Warning: 51 pages: I would not print this document! Don t worry it is mostly reading.

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1 AP Physics B Summer Assignment Warning: 51 pages: I would not print this document! Don t worry it is mostly reading. PART ONE: Read the 2 studies of the Force Concept Inventory (FCI) exam pretest found on the University of Maryland and University of Minnesota s websites. Record your thoughts on paper (no more than two pages!) Based on the data presented do you feel the FCI pre-examination is a good indicator of a students success in advanced college level physics? This exam is scheduled for the beginning of next year. I would be sure to review the Physics I notes that are attached to the bottom of the document. PART TWO: MATH PRACTICE 1. The following are ordinary physics problems. You may not know what they all mean, but you don t have to yet. Place the answer in scientific notation when appropriate and show the simplified units in the final answer. a. T s = 2π kg kg s 3 2 = 1 2 K b. = ( k 0)( 2.1 g 1 1 m0/ s) = c. 9 N m ( C)( C) F = = 2 C ( 0.32m) d = R Ω Ω p R P = 2. Problems on the AP exam often deal with variables only. Solve for the variable indicated. 2 2 e. v = v + 2a( s s ) a = o o, h. 1 = = 2 2 mgh mv, v f. T = 2π p, g g = µ I B= r = 2π r o i., g. F = G m m 1 2 g, r r = 2

2 j. 1 = = 2 2 qv mv, v 3. You will be expected to apply basic geometric principles. Solve the following geometric problems. a. What is angle? 30 θ b. How large is? θ 30 o c. The radius of a circle is 5.5 cm, i. What is the circumference in meters? ii. What is its area in square meters? 4 d. What is the area under the curve at the right? Remember the generic triangle to the right, trigonometry, and Pythagorean Theorem? Use them to solve the following problems. Set calculator to degree mode. AP Physics C, Summer Assignment 2

3 = 55 o, c = 32 m; solve for a and b. = 45 o, a = 15 m/s; solve for b and c. C. a = 250 m, b = 180 m; solve for and c. D. a =25 cm, c = 32 cm; solve for b and.

4 5. Most of the quantities in physics are vectors. This makes proficiency in vectors extremely important. A. Find the x- and y-components in the following problems. Direction is specified in degrees measured counterclockwise from east. Your calculator must be in degree mode. a. 89 at 150 o b. 990 at 20 o B. Given two component vectors solve for the resultant vector. Hint: use Pythagorean Theorem to find the hypotenuse, then use inverse tangent to solve for the angle. a. x = 600, y = 400 b. x = -32, y = 16

5 PART 3: Read your syllabus. Please bring any and all questions to the first day of school. This is the most important assignment in this packet. Textbook: Physics: 4 th edition, Wilson/Buffa Grading Policy: Product Points 70% AP Physics B Approved August 2011 Instructor: B. Weaver Tests/Exams: Formal assessments will occur after each of the 5 major AP components. Each assessment will be designed in a similar fashion to the AP exam and consist of two parts: part 1 (45 min) selected response, part 2: (45 min) free response. My goal is to mimic the intensity of the AP exam in each of our assessments. Quizzes: Each week students will be challenged with a free response question or 5 selected response questions dealing with current material. Labs: Lab portfolios will be graded on a bi-quarterly schedule. Process Points 30% Homework Class Presentations Discussions General Description from the College Board: The Physics B course includes topics in both classical and modern physics. Knowledge of algebra and basic trigonometry is required for the course; the basic ideas of calculus may be introduced in connection with physical concepts, such as acceleration and work. Understanding of the basic principles involved and the ability to apply these principles in the solution of problems will be the major goals of the course. Consequently, the course will utilize guided inquiry and student-centered learning to foster the development of critical thinking skills. Physics B will provide instruction in each of the following five content areas: Newtonian mechanics, fluid mechanics and thermal physics, electricity and magnetism, waves and optics, and atomic and nuclear physics. The Physics B course will also include a hands-on laboratory component comparable to introductory college-level physics laboratories, with a minimum of 12 student-conducted laboratory investigations representing a variety of topics covered in the course. Each student will complete a lab notebook or portfolio of lab reports. (

6 Content Outline and Breakdown found on AP Central: I. Newtonian Mechanics 35% A. Kinematics 7% 1. Motion in one dimension 2. Motion in two dimensions B. Newton s laws of motion 9% 1. Static equilibrium (first law) 2. Dynamics of a single particle (second law) 3. Systems of two or more objects (third law) C. Work, energy, power 5% 1. Work and work energy theorem 2. Forces and potential energy 3. Conservation of energy 4. Power D. Systems of particles, linear momentum 4% 1. Center of mass 2. Impulse and momentum 3. Conservation of linear momentum, collisions E. Circular motion and rotation 4% 1. Uniform circular motion 2. Torque and rotational statics F. Oscillations and gravitation 6% 1. Simple harmonic motion 2. Mass on a spring 3. Pendulum and other oscillations 4. Newton s law of gravity 5. Orbits of planets and satellites a. Circular b. General II. Fluid Mechanics and Thermal Physics 15% A. Fluid Mechanics 6% 1. Hydrostatic pressure 2. Buoyancy 3. Fluid flow continuity 4. Bernoulli s equation B. Temperature and heat 2% 1. Mechanical equivalent of heat 2. Heat transfer and thermal expansion C. Kinetic theory and thermodynamics 7% 1. Ideal gases a. Kinetic model b. Ideal gas law

7 2. Laws of thermodynamics a. First law (including processes on pv diagrams) b. Second law (including heat engines) III. Electricity and Magnetism 25% A. Electrostatics 5% 1. Charge and Coulomb s law 2. Electric field and electric potential (including point charges) B. Conductors, capacitors, dielectrics 4% 1. Electrostatics with conductors 2. Capacitors a. Capacitance b. Parallel plate C. Electric circuits 7% 1. Current, resistance, power 2. Steady-state direct current circuits with batteries and resistors only 3. Capacitors in circuits a. Steady state D. Magnetic Fields 4% 1. Forces on moving charges in magnetic fields 2. Forces on current-carrying wires in magnetic fields 3. Fields of long current-carrying wires E. Electromagnetism 5% 1. Electromagnetic induction (including Faraday s law and Lenz s law) IV. Waves and Optics 15% A. Wave motion (including sound) 5% 1. Traveling waves 2. Wave propagation 3. Standing waves 4. Superposition B. Physical optics 5% 1. Interference and diffraction 2. Dispersion of light and the electromagnetic spectrum C. Geometric optics 5% 1. Reflection and refraction 2. Mirrors 3. Lenses V. Atomic and Nuclear Physics 10%

8 A. Atomic physics and quantum effects 7% 1. Photons, the photoelectric effect, Compton scattering, x-rays 2. Atomic energy levels 3. Wave-particle duality B. Nuclear physics 3% 1. Nuclear reactions (including conservation of mass number and charge) 2. Mass energy equivalence Detailed Content and Pacing Guide (30 week breakdown to be determined by testing date) Week 1: Introduction Express the three fundamental quantities of length, mass, and time using the Systeme Internationale, SI, and British engineering units. Develop a sound problem solving procedure using diagram sketching, identifying the appropriate equation(s), writing the working equation, correct substitution, calculating, and checking the final answers. Understand how to convert from British engineering units into SI and from SI into British engineering. Apply dimensional unit analysis to a given physical equation to determine if the relationship is dimensionally correct. Understand the rules for handling significant figures and how to handle them when performing physics calculations. Carry out physics calculations using scientific notation. Perform order-of-magnitude calculations, approximations, and guesstimates. Explain the reasonableness of an answer. Become familiar with the meaning of basic physics terms, various mathematical symbols, Greek letters and Greek prefixes. Express computational results using Greek prefixes. Laboratory Objectives: Demonstrate an understanding of the reliability of a measurement. Use physics data to construct graphs. Find the slope of a graph. Demonstrate an understanding of error analysis. Completed Labs: Error Analysis Measuring with the Vernier Caliper and Micrometer Thickness of a Molecule Week 2: One Dimensional Kinematics

