensionless Quantities: The arguments of special functions, such as the trigonometric, logarithmic and exponential functions are dimensionless. A pure number is always dimensionless. ratio of similar physical quantities, such as angle as the ratio (length/length), refractive index as the ratio (speed of light in vacuum/speed of light in medium) etc., has no dimensions. Plane angle and solid angles are dimensionless. Dimensionless Constants: Such constants which do not have dimensions are called as dimensionless constants. For ex. Refractive Index , relative density, velocity of light in vacuum etc. Dimensional constants: Such constants which have dimensions are called as dimensional constants. For ex. Universal gravitational constant, electric permittivity, Coefficient of elasticity etc. Basic Quantities and Their Dimension Dimension has a specific meaning – it denotes the physical nature of a quantity Dimensions are denoted with square brackets Length [L] Mass [M] Time [T] Dimensional Analysis Technique to check the correctness of an equation or to assist in deriving an equation Dimensions (length, mass, time, combinations) can be treated as algebraic quantities add, subtract, multiply, divide Both sides of equation must have the same dimensions Any relationship can be correct only if the dimensions on both sides of the equation are the same Cannot give numerical factors: this is its limitation Dimensional Analysis …..Contd Dimensions of the 7 base quantities are – Length [L], Mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd] and amount of substance [mol]. Let any physical quantity A ,depends mass, length ,time with powers a,b,& c then : Dimensional Formula : [MaLbTc] Dimensional equation : [A]= [MaLbTc] For example, the dimensional equations of volume [V], speed [v], force [F ] and mass density [ρ] may be expressed as: [V] = [M0 L3 T0] [v] = [M0 L T–1] [F] = [M L T–2] [ρ] = [M L–3 T0] Some Important Dimensions  Checking the Dimensional Consistency of Equations Only those physical quantities which have same dimensions can be added and subtracted. This is called principle of homogeneity of dimensions. According to this principle of homogeneity a physical equation will be dimensionally correct if the dimensions of all the terms occurring on both sides of the equation are the same.  Checking the Dimensional Consistency of Equations Example: Let us consider an equation where m is the mass of the body, v its velocity, g is the acceleration due to gravity and h is the height. Check whether this equation is dimensionally correct. The dimensions of LHS are: [M] [L T–1 ]2 = [M] [ L2 T–2] = [M L2 T–2] The dimensions of RHS are: [M][L T–2] [L] = [M][L2 T–2] = [M L2 T–2] The dimensions of LHS and RHS are the same and Hence the equation is dimensionally correct.  Deducing Relation among the Physical Quantities To deduce relation among physical quantities, we should know the dependence of one quantity over others (or independent variables) and consider it as product type of dependence. Dimensionless constants cannot be obtained using this method.  Limitations of Dimensional Analysis Dimensional analysis has no information on dimensionless constants. If a quantity is dependent on trigonometric or exponential functions, this method cannot be used. In some cases, it is difficult to guess the factors while deriving the relation connecting two or more physical quantities. This method cannot be used in an equation containing two or more variables with same dimensions. It cannot be used if the physical quantity is dependent on more than three unknown variables. This method cannot be used if the physical quantity contains more than one term, say sum or difference of two terms.  