Gneral Physics I, Syllibus, By/ T.A. Eleyan1 General Physics I Instructor Tamer A. Eleyan 2009/2010

Gneral Physics I, Syllibus, By/ T.A. Eleyan2 Course Outline Textbook: the class notes beside the following textbooks: 1.Physics for Scientists and Engineers, Raymond A. Serway, 6th Edition 2.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.

Gneral Physics I, Syllibus, By/ T.A. Eleyan3  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, Syllibus, By/ T.A. Eleyan4  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, Syllibus, By/ T.A. Eleyan5 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: WeightGrade Component 10HW/Quizzes 10Attendees 20Midterm Exam 60Final Exam 100Total

Gneral Physics I, Syllibus, By/ T.A. Eleyan6 Lecture 1 Measurement & Units

Gneral Physics I, Syllibus, By/ T.A. Eleyan7 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, Syllibus, By/ T.A. Eleyan8 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 vibrations. Length The SI unit of length is Meter, which is the distance traveled by light is vacuum during a time of 1/ second.

Gneral Physics I, Syllibus, By/ T.A. Eleyan9 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, Syllibus, By/ T.A. Eleyan10 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 = km 1 ft = m = cm 1m = in = ft 1 in = m = 2.54 cm 1 mile = 5280 ft Example: Convert miles per hour to meters per second: Example: Convert miles per hour to meters per second: Questions: 1.Convert 500 millimeters into meters. 2.Convert 4.2 liters into milliliters. 3.Convert 1.45 meters into inches. 4.Convert 65 miles per hour into kilometers per second.

Gneral Physics I, Syllibus, By/ T.A. Eleyan11 Prefixes Prefixes correspond to powers of 10 Each prefix has a specific name/abbreviation PowerPrefix Abbrev petaP 10 9 giga G 10 6 mega M 10 3 kilo k centi P milli m micro  nano n pico p femto f Distance from Earth to nearest star40 Pm Mean radius of Earth6 Mm Length of a housefly5 mm Size of living cells10  m Size of an atom0.1 nm

Gneral Physics I, Syllibus, By/ T.A. Eleyan12 Dimension[L]=L[M]=M[T]=TQuantityLengthMasstime [A] = L 2 area [V]=L 3 volume [v]= L/T [v]= L/Tvelocity [a] = L/T 2 [f]=M L/T 2 Accelerationforce 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, Syllibus, By/ T.A. Eleyan13 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 at 2 is correct, where x is the distance, a is the acceleration and t is the time. 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. Solution

Gneral Physics I, Syllibus, By/ T.A. Eleyan14 Example: Suppose that the acceleration of a particle moving in circle of radius r with uniform velocity v is proportional to the r n and v m. Use the dimensional analysis to determine the power n and m. The left hand side Therefore or Solution Let us assume a is represented in this expression a = k r n v m Where k is the proportionality constant of dimensionless unit. The right hand side [ a] = L/T 2

Gneral Physics I, Syllibus, By/ T.A. Eleyan15 Hence n+m=1 and m=2 Therefore n =-1 and the acceleration a is a = k r -1 v 2 Problem: 1. Show that the expression x = vt +1/2 at 2 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, Syllibus, By/ T.A. Eleyan16 Density Every substance has a density, designated  = M/V Dimensions of density are, units (kg/m 3 ) Some examples, Substance  (10 3 kg/m 3 ) Gold19.3 Lead11.3 Aluminum 2.70 Water 1.00

Gneral Physics I, Syllibus, By/ T.A. Eleyan17 Atomic Density In dealing with macroscopic numbers of atoms (and similar In dealing with macroscopic numbers of atoms (and similar small particles) we often use a convenient quantity called Avogadro ’ s Number, N A = x atoms per mole small particles) we often use a convenient quantity called Avogadro ’ s Number, N A = x atoms per mole Commonly used mass units in regards to elements 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 = x kg What is the mass of a single carbon (C 12 ) atom ? = 2 x g/atom

Gneral Physics I, Syllibus, By/ T.A. Eleyan18 Lecture 2 Coordinate Systems & Vectors

Gneral Physics I, Syllibus, By/ T.A. Eleyan19 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, Syllibus, By/ T.A. Eleyan20 Coordinate Systems In 1 dimension, only 1 kind of system, In 1 dimension, only 1 kind of system, – Linear Coordinates (x) +/- In 2 dimensions there are two commonly used systems, In 2 dimensions there are two commonly used systems, – Cartesian Coordinates(x,y) – Polar Coordinates(r,  ) In 3 dimensions there are three commonly used systems, In 3 dimensions there are three commonly used systems, – Cartesian Coordinates(x,y,z) – Cylindrical Coordinates (r, ,z) – Spherical Coordinates(r,  )

Gneral Physics I, Syllibus, By/ T.A. Eleyan21 Cartesian coordinate system also called rectangular coordinate system also called rectangular coordinate system x and y axes x and y axes points are labeled (x,y) 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, Syllibus, By/ T.A. Eleyan22 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, Syllibus, By/ T.A. Eleyan23 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, Syllibus, By/ T.A. Eleyan24 Scalars and Vectors Scalars have magnitude only. Length, time, mass, speed and volume are examples of scalars. Vectors have magnitude and direction. The magnitude of is written Position, displacement, velocity, acceleration and force are examples of vector quantities.