9 Locate points, express angles, express rotation sense, and express distances in a Frame of Reference the Cartesian Coordinate System. Distinguish between displacement and distance and speed and velocity. Relate average speed to distance traveled and time elapsed to solve problems involving such parameters. Define acceleration and suggest means for measuring it. Distinguish between average acceleration and instantaneous acceleration. Write three general kinematics equations that involve the parameters distance, initial velocity, final velocity, acceleration, and time. Use the kinematics equations to solve motion at constant acceleration problems. Write the value of the acceleration due to gravity in both SI and British engineering units. Describe the behavior of an object in free fall when neglecting air resistance. Recognize that the equations of kinematics directly apply to bodies in free fall. Use the quadratic formula to determine the time required for a body projected vertically downward or upward to reach the ground and explain the meaning of the extraneous solution. Calculate the position and velocity at specific times for a body dropped from rest, or projected vertically downward, or projected vertically upwards with some initial velocity. Construct and analyze position and time, speed and time, and acceleration and time graphs for both kinematics and free fall. Interpret graphs relating displacement vs. time. Interpret graphs relating velocity vs. time. Determine the acceleration due to gravity by graphic interpretation of data. Laboratory Objectives: Use physics data to plot graphs. Find the slope of a graph. Realize the meaning of the slope-intercept equation of a straight line. Make displacement and time graphs from experimental data. Make velocity and time graphs from experimental data. Completed Labs Acceleration due to Gravity Graphic Analysis Uniformly Accelerated Motion Accelerated motion on an Air Track Week 3: Vectors in Physics Define scalar and vector and give examples of each. Define vector sum and resultant of two or more vectors. Use the Tip-to-Tail Method and the Parallelogram Method to find the resultant of two vectors. Determine the x and y-components of a given vector by graphical methods. Calculate the x and y-components of a given vector. Define unit vector. Write a vector in unit vector notation.

10 Calculate the magnitude and the direction of a vector when its rectangular components are given. Calculate the resultant of two or more vectors using the Component Method. Illustrate an understanding of relative motion in one and two dimensions. Laboratory Objective: Distinguish between a vector and a scalar. Completed Lab: Vector Resolution on the Force Table Week 4: Two-Dimensional Kinematics Discuss the trajectory of a projectile in the earth s gravitational field. Illustrate graphically how the motion of a horizontally projected baseball compares with that of a baseball dropped from rest. Illustrate with diagrams how the vertical motion of a baseball thrown at any angle is similar to the motion of a baseball thrown vertically. Predict the position and velocity of a projectile as a function of time when the projection angle and initial speed are given. Predict the range, maximum altitude, and time of flight for a given projectile when the initial speed and the angle of projection are given. Laboratory Objectives: Describe the motion of a projectile in two-dimensional space. Measure and compare the horizontal and vertical displacements of a projectile. To predict and verify experimentally the range of a projectile. To show that the trajectory of a projectile is parabolic. Completed Labs: Projectile Motion Two-Dimensional Motion on the Air Table Week 5: Newton s Laws of Motion Demonstrate by definition and example a clear understanding of the distinction between mass and weight. Define the units newton, pound, and slug and to be able to express them in SI and English (British engineering system) units. Draw a free-body diagram for a body or a system of bodies in motion with a constant acceleration, set the resultant force equal to the total mass times the acceleration, and solve for the unknown parameters. Demonstrate a clear understanding of Newton s First Law of Motion using examples. Demonstrate a clear understanding of Newton's Second Law of Motion using examples. Relate Newton s First and Second Laws to kinematics. State specific examples to illustrate an understanding of Newton's Third Law of Motion. Analyze the motion of both a non-accelerating and an accelerating elevator. Laboratory Objectives: Determine the relationship between the inertial mass of a body and its gravitational mass.

11 Determine experimentally the relationship between force, mass, and acceleration. Interpret and analyze a Force vs. Acceleration experimental graph. Design and conduct experiments that would show the variations in acceleration caused by a change in applied force on a given mass. Design and conduct experiments that would show the variations in acceleration caused by a change in the mass that is being accelerated. Completed Labs: The Inertial Balance Newton s Second Law of Motion Newton s Second Law on the Air Track Week 6: Applications of Newton s Laws Discuss the forces of kinetic and static friction and suggest a means of measuring them. Demonstrate by definition and example an understanding of Newton's First Law of Motion involving friction. Demonstrate by definition and example an understanding of Newton's Second Law of Motion involving friction. Make free-body diagrams for bodies or a systems of bodies in motion with a constant acceleration, set the resultant force equal to the total mass times the acceleration, and solve for the unknown parameters. Relate the force in a spring to the stretching or compressing distances. State Hooke s Law and give examples. Give a mathematical equation for Hooke s Law. Explain the meaning of the negative sign in the equation expressing Hooke s Law. Define the limiting angle of repose for the two surfaces involved along an inclined plane. Analyze the motion of a body accelerating on an inclined plane with friction. Determine the tension and acceleration on an Atwood machine. State the conditions that are necessary for uniform circular motion. Understand how acceleration is possible without a change in speed. Apply understandings of centripetal force to examples of motion in a horizontal circle. Laboratory Objectives: Demonstrate that the frictional force is independent of the area of contact Measure coefficients of static and kinetic friction using a wooden block on an inclined plane. Show a given set of surfaces that the coefficient of static friction is greater than for kinetic friction. Experimentally determine the spring constant of a spring by measuring the elongation of the spring for a given applied force. Completed Labs: Atwood s Machine Static and Kinetic Friction Hooke s Law Centripetal Force

12 Week 7: Work and Kinetic Energy Define physical work. State the conditions necessary for the performance of physical work. Define the joule as work or energy units. Write a mathematical statement for calculating the work done by a given force and demonstrate that the equation is dimensionally correct. Recognize that the area beneath Force vs. Distance curve is work done over the distance interval. Define kinetic energy. Demonstrate by example and by experiment the relationship between the performance of work and the corresponding change in kinetic energy. Calculate the kinetic energy of a body when its mass or weight is given. Discuss the Work-Energy Theorem and express it as a mathematical statement. Define power in both SI and British engineering units. Define and compare the units of the watt, kilowatt, and horsepower as they are used to measure power. Demonstrate by example an understanding of the concept of power. Laboratory Objectives: Interpret and analyze a Force vs. Elongation experimental graph for a Hooke s Law experiment. Design an experiment to demonstrate the use of the concept of power and a procedure for computation. Design and conduct an experiment to determine your horsepower output running a flight of stairs. Completed Labs: Human Horsepower on the Stairs. Work and Energy Using the Inclined Plane Hooke s Law Week 8: Potential Energy and Conservative Forces Define potential energy. Define gravitational potential energy. Write an equation that will determine the gravitational potential energy of a known mass or weight relative to a given location in space. State and write the Law of Conservation of Mechanical Energy. Include kinetic, spring potential, gravitational potential energies and work due to friction. Discuss the meaning of the expression conservative force. Understand the significance of a conservative force. Discuss the meaning of the expression nonconservative force. Understand that the gravitational field is a conservative field. Understand that the spring force is a conservative force. Understand the relationship between work, energy, and power. Laboratory Objectives:

13 Determine the potential energy of a compressed or elongated spring. Show that the change in gravitational potential energy of a mass-spring system is equal to the change in spring potential energy. Completed Lab: Conservation of Elastic and Gravitational Potential Energy Week 9: Linear Momentum and Collisions Define impulse and momentum. Discuss the relationships between Newton s Second Law, momentum, and impulse. Derive an equation illustrating the relationship of a change in momentum to the impulse. Recognize that the area beneath a force vs. time curve is impulse and that impulse is the change in momentum. Distinguish by definition between elastic, inelastic and completely inelastic collisions. Relate energy changes in elastic and completely inelastic collisions. Understand that momentum and kinetic energy are conserved in elastic collisions. Apply the Law of Conservation of Linear Momentum to problems involving colliding bodies. Use energy and momentum principles to discuss what occurs after an elastic collision has occurred. Predict the velocities of two colliding bodies after impact when the masses and initial velocities are given. Apply the Law of Conservation to solve recoil problems. Predict the scattering angles after a two-dimensional elastic collision. Understand the distinction between the center of mass and center of gravity of a system. Calculate the location of the center of mass in simple systems. Laboratory Objectives: Design an experiment to demonstrate the validity of the Law of Conservation of Linear Momentum. Design and conduct an experiment to study two-dimensional scattering collisions. Experimentally determine the center of mass of a rigid body. Design an experiment to demonstrate the validity of the Law of Conservation of Linear Momentum. Experimentally show that the total momentum of a system is the same before and after collision. Experimentally show that the impulse applied to a momentum cart equals the change in momentum of the cart. Interpret and analyze a force vs. time curve constructed from experimental data. Completed Labs: Conservation of Linear Momentum on the Air Track Ballistic Pendulum Momentum Study with Momentum Carts Momentum and Collisions in Two Dimensions on the Air Table Two Dimensional Scattering Collisions

14 Week 10: Rotational Kinematics and Energy Define and illustrate the degree, the radian, and the revolution as angular measure and be able to convert between them. State angular kinematics concepts in terms of linear kinematics terms and expressions. Define angular velocity and angular acceleration and describe procedures for measuring and expressing them. Show that all circular motion equations are dimensionally correct. Define the moment of inertia of a body and describe how this quantity and angular speed of a body determine the rotational kinetic energy. Apply the Law of Conservation of Mechanical Energy to rotating and rolling bodies. Laboratory Objective: Design an experiment to study the behavior of hoops, cylinders, and spheres rolling down an inclined plane. Completed Labs: Rotational Motion Rolling Bodies Down the Inclined Plane Week 11: Rotational Dynamics and Static Equilibrium Determine the factors that cause a body to rotate when force is applied to it. Determine the conditions necessary for a body to be in rotational equilibrium. Define equilibrium. State whether the resulting torque is positive or negative by convention when a force is applied to an extended body pivoted at some point. Calculate the resultant torque about any point given the magnitude and the position of the forces applied to an extended body. Demonstrate an understanding of Static Equilibrium through example. Discuss the difference between Static Equilibrium, Static Rotational, Equilibrium, Translational Equilibrium, and Rotational Equilibrium. Make a mathematical statement of the First and Second Conditions for Equilibrium and give several physical examples. Apply the First Condition of Equilibrium to write two equations involving components of given vectors along the x-axis and the y-axis of a frame of reference. Solve for unknown forces in a static system by applying the First and the Second Conditions of Equilibrium. Solve simultaneous equations derived from the First and Second Conditions for unknown forces. Clearly demonstrate an understanding of torque and apply this understanding to various applications. Define angular momentum and give at least two examples illustrating application. State and apply the Law of Conservation of Angular Momentum. Define rotational work and rotational power using torque and derive equations for their computation in applied situations.

15 Laboratory Objectives: Experimentally locate the center of mass of a body. Design and conduct an experiment to study torque and force methods of solving problems involving bodies in static equilibrium. Experimentally study the forces and torques involved in the equilibrium of a simple crane. Design and conduct an experiment to measure the forces in a ladder. Design an experiment to determine an unknown weight using a meterstick, a support, and a known weight. Completed Labs: Parallel Forces Center of Mass and Equilibrium The Boom Crane The Ladder Week 12: Gravity Use Newton's Universal Law of Gravitation to derive the acceleration due to gravity for the surface of the earth and for the surfaces of other planets when the radii and the masses of the planets are given. Use Newton's Universal Law and Newton's Second Law of Motion to express weight for any location in the universe. Determine mass from weight or weight from mass where a value for the acceleration due to gravity is known. Describe an experiment that would measure the Universal Gravitational Constant. Find the acceleration due to gravity for various positions on the surface and above the surface of the earth. Apply centripetal force and gravitation to satellite motion. State Kepler s Three Laws of Planetary Motion. Use Kepler s Third Law to relate the radius of an orbit to its period. Calculate the acceleration due to gravity on the surfaces of various planets and moons. Calculate escape speeds from the surfaces of various planets and moons. Laboratory Objectives: Design a laboratory experiment to measure the mass of an unknown object moving in a horizontal circle of radius R. Be able to plot and analyze a force vs. period and a force vs. period squared graph for a body moving in a horizontal circle. Completed Lab: Centripetal Force Week 13: Oscillations About Equilibrium Describe oscillatory and simple harmonic motion (SHM) through examples. Define the parameters of oscillatory motion. Describe the relationships between force and displacement in oscillatory motion.

16 Describe and illustrate how the magnitude and direction of velocity varies as a function of time in oscillatory motion. Describe and illustrate how the magnitude and direction of acceleration varies as a function of time in oscillatory motion. Calculate the frequency or period when the position and acceleration of an object are given at any instant during oscillatory motion. Determine the period and total energy of a simple pendulum undergoing oscillatory motion. Laboratory Objectives: Design and conduct an experiment using different mass and springs having different spring constants to demonstrate the dependence of frequency in simple harmonic motion upon mass and spring constant. Design and conduct an experiment to measure the acceleration due to gravity with a simple pendulum. Experimentally evaluate potential energy and kinetic energy of an oscillating spring-mass system as a function of displacement. Analyze the potential energy and kinetic energy vs. displacement graphs of an oscillating pendulum. Completed Labs: The Simple Pendulum The Acceleration Due to Gravity on the Pendulum Simple Harmonic Motion on a Spring The Compound Pendulum Week 14: Waves and Sound Distinguish between the physiological and physical definitions of sound. State ways of approximating the speed of sound in liquids and gases knowing the speed of sound in air. Describe and illustrate transverse and longitudinal wave motion. Describe, relate, and illustrate the meaning of frequency, speed, and wavelength as they apply to wave motion. Understand the principles of reflection, refraction, dispersion, and diffraction as they relate to mechanical waves. Understand the mathematics of traveling and standing waves. Calculate the intensity level in decibels for a sound wave whose intensity is given in watts per square meter. Relate the energy of a sound wave to its intensity. Distinguish between harmonics and overtones as they apply to a vibrating system with fixed end points. Use the superposition principle and determine the resultant wave when two waves merge. Use the Doppler effect to predict the apparent change in sound frequency that occurs as a result of relative motion between a source and an observer. Define resonance and site examples. Discuss the origin and significance of beats.