Exercise The period P of a swinging pendulum depends only on the length of the pendulum d and the acceleration of gravity g. Which of the following formulas for P could be correct ? d d (a) P = 2p (dg)2 (b) P = 2p (c) P = 2p g g Given: d has units of length (L) and g has units of (L / T 2). Solution Realize that the left hand side P has units of time (T ) Try the first equation 2 (a) æç L × L ö÷ L4 = 4 ¹T Not Right !! è T2 ø T d d P = 2p (dg ) (b) P = 2p (c) P = 2p 2 (a) g g Solution Try the second equation L (b) = T2 ¹ T Not Right !! L T2 d d P = 2p (dg ) (b) P = 2p P = 2p 2 (a) (c) g g Solution ! Try the third equation L This has the correct units!! (c) = T2 =T L This must be the answer!! T2 d d P = 2p (dg ) (b) P = 2p (c) P = 2p 2 (a) g g Uncertainty in Measurements There is uncertainty in every measurement – this uncertainty carries over through the calculations May be due to the apparatus, the experimenter, and/or the number of measurements made Need a technique to account for this uncertainty We will use rules for significant figures to approximate the uncertainty in results of calculations  Class Exercise 1. Derive the dimensional formula of following Quantity & write down their dimensions (i) Density (ii) Power (iii) Co-efficient of viscosity (iv) Angle 2. Explain which of the following pair of physical quantities have the same dimension (i) Work &Power (ii) Stress & Pressure (iii) Momentum &Impulse 3. Check the correctness of the following equation on the basis of dimensional ! analysis, / =. Here V is the velocity of sound, E is the elasticity and d is " the density of the medium 4. Using Principle of Homogeneity of dimensions, check the correctness of equation, h = 2Td /rgCos0 (T= surface tension, D= density, r =radius, g=acceleration due to gravity). # 5. Check the accuracy of the following relations (i) 1 = 23ℎ + 25 $ ; (ii) 5 % − $ $ $ 6 = 278. Vectors and Scalars  Scalar and Vectors nA SCALAR is any quantity in physics nA VECTOR is any quantity in that has MAGNITUDE, but NOT a physics that has BOTH direction associated with it. MAGNITUDE and DIRECTION. Scalar Quantity Description Mass The amount of matter in an object (e.g., 50 kg). Temperature A measure of thermal energy (e.g., 25°C). Time Duration of events (e.g., 10 seconds). Energy Capacity to do work (e.g., 500 Joules). Speed Rate of motion without direction (e.g., 60 km/h). Distance Total length of the path traveled (e.g., 200 meters). Work Energy transferred by force over distance (e.g., 100 Joules). Power Rate of doing work (e.g., 100 Watts). Density Mass per unit volume (e.g., 1.2 g/cm³). Charge Amount of electric charge (e.g., 5 Coulombs). Pressure Force per unit area (e.g., 101,325 Pascals). Volume Space occupied by an object (e.g., 2 cubic meters). Potential Energy Energy stored due to position (e.g., 50 Joules).  Vector Quantity Description The shortest path between two points in a given direction Displacement (e.g., 30 m east). Velocity Speed with direction (e.g., 20 m/s west). A push or pull with magnitude and direction (e.g., 15 N Force downward). Rate of change of velocity in a particular direction (e.g., 9.8 Acceleration m/s² downward). Momentum Product of mass and velocity (e.g., 100 kg·m/s north). Weight Gravitational force acting on an object (e.g., 9.8 N downward). Electric Field Force per unit charge with direction (e.g., 50 N/C upward). Magnetic force per unit charge moving in a field (e.g., 0.1 Tesla Magnetic Field north). Torque Rotational force with direction (e.g., 10 Nm clockwise). Angular Velocity Rate of rotation about an axis (e.g., 3 rad/s counterclockwise). Flow of electric charge in a specific direction (e.g., 5 A Current eastward).  Vector Calculations VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them. VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, subtract them. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E - 30 m, W 24.5 m, E Problem: Car moves 80 Km east, then 30 Km west. After that it moves again 50 Km east. Calculate its displacement relative to where it started? 