Gneral Physics I, Syllibus, By/ T.A. Eleyan25 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, Syllibus, By/ T.A. Eleyan26 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, Syllibus, By/ T.A. Eleyan27 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, Syllibus, By/ T.A. Eleyan28 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, Syllibus, By/ T.A. Eleyan29 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, Syllibus, By/ T.A. Eleyan30 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, Syllibus, By/ T.A. Eleyan31 Vector Subtraction Special case of vector addition If A – B, then use A+(-B) Continue with standard vector addition procedure

Gneral Physics I, Syllibus, By/ T.A. Eleyan32 Components of a Vector These are the projections of the vector along the x- and y-axes

Gneral Physics I, Syllibus, By/ T.A. Eleyan33 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, Syllibus, By/ T.A. Eleyan34 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 R x : (4)Add all the y-components This gives Ry

Gneral Physics I, Syllibus, By/ T.A. Eleyan35 (5)find the magnitude of the Resultant Use the inverse tangent function to find the direction of R:

Gneral Physics I, Syllibus, By/ T.A. Eleyan36 Unit Vectors A Unit Vector is a vector having length 1 and no units A Unit Vector is a vector having length 1 and no units It is used to specify a direction. It is used to specify a direction. Unit vector u points in the direction of U Unit vector u points in the direction of U –Often denoted with a “ hat ” : u = û U = |U| û û x y z i j k i, j, k l Useful examples are the cartesian unit vectors [ i, j, k ]  Point in the direction of the x, y and z axes. R = r x i + r y j + r z k

Gneral Physics I, Syllibus, By/ T.A. Eleyan37 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, Syllibus, By/ T.A. Eleyan38 Product of a vector 1-The scalar product (dot product ) There are two different ways in which we can usefully define the multiplication of two vectors 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, Syllibus, By/ T.A. Eleyan39 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, Syllibus, By/ T.A. Eleyan40

Gneral Physics I, Syllibus, By/ T.A. Eleyan41 2- The vector product (cross product) Special cases of the cross product

Gneral Physics I, Syllibus, By/ T.A. Eleyan42

Gneral Physics I, Syllibus, By/ T.A. Eleyan43 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, Syllibus, By/ T.A. Eleyan44 Lecture 3 Discussion

Gneral Physics I, Syllibus, By/ T.A. Eleyan45 Solution and [1] The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position s = ka m t n where, k is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if if m = 1 and n = 2

Gneral Physics I, Syllibus, By/ T.A. Eleyan46 Solution: [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?

Gneral Physics I, Syllibus, By/ T.A. Eleyan47 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: [3] A solid piece of lead has a mass of g and a volume of 2.10 cm3. From these data, calculate the density of lead in SI units (kg/ m3).

Gneral Physics I, Syllibus, By/ T.A. Eleyan48 Solution: then a) Solution: [4] If the rectangular coordinates of a point are given by (2, y) and its polar coordinates are ( r, 30°), determine y and r. [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.

Gneral Physics I, Syllibus, By/ T.A. Eleyan49 b For (2,-4) the polar coordinate is (2,2√5) since

Gneral Physics I, Syllibus, By/ T.A. Eleyan50 [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, Syllibus, By/ T.A. Eleyan51 Solution: [6] Given the vectors A = 2.00 i j and B = 3.00 i j, (a) draw the vector sum, C = A + B and the vector difference D = A - B. (b) Calculate C and D, first in terms of unit vectors and then in terms of polar coordinates, with angles measured with respect to the, +x axis.

Gneral Physics I, Syllibus, By/ T.A. Eleyan52 Solution: Direction of A+B Direction of A-B [7] Consider the two vectors A = 3 i - 2 j and B = i - 4 j. Calculate (a) A + B, (b) A - B, (c) │A + B│, (d) │A - B│, and (e) the directions of A + B and A - B.

Gneral Physics I, Syllibus, By/ T.A. Eleyan53 Solution: a) b) c) [8] The vector A has x, y, and, z components of 8.00, 12.0, and units, respectively. (a) Write a vector expression for A in unit vector notation. (b) Obtain a unit vector expression for a vector B four time the length of A pointing in the same direction as A. (c) Obtain a unit vector expression for a vector C three times the length of A pointing in the direction opposite the direction of A.