17 Laboratory Objectives: Experimentally measure the wavelength of a sound wave using a tuning fork and resonance tube. Determine the speed of sound in air from the wavelength measured in the resonance tube and the frequency of the tuning fork. Compare the experimental speed of sound to the accepted value. Suggest possible experiments to study and measure the frequency and wavelengths of sound waves in air. Completed Labs: Ripple Tank Speed of Sound in Air: Resonance Tube Standing Waves in Strings Week 15: Fluids Define the properties of fluids. In terms of structure, distinguish the differences between solids, liquids, gases, and plasma. Distinguish between mass density, weight density, and specific gravity. Define the concept of pressure and that of absolute pressure. State and apply Pascal s Principle to physical situations. State Archimedes Principle and its relationship it to buoyancy. Apply Archimedes s Principle to floating and submersed bodies in a fluid. Understand the use of the equation of continuity and its application to ideal fluids. Understand the effect of friction on fluid flow through a horizontal pipe of circular cross section. Understand Bernoulli s equation and its application to ideal fluids. Demonstrate an understanding of the workings of an airfoil. Discuss surface tension and viscosity. Write Poiseuille s equation and give several examples. Laboratory Objectives: Determine the density of several solid objects whose density is greater than water. Determine the density of a solid material whose density is less than water. Determine the density of several liquids. Design an experiment to measure buoyant effects in fluids. Design and conduct an experiment to study and measure liquid viscosity. Completed Labs: Density and Specific Gravity Archimedes Principle Viscosity Week 16: Temperature and Heat Objectives: Understand the distinction between temperature and thermal energy.

18 Demonstrate an understanding of the difference between a specific temperature and a temperature interval. Demonstrate working skill with the Celsius, Fahrenheit, Kelvin, and Rankine temperature scales and the inner conversion between them. Predict the change in the length of a metal rod of known length and material as the rod is heated through a known temperature range. Develop a method to determine a relationship for the change in area of a sheet of a material as it is heated or cooled over a given temperature range. Predict the volume overflow when a container of known volume and material filled with a given liquid is heated over a given temperature interval. Understand that heat is an energy form. Represent the heat gained or lost in a given process in terms of calories, joules, and BTU's. Give two or more examples illustrating the distinction between quantity of heat and the temperature of a material. Demonstrate by example and by experiment an understanding of specific heat capacity and its distinction from heat capacity. Explain practical advantages or disadvantages of metals with large specific heat capacities. Demonstrate by two examples in each case the application of heat transfer by radiation, conduction, and convection currents. Laboratory Objectives: Design and conduct an experiment to determine the coefficient of linear expansion of a metal. Design and conduct an experiment to measure the specific heat of metals and liquids. Completed Labs: Cooling Curve Linear Thermal Expansion Specific Heat of Solids and Liquids Week 17: Phases and Phase Changes Distinguish between an ideal gas and a real gas, giving reasons why some gases closely approximate the ideal condition. Demonstrate by example an understanding of (1) Boyle's Law, (2) The Law of Charles, and (3) the Ideal Gas Law. Explain Boyle s and Charles Laws in terms of the Kinetic Theory of Gases. Apply the Law of Conservation of Energy to a given process in order to determine unknown parameters such as mass, specific heat, temperature, or latent heats of fusion or vaporization. Describe the changes that take place during phase changes in terms of atomic and molecular structure of matter. Laboratory Objectives: Design and conduct an experiment to measure the latent heat of fusion and the latent heat of vaporization for a given substance.

19 Design an experiment that will measure the vapor pressure of a liquid at a given temperature. Construct and interpret P, V, and T graphs for a given gas. Completed Labs: Boyle s Law Heat of Fusion Heat of Vaporization Week 18: The Laws of Thermodynamics Define a thermodynamic system. Differentiate between state and phase. Give two examples in which the internal energy of a system can be changed. State the First Law of Thermodynamics, give two examples in which the law is demonstrated, and represent the first law mathematically. Define and give illustrated examples of each of the following thermodynamic processes: (a) adiabatic, (b) isochoric, (c) isothermal, and (d) isobaric. Explain the significance of a P-V diagram in describing (a) adiabatic, (b) isochoric, (c) isothermal, and (d) isobaric thermodynamic processes. State the Second Law of Thermodynamics. Define the Entropy of a system. Explain the operation and the limitations of the efficiency of a heat engine. Determine the efficiency of a heat engine in terms of heat input and heat output. Determine the efficiency of a heat engine in terms of input temperature and output temperature. Differentiate between Carnot Efficiency and actual efficiency as applied to heat engines. Laboratory Objectives: Devise an experiment to measure the work done in an isothermal process. Interpret and make calculations from PV diagrams. Completed Lab: Phase Change Week 19: Electric Charges, Forces, and Fields Discuss the nature of electrical charge. Understand charge quantization. Recognize that all charges are multiple of the fundamental unit of charge. Describe and illustrate Millikan's Oil-Drop Experiment and its significance in the history of the development of physics. Demonstrate that charge is conserved. State the Law of Conservation of Electrical Charge. Distinguish between an insulator and a conductor using examples. State Coulomb's Law and express it in terms of an equation. Apply Coulomb's Law to physical situations involving systems of point charges using the principle of superposition

20 Define the electrical field in terms of an isolated point charge. Calculate the magnitude and the direction of the force that would act on a test charge placed at a given point in an electric field. Write a mathematical expression to determine the electrical field at a given point in space. Calculate the electric field of a system of charge distributions using then principle of superposition. Describe the behavior of a charged particle in a parallel plate capacitor. Explain how to charge a body by induction. Laboratory Objectives: Discover the electrical properties of conductors and insulators. Devise an experiment to measure the charge on an electron. Determine the shape of the electrical field around a conductor. Sketch the electric field pattern between point charges and charged objects. Completed Labs: Mapping the Electrical Field Charge on an Electron Week 20: Electrical Potential and Energy Potential Energy Distinguish by definition and example between potential energy, electric potential, and electric potential difference. Define the volt. Define the electron volt, ev, and be able to express energy in terms of this unit. Calculate the potential energy of a known charge at a given distance from another known charge and state whether the potential energy is positive or negative. Determine the electric potential at any point due to a charge of known magnitude. Calculate the electric potential at a point in the neighborhood of a number of isolated charges. Determine the force exerted on a given charge placed between two oppositely charged parallel plates of known separation and potential difference. Define the dielectric strength of a material and describe the part it plays in limiting the charge that can be placed on a conductor. Discuss the effects of the size and the shape of a conductor on its ability to store a charge. Derive a relationship between applied voltage, capacitance, and total charge. Find the capacitance of a parallel-plate capacitor when the area of the plates is given, and they are separated by a medium of a known dielectric constant. Define and calculate the energy of a charged capacitor. Laboratory Objective: Experimentally determine charge and voltage relationships for a parallel plate capacitor. Completed Labs: Equipotentials and Electric Fields Capacitance Week 21: Electric Current and Direct-Current Circuits

