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## MECHANICS

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**General Physics I**MECHANICS 2009/2010 Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Textbook: the class notes beside the following textbooks:**• Physics for Scientists and Engineers, Raymond A. Serway, 6th Edition • University Physics, Sears, Zemansky and Young • Physics and Measurement: Standards of Length, Mass, and Time, Density and Atomic Mass, Dimensional Analysis, Conversion of Units. • Motion in One Dimension: Position, Velocity, and Speed, Instantaneous Velocity and Speed, Acceleration, Motion Diagrams, One-Dimensional Motion with Constant Acceleration, Freely Falling Objects, Kinematics Equations Derived from Calculus. • Vectors: Coordinate Systems, Vector and Scalar Quantities, Some Properties of Vectors, Components of a Vector and Unit Vectors, The Scalar Product of Two Vectors, The Vector Product of Two Vectors. Course Outline Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Motion in Two Dimensions: The Position, Velocity, and**Acceleration • Vectors, Two-Dimensional Motion with Constant Acceleration, Projectile Motion, Uniform Circular Motion, Tangential and Radial Acceleration, Relative Velocity and Relative Acceleration. • The Laws of Motion: The Concept of Force. Newton's First Law and Inertial Frames, Newton's Second Law, The Gravitational Force and Weight, Newton's Third Law, Some Applications of Newton's Laws, Forces of Friction. • Energy and Energy Transfer: Systems and Environments, Work Done by a Constant Force, Work Done by a Varying Force, Kinetic Energy and the Work-Kinetic Energy Theorem, The Non-Isolated System, Conservation of Energy, • Situations Involving Kinetic Friction, Power. • Potential Energy: Potential Energy of a System, The Isolated System, Conservation of Mechanical Energy, Conservative and Nonconservative Forces, Changes in Mechanical Energy for Nonconservative Forces, Relationship Between Conservative Forces and Potential Energy, Energy Diagrams and Equilibrium of a System. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Linear Momentum and Collisions: Linear Momentum and Its**Conservation, • Impulse and Momentum, Collisions in One Dimension, Two-Dimensional Collisions, The Center of Mass, Motion of a System of Particles. • Universal Gravitation: Newton's Law of Universal Gravitation, Measuring the Gravitational Constant, Free-Fall Acceleration and the Gravitational Force, • Kepler's Laws and the Motion of Planets, The Gravitational Field, Gravitational Potential Energy, Energy Considerations in Planetary and Satellite Motion. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**GRADING POLICY**Your grade will be judged on your performance in Home work, Quizzes, tow tests and the Lab. Points will be allocated to each of these in the following manner: GRADING SCALE: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Lecture 1**Measurement & Units Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Physical quantities (in mechanics)**Basic quantities : in mechanics the three fundamental quantities are Length (L), mass (M), time (T) Derived quantities : all other physical quantities in mechanics can be expressed in term of basic quantities Area Volume Velocity Acceleration Force Momentum Work ….. ….. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Mass**The SI unit of mass is the Kilogram, which is defined as the mass of a specific platinum-iridium alloy cylinder. Time The SI unit of time is the Second, which is the time required for a cesium-133 atom to undergo 9192631770 vibrations. Length The SI unit of length is Meter, which is the distance traveled by light is vacuum during a time of 1/2999792458 second. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Systems of Units**• SI units (International System of Units): • length: meter (m), mass: kilogram (kg), time: second (s) • *This system is also referred to as the mks system for meter-kilogram-second. • Gaussian units • length: centimeter (cm), mass: gram (g), time: second (s) • *This system is also referred to as the cgs system for centimeter-gram-second. • British engineering system: • Length: inches, feet, miles, mass: slugs (pounds), time: seconds We will use mostly SI units, but you may run across some problems using British units. You should know how to convert back & forth. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Conversions**When units are not consistent, you may need to convert to appropriate ones. Units can be treated like algebraic quantities that can cancel each other out. 1 mile = 1609 m = 1.609 km 1 ft = 0.3048 m = 30.48 cm 1m = 39.37 in = 3.281 ft 1 in = 0.0254 m = 2.54 cm 1 mile = 5280 ft • Questions: • Convert 500 millimeters into meters. • Convert 4.2 liters into milliliters. • Convert 1.45 meters into inches. • Convert 65 miles per hour into kilometers per second. Example: Convert miles per hour to meters per second: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Prefixes**Prefixes correspond to powers of 10 Each prefix has a specific name/abbreviation Power Prefix Abbrev. 