Gneral Physics I, Syllibus, By/ T.A. Eleyan54 Solution: [9] Three displacement vectors of a croquet ball are shown in Figure, where A = 20.0 units, B = 40.0 units, and C = 30.0 units. Find the magnitude and direction of the resultant displacement

Gneral Physics I, Syllibus, By/ T.A. Eleyan55 [10] Find the magnitude and the direction of resultant force Solution:

Gneral Physics I, Syllibus, By/ T.A. Eleyan56 Lecture 4 One Dimensional Kinematics

Gneral Physics I, Syllibus, By/ T.A. Eleyan57 In lecture this we discuss motion in one dimension. We introduce definitions for displacement, velocity and acceleration, and derive equations of motion for bodies moving in one dimension with constant acceleration. We apply these equations to the situation of a body moving under the influence of gravity alone. In lecture this we discuss motion in one dimension. We introduce definitions for displacement, velocity and acceleration, and derive equations of motion for bodies moving in one dimension with constant acceleration. We apply these equations to the situation of a body moving under the influence of gravity alone. One Dimensional Kinematics** **One dimensional kinematics refers to motion along a straight line.

Gneral Physics I, Syllibus, By/ T.A. Eleyan58 Kinematics is that branch of physics which involves the description of motion, without examining the forces which produce the motion. Dynamics, on the other hand, involves an examination of both a description of motion and the forces which produce it.

Gneral Physics I, Syllibus, By/ T.A. Eleyan59 Distance and displacement Displacement is defined as the change in position of an object. x f = final value of x, x i = initial value of xx f = final value of x, x i = initial value of x Change can be positive, negative or zero.Change can be positive, negative or zero. Displacement is a vectorDisplacement is a vector Distance is the total length of travel. It is always positive. It is measured by the meter. ‘  ’ (Delta)=change

Gneral Physics I, Syllibus, By/ T.A. Eleyan60 Distance or Displacement? Distance may be, but is not necessarily, the magnitude of the displacement Distance Displacement

Gneral Physics I, Syllibus, By/ T.A. Eleyan61 Example, starting with x i = 60 m and ending at x f = 150 m, the displacement is Δ x = x f - x i = 150 m - 60 m = 90 m Δ x > 0 since x i < x f. Example, starting with x i = 150 m and ending at x f = 60 m, the displacement is Δx = x f - x i = 60 m m = -90 m Δx x f.

Gneral Physics I, Syllibus, By/ T.A. Eleyan62 Average Speed and Velocity Speed and velocity are not the same in physics! Speed is rate of change of distance: (always positive) (positive, negative or zero) velocity is a vector Here we are just giving the ‘x-component’ of velocity, assuming the other components are either zero or irrelevant to our present discussion Velocity is rate of change of displacement:

Gneral Physics I, Syllibus, By/ T.A. Eleyan63 The Position - Time graph The average velocity between two times is the slope of the straight line connecting those two points. average velocity from 0 to 3 sec is positive average velocity from 2 to 3 sec is negative

Gneral Physics I, Syllibus, By/ T.A. Eleyan64 Instantaneous Velocity The velocity at one instant in time is known as the instantaneous velocity and is found by taking the average velocity for smaller and smaller time intervals: The speedometer indicates instantaneous velocity (  t  1 s). On an x vs t plot, the slope of the line tangent to the curve at a point in time is the instantaneous velocity at that time.

Gneral Physics I, Syllibus, By/ T.A. Eleyan65 The average velocity of a particle is defined as the ratio of the displacement to the time interval. The instantaneous velocity of a particle is defined as the limit of the average velocity as the time interval approaches zero.

Gneral Physics I, Syllibus, By/ T.A. Eleyan66 Example : A Particle movies along the x-axis. According to the expression 1- determine the displacement of the particle in the time interval t=0s to t=1s x(0)=0 and x(1)=-2 m 2- Calculate the average velocity in the time interval t=0 to t=1 3- Find the instantaneous velocity of the particle at t=2.5 second At t=2.5s,

Gneral Physics I, Syllibus, By/ T.A. Eleyan67 Acceleration Often, velocity is not constant, rather it changes with time. The rate of change of velocity is known as acceleration. This is the average acceleration. Acceleration is a vector Acceleration is a vector The unit of acceleration is: m/s 2 The unit of acceleration is: m/s 2 positive, negative or zero

Gneral Physics I, Syllibus, By/ T.A. Eleyan68 Instantaneous Acceleration If we wish to know the instantaneous acceleration, we once again let  t  0:

Gneral Physics I, Syllibus, By/ T.A. Eleyan69 The Velocity - Time graph Graphically, acceleration can be found from the slope of a velocity vs. time curve. For these curves, the average acceleration and the instantaneous acceleration are the same, because the acceleration is constant.

Gneral Physics I, Syllibus, By/ T.A. Eleyan70 Example A car moves from a position of +4.0 m to a position of –1.0 m in 2.0 sec. The initial velocity of the car is –4.0 m/s and the final velocity is –1 m/s. (a) What is the displacement of the car? (b) What is the average velocity of the car? (c) What is the average acceleration of the car? Answer: (a)  x = x f – x i = –1.0 m – (+4.0 m) = – 5 m (b) v av =  x/  t = (– 5.0 m)/(2.0 s) = – 2.5 m/s (c)

Gneral Physics I, Syllibus, By/ T.A. Eleyan71 Example : The velocity of a particle moving according to the expression 1) Find the average acceleration in the time interval t=0 to t=2s 2) Determine the acceleration at t=2s At, t=2 then