21 Define the ampere as the unit of electrical current. Distinguish between conventional flow and electron flow. State Ohm s Law for electrical components. Define the unit of resistance, the ohm. Calculate the resistance across a bank of resistors in series, parallel, and combined. Discuss emf and its role in DC electrical theory. Distinguish between emf and potential difference. Define and describe voltage, current, and equivalent resistance for resistors connected in series, parallel, and combined. State Ohm's Law for an entire electrical circuit and apply it to the solution of electrical problems involving internal battery resistance and total resistance of the circuit. Calculate the total resistance of an entire DC circuit. Compute power loss in a given DC circuit. Determine the terminal voltage, given the emf of a battery, its internal resistance, and the load resistance. Determine the potential drop across a resistance carrying a given current. Define the factors that determine the resistance of a given wire. Calculate the resistance of a wire given its resistivity, length, and radius. Relate the potential difference across a resistor carrying a current to its energy loss. Define the watt as the unit of electrical power. Determine the power loss across a given current carrying resistance. Write and apply Kirchhoff's Rules for electrical networks in the determination of unknown currents. Analyze multiloop circuits using Ohm s Law and Kirchhoff s Rules. Calculate the equivalent capacitance of a number of capacitors arranged in (1) series, (2) parallel, and (3) series and parallel combination. Understand how to use ammeters, voltmeters, galvanometers, and the Wheatstone bridge. Laboratory Objectives: Experimentally, demonstrate Ohm's Law with a voltmeter, an ammeter, a rheostat, a source of emf, and appropriate lead wires and draw a schematic diagram of an electrical set-up, using appropriate symbols for the electrical equipment used. Design an experiment to measure the resistivity of a conductor. Design an experiment to measure the power loss across a resistance. Design and conduct an experiment using two loops, resistors in each loop, and several seats of emf. Design and conduct an experiment to find the resistance of unknown resistors using the Wheatstone Bridge. Experimentally determine charge and voltage relationships for capacitors in series, parallel, and combined networks. Design and conduct an experiment to measure the time constant in an RC circuit. Completed Labs: Ohm s Law Kirchhoff s Rules Wheatstone Bridge

22 RC Time Constant Week 22: Magnetism Define the tesla. Discuss the basic features and properties of the earth s magnetic field. Write the basic law of magnetic force and apply it to physical situations. Use the right-hand-rule in determining the direction of magnetic forces on a positive particle. Explain the magnetic field in terms of the force acting on a charged particle. Discuss the motion in circular orbits of charged particles in uniform magnetic fields. Understand that the provider of the centripetal force on a charged particle is the magnetic field. Understand the operation and use of the mass spectrograph. Find the force on a current-carrying wire placed in a known magnetic field. Determine the magnetic field at a known distance from a current carrying wire. Find the magnetic field at the center of a current loop or coil of N-turns. Calculate the magnetic field in the interior of a solenoid and a toroid of N-turns. Define permeability and the role it plays in defining the magnetic field in magnetic materials. Define relative permeability. Use the right-hand-rule in determining the direction of magnetic field about a currentcarrying wire. Calculate the magnetic force on a current-carrying wire. Calculate the magnetic torque on a coil of area A having N turns of wire carrying current I when it is orientated in a known magnetic field of strength B. Determine the torque on a solenoid that is free to rotate in a known magnetic field. Laboratory Objectives: Explain how to determine the direction of a magnetic field using a compass. Map the magnetic field around bar and horseshoe magnets. Design and conduct an experiment to measure the magnetic field around a current carrying wire. Completed Labs: Mapping Magnetic Fields Magnetic Fields Around Current-Carrying Wires Week 23: Magnetic Flux and Faraday s Law of Induction Define magnetic flux giving its units. Explain how changing magnetic flux through a single loop creates induced emf s. Predict the polarity of an induced emf. Describe ways in which magnetic flux can change. Discuss induced emf and current. Write Faraday s Law of Induction and apply it to induced emf through a loop. State Lenz s Law and use it to determine the direction of an induced current.

23 Define inductance. Describe the main components of a DC motor and generator. Calculate the instantaneous and maximum emf and current generated by a simple generator. Explain how back emf reduces the net voltage delivered by a generator. Find the characteristic time interval in an RL circuit. Describe how to find the energy stored in an inductor. Laboratory Objectives: Determine the direction of the emf induced in a loop of wire moving through a magnetic field. To study the relationship between a magnetic field and the electric potential that can be induced by the field. Completed Labs: Induced Electrical Potential Simple Motors Week 24: Electromagnetic Waves Discuss various ways the speed of light was experimentally determined. Understand the behavior of electromagnetic waves. Differentiate between radio, TV, microwaves, infrared, visible, ultraviolet, x-rays, and gamma rays as forms of electromagnetic radiations. Discuss the sources of radio, TV, microwaves, infrared, visible, ultraviolet, x-rays, and gamma rays. Describe the phenomenon of polarization. Laboratory Objectives: Design and conduct an experiment to study optically active liquids. Design and conduct an experiment to study polarization by scattering. Design and conduct an experiment to study polarization by reflection. Completed Lab: Polarization of Light Week 25: Geometrical Optics Define and discuss the concepts of wave fronts and rays. Understand the law of reflection. Describe the characteristics of plane mirrors. Demonstrate an understanding of the nature of the images formed by plane mirrors. Distinguish between virtual and real images. Define magnification in terms of image height and object height. Distinguish between plane mirrors and spherical mirrors. Understand the characteristics of converging and diverging mirrors. Describe the images formed by converging and diverging mirrors. Use ray-tracing techniques to construct images formed by spherical mirrors. Define the focal length of a spherical mirror.

24 Calculate the magnification of a spherical mirror. Define index of refraction. State Snell s law and use it to predict the path of a light ray as it passes from one medium into another. Calculate the speed of light in a medium given the index of refraction. Determine the wavelength of light in a medium given the index of refraction. Understand total internal reflection and the application of fiber optics. Define critical angle. Use ray-tracing techniques to construct images formed by lenses. Distinguish between converging and diverging lenses. Understand the characteristics of converging and diverging lenses. Describe the images formed by converging and diverging lenses. Define the focal length of a lens. Use the thin lens equation to solve problems. Calculate the magnification of a thin lens. Understand the sign convention for thin lens calculations. Discuss lens aberration. Apply the thin-lens equation to solve for unknown parameters related to the construction of lenses. Discuss dispersion and its effects. Laboratory Objectives: Investigate the positions and characteristics of images produced by plane and curved mirrors. Demonstrate that rays traveling from air into a transparent medium are refracted at the boundary. Determine the index of refraction of a medium from direct measurement of incident and refracted angles. Design and conduct an experiment to determine the velocity of light in a medium. Observe the positions and characteristics of images produced by convex and concave lenses. Design an experiment that would give the magnification of a given lens for a given distance. Completed Labs: Mirror Optics Reflection, Refraction, and Snell s Law Index of Refraction of Glass Focal Length of Lenses Week 26: Optical Instruments Describe the optics of the human eye. Discuss some common visual defects and explain how they may be corrected. Distinguish between lateral and angular magnification in the microscope. Describe the simple and the compound microscope and their magnifications. Distinguish between refractive and reflecting telescopes.

25 Discuss the advantages to refractive and reflecting telescopes. Explain with diagrams the operation of a simple and compound microscope, a refractive and reflective telescope, a camera, and a projector. Determine the focal length of a lens system. Predict mathematically the nature, size, and location of images formed by optical systems. Laboratory Objective: Design an experiment to study the magnification of a simple and a compound microscope. Completed Labs: Simple Magnifier Simple Telescope Week 27: Physical Optics: Interference and Diffraction Discuss Young s experiment. Explain how the phenomena of diffraction and interference demonstrate the wave nature of light. Give graphic examples of constructive and destructive interference. Describe how thin films produce colorful displays. Understand the interference patterns produced by light reflecting off the two surfaces of a thin film. Understand how anti-reflective coatings work. Discuss Newton s rings. Define diffraction. Discuss single-slit diffraction. Explain how diffraction gratings are used in spectroscopy. Derive the diffraction grating equation. Apply the diffraction grating equation to solve problems involving diffraction gratings. Define the phenomenon of polarization. Laboratory Objectives: Set up a model of Young s experiment in the lab to determine the wavelength of a given source of monochromatic light. Design and conduct an experiment using a diffraction grating to measure the wavelength of a monochromatic light source such as a laser pen. Completed Labs: Diffraction of Light Interference of Light Week 28: Quantum Physics Discuss the Einstein mass-energy relationship and use it to the energies released in mass to energy conversions. Calculate the energies involved in pair annihilation using the Einstein mass-energy equation.