1015 peta P 109 giga G 106 mega M 103 kilo k 10-2 centi P 10-3 milli m 10-6 micro m 10-9 nano n 10-12 pico p 10-15 femto f Distance from Earth to nearest star 40 Pm Mean radius of Earth 6 Mm Length of a housefly 5 mm Size of living cells 10 mm Size of an atom 0.1 nm Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Dimensional Analysis**Definition: The Dimension is the qualitative nature of a physical quantity (length, mass, time). brackets [ ] denote the dimension or units of a physical quantity: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Idea:Dimensional analysis can be used to derive or check**formulas by treating dimensions as algebraic quantities. Quantities can be added or subtracted only if they have the same dimensions, and quantities on two sides of an equation must have the same dimensions Example : Using the dimensional analysis check that this equation x = ½ at2 is correct, where x is the distance, a is the acceleration and t is the time. Solution left hand side right hand side This equation is correct because the dimension of the left and right side of the equation have the same dimensions. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Example:**Suppose that the acceleration of a particle moving in circle of radius r with uniform velocity v is proportional to the rn and vm. Use the dimensional analysis to determine the power n and m. Solution Let us assume a is represented in this expression a = k rnvm Where k is the proportionality constant of dimensionless unit. The right hand side [a] = L/T2 The left hand side Therefore or Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Hence**n+m=1 and m=2 Therefore n =-1 and the acceleration a is a = k r -1v2 Problem: 1. Show that the expression x = vt +1/2 at2 is dimensionally correct , where x is coordinate and has unit of length, v is velocity, a is acceleration and t is the time. 2. Show that the period T of a simple pendulum is measured in time unit given by Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Density**Every substance has a density, designated = M/V Dimensions of density are, units (kg/m3) • Some examples, Substance (103 kg/m3) Gold 19.3 Lead 11.3 Aluminum 2.70 Water 1.00 Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Atomic Density**In dealing with macroscopic numbers of atoms (and similar small particles) we often use a convenient quantity called Avogadro’s Number, NA = 6.023 x 1023 atoms per mole Commonly used mass units in regards to elements 1. Molar Mass = mass in grams of one mole of the substance (averaging over natural isotope occurrences) 2. Atomic Mass = mass in u (a.m.u.) of one atom of a substance. It is approximately the total number of protons and neutrons in one atom of that substance. 1u = 1.660 538 7 x 10-27 kg What is the mass of a single carbon (C12) atom ? = 2 x 10-23 g/atom Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Lecture 2**Coordinate Systems & Vectors Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Coordinate Systems and Frames of**Reference The location of a point on a line can be described by one coordinate; a point on a plane can be described by two coordinates; a point in a three dimensional volume can be described by three coordinates. In general, the number of coordinates equals the number of dimensions. A coordinate system consists of: 1. a fixed reference point (origin) 2. a set of axes with specified directions and scales 3. instructions that specify how to label a point in space relative to the origin and axes Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Coordinate Systems**• In 1 dimension, only 1 kind of system, • Linear Coordinates (x) +/- • In 2 dimensions there are two commonly used systems, • Cartesian Coordinates (x,y) • Polar Coordinates (r,q) • In 3 dimensions there are three commonly used systems, • Cartesian Coordinates (x,y,z) • Cylindrical Coordinates (r,q,z) • Spherical Coordinates (r,q,f) Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Cartesian coordinate system**• also called rectangular coordinate system • x and y axes • points are labeled (x,y) Plane polar coordinate system • origin and reference line are noted • point is distance r from the origin in the direction of angle • points are labeled (r,) Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**The relation between coordinates**Furthermore, it follows that Problem: A point is located in polar coordinate system by the coordinate and . Find the x and y coordinates of this point, assuming the two coordinate systems have the same origin. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Example :**The Cartesian coordinates of a point are given by (x,y)= (-3.5,-2.5) meter. Find the polar coordinate of this point. Solution: Note that you must use the signs of x and y to find that is in the third quadrant of coordinate system. That is not 36 Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Scalars and Vectors**Scalarshavemagnitudeonly. Length, time, mass, speed and volume are examples of scalars. Vectorshavemagnitudeanddirection. The magnitude ofis written Position, displacement, velocity, acceleration and force are examples of vector quantities. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Properties of Vectors**Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Negative Vectors**Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) Multiplication or division of a vector by a scalar results in a vector for which (a) only the magnitude changes if the scalar is positive (b) the magnitude changes and the direction is reversed if the scalar is negative. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Adding Vectors**When adding vectors, their directions must be taken into account and units must be the same First: Graphical Methods Second: Algebraic Methods Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Adding Vectors Graphically (Triangle Method)**Continue drawing the vectors “tip-to-tail” The resultant is drawn from the origin of A to the end of the last vector Measure the length of R and its angle Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**When you have many vectors, just keep repeating the process**until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Alternative Graphical Method (Parallelogram Method)**When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin The remaining sides of the parallelogram are sketched to determine the diagonal, R Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Vector Subtraction**Special case of vector addition If A–B, then use A+(-B) Continue with standard vector addition procedure Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Components of a Vector**These are the projections of the vector along the x- and y-axes Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**The x-component of a vector is the projection along the**x-axis The y-component of a vector is the projection along the y-axis Then, Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Adding Vectors Algebraically**(1)Choose a coordinate system and sketch the vectors (2)Find the x- and y-components of all the vector (3)Add all the x-components This gives Rx: (4)Add all the y-components This gives Ry Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**(5)find the magnitude of the Resultant**Use the inverse tangent function to find the direction of R: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**U = |U| û**û • Useful examples are the cartesian unit vectors [i, j, k] • Point in the direction of the x, y and z axes. R = rx i + ry j + rz k y j x i k z Unit Vectors • A Unit Vector is a vector having length 1 and no units • It is used to specify a direction. • Unit vector u points in the direction of U • Often denoted with a “hat”: u = û Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Example :**A particle undergoes three consecutive displacements given by Find the resultant displacement of the particle Solution: The resultant displacement has component The magnitude is Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Product of a vector**There are two different ways in which we can usefully define the multiplication of two vectors 1-The scalar product (dot product ) Each of the lengths |A| and |B| is a number and is number, so A.B is not a vector but a number or scalar. This is why it's called the scalar product. Special cases of the dot product Since i and j and k are all one unit in length and they are all mutually perpendicular, we have i.i = j.j = k.k = 1 and i.j = j.i = i.k = k.i = j.k = k.j = 0. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**The angle between the two vector**If A and B both have x,y and z components, we express them in the form Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**2- The vector product**(cross product) Special cases of the cross product Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Problem 1: Find the sum of two vectors A and B lying in the**xy plane and given by Problem 2: A particle undergoes three consecutive displacements : Find the components of the resultant displacement and its magnitude. Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**Lecture 3**Discussion Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**[1] The position of a particle moving under uniform**acceleration is some function of time and the acceleration. Suppose we write this position s = kam tn where , k is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if if m = 1 and n = 2 Solution and Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**[2] Newton’s law of universal gravitation is represented**by Here F is the magnitude of the gravitational force exerted by one small object on another , M and m are the masses of the objects, and r is a distance. Force has the SI units kg ·m/ s2. What are the SI units of the proportionality constant G? Solution: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**[3] A solid piece of lead has a mass of 23.94 g and a volume**of 2.10 cm3. From these data, calculate the density of lead in SI units (kg/ m3). Solution • One centimeter (cm) equals 0.01 m. • One kilometer (km) equals 1000 m. • One inch equals 2.54 cm • One foot equals 30 cm… Example: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**[4] If the rectangular coordinates of a point are given by**(2, y) and its polar coordinates are ( r , 30°), determine y and r. Solution: then then [4] Two points in the xy plane have Cartesian coordinates (2.00, -4.00) m and ( -3.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates. a) Solution: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**b**For (2,-4) the polar coordinate is (2,2√5) since Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)**[5] Vector A has a magnitude of 8.00 units and makes an**angle of 45.0 ° with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical methods, find (a) the vector sum A + B and (b) the vector difference A - B. Solution: Gneral Physics I, Lecture Note, Part 1 (Lecture 1-11)