100 Km , East  Non-Collinear Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. N # North W of N E of N # South N of E # East N of W E W # West S of W S of E W of S E of S S Non-Collinear Vectors Example: A boat moves north with a velocity of 15 m/s in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W Rv = 82 + 152 = 17 m / s 15 m/s, N 8 Rv q Tanq = = 0.5333 15 q = Tan -1 (0.5333) = 28.1! The Final Answer : 17 m/s, @ 28.1 degrees West of North  Non-Collinear Vectors Example : A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. x 32 y X = 53.85 m/s , east. 63.5 m/s Y= 33.64 m/s , south. Example : R = 14.3 km Angle : 65° N of E  Addition of Vectors:Graphical Methods The parallelogram method may also be used; here again the vectors must be “tail-to-tip.” Trigonometric Identities opposite adjacent sinq = cosq = hypotenuse hypotenuse opposite tanq = adjacent Pythagoras' Theorem : hyp = opp + adj 2 2 2 a b c Sine Rule : = = sin A sin B sin C Cosine Rule : a 2 = b 2 + c 2 - 2b × c × cos A  4.0 N 5.0 N Vectors at 0o R= 9.0 N R= 3.6 N 5.0 N Vectors at 45o 4.0 N R= 6.4 N 5.0 N 4.0 N Vectors at 90o  R= 8.3 N 5.0 N Vectors at 135o 4.0 N 4.0 N 5.0 N Vectors at 180o R= 1.0 N Unit Vector Notation =3j J = vector of magnitude “1” in the “y” direction = 4i i = vector of magnitude “1” in the “x” direction The hypotenuse in Physics is called the RESULTANT or VECTOR SUM. The LEGS of the triangle are A = 4iˆ + 3 ˆj called the COMPONENTS 3j Vertical Component NOTE: When drawing a right triangle that 4i conveys some type of motion, you MUST draw Horizontal Component your components HEAD TO TOE. Unit Vector Notation iˆ - unit vector = 1 in the + x direction The proper terminology is to use the “hat” instead of the arrow. So ˆj - unit vector = 1 in the + y direction we have i-hat, j-hat, and k-hat kˆ - unit vector = 1 in the + z direction which are used to describe any type of motion in 3D space. How would you write vectors J and K in unit vector notation? J = 2iˆ + 4 ˆj K = 2iˆ - 5 ˆj -5 Example: A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 12 m, W R = 14 2 + 232 = 26.93m 26.93m @ 31.3! NofE 6 m, S R 14 m, N 14 26.93m < 31.3 20 m, N Tanq = =.6087 q 23 23iˆ m + 14 ˆj m q = Tan -1 (0.6087) = 31.3! 23 m, E 35 m, E Example: A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W 15 m/s, N Rv = 82 + 152 = 17 m / s 17 m / s @ 28.1! WofN Rv 8 q Tanq = = 0.5333 17 m / s < 118.1! 15 q = Tan -1 (0.5333) = 28.1! - 8iˆ m / s + 15 ˆj m / s  Classwork 1 Find the resultant of the following two displacements: 2.0 m at 40o and 4.0 m at 127o, the angles being taken relative to the +x-axis, as is customary. Give your answer to two significant figures. (a) using rectangular method (b) using cosine rule 2 3 Classwork 1 Part (a)  Classwork 1 Part (a) m 4.0 53o R o 40o 93 m 2.0 9 $ = 4$ + 2$ − 2×4×2×cos(93) =4.6 m Classwork 2 Adjacent = Fx Opposite = Fy Hypotenuse = 60 0 = 40o Classwork 3 (a) (b)  Class Exercise (1) (2) (3) (4) (5) Differentiation of Vectors Differentiation Rules (General Formulas) Differentiation Rules (General Formulas) Differentiation Rules (Trigonometric Functions) Differentiation Rules (Others) Differentiation Rules (Inverse Trigonometric Functions) Class Examples Class Examples Solution Class Examples Class Examples Solution Class Examples Solution Class Examples Solution Example 1 Example 2 Example 3 Solution Example 3 Solution Example 3 Class Exercise Kinematics Kinematics Kinematics involves the study of how things move. It is only concerned with the motion itself, not the forces that cause this motion. The kinematics of an object is described in terms of its distance, displacement, speed, velocity, acceleration. Kinematics MECHANICS STATICS DYNAMICS (objects in equilibrium) (objects in motion) KINEMATICS KINETICS (geometric aspects of (analysis of forces causing motion) motion)  Kinematics …………. Contd Distance is a scalar quantity. The distance a body has travelled is literally the amount of ‘ground’ it has covered during its motion. Displacement is a vector quantity. Displacement describes how far a body is from its starting point and in what direction. Distance and displacement are measured in metres, m. Speed is a scalar quantity. The speed of a body relates to how fast the body is travelling. Velocity is a vector quantity. The velocity of a body relates to how fast the body is travelling and in what direction. It is the rate at which a body changes its position. Speed and velocity are measured in metres per second, ms–1. Acceleration can be a scalar or a vector quantity. Acceleration is the rate of change of speed or velocity. It is measured in metres per second per second, ms–2. Negative acceleration is often called deceleration or retardation.  