26 Define blackbody radiation. Discuss the Ultraviolet Catastrophe. Use the Planck equation to calculate the energies associated with frequency and wavelength of light. Discuss the role of photons as energy carriers. Describe the photoelectric effect. Use the Einstein photoelectric equation to calculate energies of photoelectrons. Relate the photoelectric effect to stopping potential and threshold or cutoff frequency. Discuss the importance of the Compton effect. Discuss the de Broglie hypothesis and state the circumstances under which the wave nature of matter is observed. Calculate the wavelengths of matter waves. Explain the Davisson and Germer experiment. Discuss the Heisenberg Uncertainty Principle and its role in the microscopic and macroscopic worlds. Laboratory Objectives: Experimentally study the bright line spectra of several elements with a spectroscope Calibrate a spectroscope. Completed Lab: Spectroscope and Atomic Spectra Week 29: Atomic Physics Demonstrate an understanding of the Plum Pudding Model and the Rutherford model of the atom. Discuss the Rutherford, Geiger, and Marsden scattering experiment. Demonstrate with appropriate diagrams an understanding of emission spectra. Sketch diagrams for the Lyman, Balmer, Paschen, Brackett, and Pfund Series. Write Bohr s First Postulate and use it to verify standing de Broglie waves. Write and illustrate the meaning of Bohr s Second Postulate. Calculate the energy emitted per photon per Bohr orbit quantum jump. Discuss the quantum mechanical nature of the hydrogen atom. Discuss the role of quantum numbers in atomic structure Predict the structure of various atoms. Discuss the production of continuous and characteristic x-rays. Calculate voltages required to produce x-rays in an x-ray tube. Laboratory Objectives: Determine the average spacing between the lines of a diffraction grating by using known wavelengths. Measure the wavelengths of the visible spectrum of helium. Completed Lab: Diffraction Grating Measurement of the Wavelength of Light Week 30: Nuclear Physics and Nuclear Radiation

27 Discuss the Thomson and Rutherford atomic models. Discuss or write statements demonstrating an understanding mass, charge, and size of a nucleus. Write nuclear symbols. Determine proton and neutron populations in nuclei. Understand of the equivalence of mass and energy by interchanging kilograms, atomic mass units, joules and electron volts. Discuss radioactivity relating alpha and beta particles and gamma rays. Write a brief description of alpha particles, beta particles, and gamma rays, listing their properties. Demonstrate an understanding of radioactive decay complete with balanced general and specific equations. Calculate the activity and half-life of a given radioactive isotope. Discuss the binding energy curve. Write balanced nuclear decay equations. Calculate Q-values for alpha decay. Calculate the Q-values for elementary nuclear reactions. Distinguish between nuclear fission and fusion. Discuss the design and operation of a nuclear reactor. Laboratory Objectives: Demonstrate that the number of nuclei not yet decayed and the rate of decay are exponential. Calculate the decay constant. Calculate the half-life. Use three-cycle semi log graph paper to plot the number counts vs. time. Analyze and discuss the number of counts vs. time graph. Completed Labs: Radioactive Decay Simulated Radioactive Decay using Dice Nuclei

28 Review Notes from Physics I (not mandatory) Scalars and Vectors Physics is a mathematical science. The underlying concepts and principles have a mathematical basis. Throughout the course of our study of physics, we will encounter a variety of concepts that have a mathematical basis associated with them. While our emphasis will often be upon the conceptual nature of physics, we will give considerable and persistent attention to its mathematical aspect. The motion of objects can be described by words. Even a person without a background in physics has a collection of words that can be used to describe moving objects. Words and phrases such as going fast, stopped, slowing down, speeding up, and turning provide a sufficient vocabulary for describing the motion of objects. In physics, we use these words and many more. We will be expanding upon this vocabulary list with words such as distance, displacement, speed, velocity, and acceleration. As we will soon see, these words are associated with mathematical quantities that have strict definitions. The mathematical quantities that are used to describe the motion of objects can be divided into two categories. The quantity is either a vector or a scalar. These two categories can be distinguished from one another by their distinct definitions: Scalars are quantities that are fully described by a magnitude (or numerical value) alone. Vectors are quantities that are fully described by both a magnitude and a direction. The remainder of this lesson will focus on several examples of vector and scalar quantities (distance, displacement, speed, velocity, and acceleration). As you proceed through the lesson, give careful attention to the vector and scalar nature of each quantity. As we proceed through other units at The Physics Classroom Tutorial and become introduced to new mathematical quantities, the discussion will often begin by identifying the new quantity as being either a vector or a scalar. Scalar: Scalars are used to describe one dimensional quantities, that is, quantities which require only one number to completely describe them. Examples of scalar quantities are: Temperature Time Speed Mass Location Along a Line (1D) Vector: Vectors are used to describe multi-dimensional quantities. Multi-dimensional quantities are those which require more than one number to completely describe them. Vectors, unlike scalars, have two characteristics, magnitude and direction. Examples of vector quantities are: Location in a Plane (2D) Location in Space (3D) Velocity Acceleration Force The Meaning of Shape for a p-t Graph Our study of 1-dimensional kinematics has been concerned with the multiple means by which the motion of objects can be represented. Such means include the use of words, the use of diagrams, the use of numbers, the use of equations, and the use of graphs. Lesson 3 focuses on the use of position vs. time graphs to describe motion. As we will learn, the specific features of the motion of objects are demonstrated by the shape and the slope of the lines on a

29 position vs. time graph. The first part of this lesson involves a study of the relationship between the shape of a p-t graph and the motion of the object. To begin, consider a car moving with a constant, rightward (+) velocity - say of +10 m/s. If the position-time data for such a car were graphed, then the resulting graph would look like the graph at the right. Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph. Now consider a car moving with a rightward (+), changing velocity - that is, a car that is moving rightward but speeding up or accelerating. If the position-time data for such a car were graphed, then the resulting graph would look like the graph at the right. Note that a motion described as a changing, positive velocity results in a line of changing and positive slope when plotted as a positiontime graph. The position vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - are depicted as follows. Constant Velocity Positive Velocity Positive Velocity Changing Velocity (acceleration) The Importance of Slope The shapes of the position versus time graphs for these two basic types of motion - constant velocity motion and accelerated motion (i.e., changing velocity) - reveal an important principle. The principle is that the slope of the line on a position-time graph reveals useful information about the velocity of the object. It is often said, "As the slope goes, so goes the velocity." Whatever characteristics the velocity has, the slope will exhibit the same (and vice versa). If the velocity is constant, then the slope is constant (i.e., a straight line). If the velocity is changing, then the slope is changing (i.e., a curved line). If the velocity is positive, then the slope is positive (i.e., moving upwards and to the right). This very principle can be extended to any motion conceivable.

30 Consider the graphs below as example applications of this principle concerning the slope of the line on a position versus time graph. The graph on the left is representative of an object that is moving with a positive velocity (as denoted by the positive slope), a constant velocity (as denoted by the constant slope) and a small velocity (as denoted by the small slope). The graph on the right has similar features - there is a constant, positive velocity (as denoted by the constant, positive slope). However, the slope of the graph on the right is larger than that on the left. This larger slope is indicative of a larger velocity. The object represented by the graph on the right is traveling faster than the object represented by the graph on the left. The principle of slope can be used to extract relevant motion characteristics from a position vs. time graph. As the slope goes, so goes the velocity. Slow, Rightward(+) Constant Velocity Fast, Rightward(+) Constant Velocity Consider the graphs below as another application of this principle of slope. The graph on the left is representative of an object that is moving with a negative velocity (as denoted by the negative slope), a constant velocity (as denoted by the constant slope) and a small velocity (as denoted by the small slope). The graph on the right has similar features - there is a constant, negative velocity (as denoted by the constant, negative slope). However, the slope of the graph on the right is larger than that on the left. Once more, this larger slope is indicative of a larger velocity. The object represented by the graph on the right is traveling faster than the object represented by the graph on the left. Slow, Leftward(-) Constant Velocity Fast, Leftward(-) Constant Velocity As a final application of this principle of slope, consider the two graphs below. Both graphs show plotted points forming a curved line. Curved lines have changing slope; they may start with a very small slope and begin curving sharply (either upwards or downwards) towards a large slope. In either case, the curved line of changing slope is a sign of accelerated motion (i.e., changing velocity). Applying the principle of slope to the graph on the left, one would conclude that the object depicted by the graph is moving with a negative velocity (since the slope is negative ). Furthermore, the object is starting with a small velocity (the slope starts out with a small slope) and finishes with a large velocity (the slope becomes large). That would mean that this object is moving in the negative direction and speeding up (the small velocity turns into a larger velocity). This is an example of negative acceleration - moving in the negative direction and speeding up. The graph on the right also depicts an object with negative velocity (since there is a negative slope). The object begins with a high velocity (the slope is initially large) and finishes with a small velocity (since the slope becomes smaller). So this object is moving in the negative direction and slowing down. This is an example of positive acceleration.

31 Negative (-) Velocity Slow to Fast Leftward (-) Velocity Fast to Slow The principle of slope is an incredibly useful principle for extracting relevant information about the motion of objects as described by their position vs. time graph. Once you've practiced the principle a few times, it becomes a very natural means of analyzing position-time graphs. How do I determine relative velocity? To determine relative velocity, first choose a frame of reference (a fixed point and a set of directions) and measure velocities relative to the fixed point. Usually the reference point is the ground and the directions are compass points. Relative velocity is the velocity of the object after deducting the velocity of the observer. EXAMPLE 1 An oarsman can row his boat 3 mph in still water. He sets out on the Illinois River, which flows at 5 mph. We are interested in what an observer on shore measures. When the man heads the boat directly downstream and rows as fast as he can, which direction does the observer on shore see the boat going? When the man heads the boat directly downstream and rows as fast as he can, how fast does the observer on shore see the boat going? ANSWER TO EXAMPLE 1 velocity relative to shore = velocity of water relative to shore + velocity of boat relative to water velocity relative to shore = 5 mph [downstream] + 3 mph [downstream] = 8 mph [downstream] EXAMPLE 2 An aircraft has a speed and direction relative to the air (wind) in which it is traveling, and the wind has a speed and direction relative to the ground. How do you determine the velocity of the aircraft relative to the ground? ANSWER TO EXAMPLE 2 Ground Velocity = Airspeed and heading + Wind Speed and heading EXAMPLE 3 An observer sitting on shore sees a canoe traveling 5.0 m/s east, and a sailboat traveling 15.0 m/s west. What is the velocity of the sailboat as observed on the canoe? ANSWER TO EXAMPLE 3

32 R = v object - v observer R = v sailboat - v canoe R = 15.0 [W] [E] R = 15.0 [W] {W} R = 20.0 m/s west The Kinematic Equations There are a variety of quantities associated with the motion of objects - displacement (and distance), velocity (and speed), acceleration, and time. Knowledge of each of these quantities provides descriptive information about an object's motion. For example, if a car is known to move with a constant velocity of 22.0 m/s, North for 12.0 seconds for a northward displacement of 264 meters, then the motion of the car is fully described. And if a second car is known to accelerate from a rest position with an eastward acceleration of 3.0 m/s 2 for a time of 8.0 seconds, providing a final velocity of 24 m/s, East and an eastward displacement of 96 meters, then the motion of this car is fully described. These two statements provide a complete description of the motion of an object. However, such completeness is not always known. It is often the case that only a few parameters of an object's motion are known, while the rest are unknown. For example as you approach the stoplight, you might know that your car has a velocity of 22 m/s, East and is capable of a skidding acceleration of 8.0 m/s 2, West. However you do not know the displacement that your car would experience if you were to slam on your brakes and skid to a stop; and you do not know the time required to skid to a stop. In such an instance as this, the unknown parameters can be determined using physics principles and mathematical equations (the kinematic equations). The kinematic equations are a set of four equations that can be utilized to predict unknown information about an object's motion if other information is known. The equations can be utilized for any motion that can be described as being either a constant velocity motion (an acceleration of 0 m/s/s) or a constant acceleration motion. They can never be used over any time period during which the acceleration is changing. Each of the kinematic equations include four variables. If the values of three of the four variables are known, then the value of the fourth variable can be calculated. In this manner, the kinematic equations provide a useful means of predicting information about an object's motion if other information is known. For example, if the acceleration value and the initial and final velocity values of a skidding car is known, then the displacement of the car and the time can be predicted using the kinematic equations. Lesson 6 of this unit will focus upon the use of the kinematic equations to predict the numerical values of unknown quantities for an object's motion. The four kinematic equations that describe an object's motion are: There are a variety of symbols used in the above equations. Each symbol has its own specific meaning. The symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stands for the acceleration of the object. And the symbol v stands for the velocity of the object; a subscript of i after the v (as in v i ) indicates that the velocity value is the initial velocity value and a subscript of f (as in v f ) indicates that the velocity value is the final velocity value.

33 Each of these four equations appropriately describes the mathematical relationship between the parameters of an object's motion. As such, they can be used to predict unknown information about an object's motion if other information is known. Gravity: How Fast? and How Far? Free-falling objects are in a state of acceleration. Specifically, they are accelerating at a rate of 9.8 m/s/s. This is to say that the velocity of a free-falling object is changing by 9.8 m/s every second. If dropped from a position of rest, the object will be traveling 9.8 m/s (approximately 10 m/s) at the end of the first second, 19.6 m/s (approximately 20 m/s) at the end of the second second, 29.4 m/s (approximately 30 m/s) at the end of the third second, etc. Thus, the velocity of a free-falling object that has been dropped from a position of rest is dependent upon the time that it has fallen. The formula for determining the velocity of a falling object after a time of t seconds is v f = g * t where g is the acceleration of gravity. The value for g on Earth is 9.8 m/s/s. The above equation can be used to calculate the velocity of the object after any given amount of time when dropped from rest. Example calculations for the velocity of a free-falling object after six and eight seconds are shown below. Example Calculations: At t = 6 s v f = (9.8 m/s 2 ) * (6 s) = 58.8 m/s At t = 8 s v f = (9.8 m/s 2 ) * (8 s) = 78.4 m/s The distance that a free-falling object has fallen from a position of rest is also dependent upon the time of fall. This distance can be computed by use of a formula; the distance fallen after a time of t seconds is given by the formula. d = 0.5 * g * t 2 where g is the acceleration of gravity (9.8 m/s/s on Earth). Example calculations for the distance fallen by a freefalling object after one and two seconds are shown below. Example Calculations: At t = 1 s d = (0.5) * (9.8 m/s 2 ) * (1 s) 2 = 4.9 m At t = 2 s d = (0.5) * (9.8 m/s 2 ) * (2 s) 2 = 19.6 m At t = 5 s d = (0.5) * (9.8 m/s 2 ) * (5 s) 2 = 123 m (rounded from m) The diagram below (not drawn to scale) shows the results of several distance calculations for a free-falling object dropped from a position of rest. Newton's First Law