Formulae for constant acceleration If a particle is moving in a straight line with a constant acceleration then there are five equations of motion that can be used to determine missing quantities. v = u + at Where s = displacement in metres s = 21 (u + v )t u = initial velocity in ms–1 v = final velocity in ms–1 s = ut + 21 at 2 a = acceleration in ms–2 s = vt - 21 at 2 t = time taken in seconds v 2 = u 2 + 2as These are sometimes called the suvat formulae. For vertical motion acceleration due to gravity is g, 9.8ms–2. Derivation of v2 = u2 + 2as v = u + at 1 æu+vö s=ç ÷ t 2 è 2 ø From equation 1, make t the subject of formular (v - u ) t= 3 a Substituting 3 into equation 2 æ u + v öæ v - u ö s=ç ÷ç ÷ è 2 øè a ø 2as = (u + v )(v - u ) 2as = v 2 - u 2 So v 2 = u 2 + 2as  Example 1 A stone is thrown vertically upwards with a speed of 10 ms–1 from a point 15 m above the ground. Find the maximum height above the ground that the stone reaches and find the time taken for the stone to reach the ground. Solution u = 10 a = –g (Take g to be 9.8) The question is asking for s when v = 0, and for t when s = 15. To calculate s when v = 0, given u and a requires the use of v2 = u2 + 2as. 02 = 102 + 2(–9.8)(s) ­ 19.6s = 100 \s = 5.10 (to 3 s.f.) \Therefore, the maximum height above the ground that the stone reaches is 20.1 m (to 3 s.f.).  Solution of Example 1 1 To calculate t when s = –15, given u and a requires the use of s = ut + at 2. 2 1 –15 = 10t + (–9.8)t 2 2 –15 = 10t – 4.9t2 4.9t2 –10t – 15 = 0 Using -b ± b - 4ac 2 gives the solution 2a t = 3.05 or t = –1.01 (to 3 s.f.) Therefore the stone reaches the ground after 3.05 s (to 3 s.f.).  Class exercise A particle moves in a horizontal line from a point A to a point C, via point B. It has a constant acceleration of 1 ms–2 and passes point B after 6 seconds and1 point C after a further 4 seconds. Its velocity at C is 50 ms–1. Calculate the velocity 2 at A and the distances AB and BC. A ball falls off a cliff and lands on the beach 3.2 seconds later. How high is the cliff? A ball is thrown vertically upwards with an initial velocity of u ms–1 from a point 1.2 m above the ground. It reaches a maximum height of 23 m above the ground. Calculate a) the initial velocity b) the velocity with which the ball strikes the ground c) the total time the ball is in the air. Rectilinear Kinematics: Continuous Motion Rectilinear Kinematics: One-dimensional motion. A particle’s position, velocity, and acceleration define its kinematic state. A particle is an idealization of a body where mass is concentrated at a point. A particle has mass but negligible size and shape. A particle has location but not rotation. Position: A single coordinate axis, s, defines the location of a particle. Displacement: Change in position. Ds = s '- s  Rectilinear Kinematics: Continuous Motion Special Case: Constant Acceleration a = ac Velocity as a Function of Time: Integrate ac = dv dt , assuming v = v0 when t = 0 v t òv dv= ò0 ac dt 0 Þ v = v0 + act ( EQ 12-4 ) Position as a Function of Time: Integrate v = ds dt = v0 + act , assuming s = s0 when t = 0 s t òs ds = ò ( v0 + act ) dt Þ s = s0 + v0t + 12 act 2 ( EQ 12-5) 0 0 Velocity as a Function of Integrate vdv = ac ds, assuming v = v0 when s = s0 Position: v s òv vdv= òs ac ds Þ v 2 = v02 + 2ac ( s - s0 ) ( EQ 12-6 ) 0 0  Example 2 (b) Have v(t), and want s(t) (a) Have a(t), and want v(t) Example 3 The co-ordinates of a moving particle at any time t are given by x = ct2 and y = bt2. The speed of the particle is given by  Class Exercise 1 2  Class Exercise 3 4 A particle starts moving from rest along a circle of radius 4 m with a constant tangential acceleration of 3 m/s2. The total acceleration of the particle at t = 2 seconds makes an angle ! with the tangential acceleration, where ! is equal to