34 In a previous chapter of study, the variety of ways by which motion can be described (words, graphs, diagrams, numbers, etc.) was discussed. In this unit (Newton's Laws of Motion), the ways in which motion can be explained will be discussed. Isaac Newton (a 17th century scientist) put forth a variety of laws that explain why objects move (or don't move) as they do. These three laws have become known as Newton's three laws of motion. The focus of Lesson 1 is Newton's first law of motion - sometimes referred to as the law of inertia. Newton's second law of motion can be formally stated as follows: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. This verbal statement can be expressed in equation form as follows: a = F net / m The above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration. Newton's Third Law F net = m * a A force is a push or a pull upon an object that results from its interaction with another object. Forces result from interactions! Some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces are the result of action-at-a-distance interactions (gravitational, electrical, and magnetic forces). According to Newton, whenever objects A and B interact with each other, they exert forces upon each other. When you sit in your chair, your body exerts a downward force on the chair and the chair exerts an upward force on your body. There are two forces resulting from this interaction - a force on the chair and a force on your body. These two forces are called action and reaction forces and are the subject of Newton's third law of motion. Formally stated, Newton's third law is: For every action, there is an equal and opposite reaction. Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universality of gravity. Newton's place in the Gravity Hall of Fame is not due to his discovery of gravity, but rather due to his discovery that gravitation is universal. ALL objects attract each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance that separates their centers. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as Since the gravitational force is directly proportional to the mass of both interacting objects, more massive objects will attract each other with a greater gravitational force. So as the mass of either object increases, the force of gravitational attraction between them also increases. If the mass of one of the objects is doubled, then the force of gravity between them is doubled. If the mass of one of the objects is tripled, then the force of gravity between them

35 is tripled. If the mass of both of the objects is doubled, then the force of gravity between them is quadrupled; and so on. Since gravitational force is inversely proportional to the square of the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases. If the separation distance between two objects is doubled (increased by a factor of 2), then the force of gravitational attraction is decreased by a factor of 4 (2 raised to the second power). If the separation distance between any two objects is tripled (increased by a factor of 3), then the force of gravitational attraction is decreased by a factor of 9 (3 raised to the second power). The proportionalities expressed by Newton's universal law of gravitation are represented graphically by the following illustration. Observe how the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation. Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below. Momentum Conservation Principle One of the most powerful laws in physics is the law of momentum conservation. The law of momentum conservation can be stated as follows. For a collision occurring between object 1 and object 2 in an isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2.

36 The above statement tells us that the total momentum of a collection of objects (a system) is conserved - that is, the total amount of momentum is a constant or unchanging value. This law of momentum conservation will be the focus of the remainder of Lesson 2. To understand the basis of momentum conservation, let's begin with a short logical proof. Consider a collision between two objects - object 1 and object 2. For such a collision, the forces acting between the two objects are equal in magnitude and opposite in direction (Newton's third law). This statement can be expressed in equation form as follows. The forces act between the two objects for a given amount of time. In some cases, the time is long; in other cases the time is short. Regardless of how long the time is, it can be said that the time that the force acts upon object 1 is equal to the time that the force acts upon object 2. This is merely logical. Forces result from interactions (or contact) between two objects. If object 1 contacts object 2 for seconds, then object 2 must be contacting object 1 for the same amount of time (0.050 seconds). As an equation, this can be stated as Since the forces between the two objects are equal in magnitude and opposite in direction, and since the times for which these forces act are equal in magnitude, it follows that the impulses experienced by the two objects are also equal in magnitude and opposite in direction. As an equation, this can be stated as But the impulse experienced by an object is equal to the change in momentum of that object (the impulsemomentum change theorem). Thus, since each object experiences equal and opposite impulses, it follows logically that they must also experience equal and opposite momentum changes. As an equation, this can be stated as The Total Mechanical Energy As already mentioned, the mechanical energy of an object can be the result of its motion (i.e., kinetic energy) and/or the result of its stored energy of position (i.e., potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME). TME = PE + KE As discussed earlier, there are two forms of potential energy discussed in our course - gravitational potential energy and elastic potential energy. Given this fact, the above equation can be rewritten: TME = PE grav + PE spring + KE The diagram below depicts the motion of Li Ping Phar (esteemed Chinese ski jumper) as she glides down the hill and makes one of her record-setting jumps.

37 The total mechanical energy of Li Ping Phar is the sum of the potential and kinetic energies. The two forms of energy sum up to Joules. Notice also that the total mechanical energy of Li Ping Phar is a constant value throughout her motion. There are conditions under which the total mechanical energy will be a constant value and conditions under which it will be a changing value. For now, merely remember that total mechanical energy is the energy possessed by an object due to either its motion or its stored energy of position. The total amount of mechanical energy is merely the sum of these two forms of energy. And finally, an object with mechanical energy is able to do work on another object. Coulomb's Law The interaction between charged objects is a non-contact force that acts over some distance of separation. Charge, charge and distance. Every electrical interaction involves a force that highlights the importance of these three variables. Whether it is a plastic golf tube attracting paper bits, two like-charged balloons repelling or a charged Styrofoam plate interacting with electrons in a piece of aluminum, there is always two charges and a distance between them as the three critical variables that influence the strength of the interaction. In this section of Lesson 3, we will explore the importance of these three variables. Force as a Vector Quantity The electrical force, like all forces, is typically expressed using the unit Newton. Being a force, the strength of the electrical interaction is a vector quantity that has both magnitude and direction. The direction of the electrical force is dependent upon whether the charged objects are charged with like charge or opposite charge and upon their spatial orientation. By knowing the type of charge on the two objects, the direction of the force on either one of them can be determined with a little reasoning. In the diagram below, objects A and B have like charge causing them to repel each other. Thus, the force on object A is directed leftward (away from B) and the force on object B is directed rightward (away from A). On the other hand, objects C and D have opposite charge causing them to attract each other. Thus, the force on object C is directed rightward (toward object D) and the force on object D is directed leftward (toward object C). When it comes to the electrical force vector, perhaps the best way to determine the direction of it is to apply the fundamental rules of charge interaction (opposites attract and likes repel) using a little reasoning. Electrical force also has a magnitude or strength. Like most types of forces, there are a variety of factors that influence the magnitude of the electrical force. Two like-charged balloons will repel each other and the strength of their repulsive force can be altered by changing three variables. First, the quantity of charge on one of the balloons will affect the strength of the repulsive force. The more charged a balloon is, the greater the repulsive force. Second, the quantity of charge on the second balloon will affect the strength of the repulsive force. Gently rub two balloons

38 with animal fur and they repel a little. Rub the two balloons vigorously to impart more charge to both of them, and they repel a lot. Finally, the distance between the two balloons will have a significant and noticeable effect upon the repulsive force. The electrical force is strongest when the balloons are closest together. Decreasing the separation distance increases the force. The magnitude of the force and the distance between the two balloons is said to be inversely related. Coulomb's Law Equation The quantitative expression for the effect of these three variables on electric force is known as Coulomb's law. Coulomb's law states that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the separation distance between the two objects. In equation form, Coulomb's law can be stated as where Q 1 represents the quantity of charge on object 1 (in Coulombs), Q 2 represents the quantity of charge on object 2 (in Coulombs), and d represents the distance of separation between the two objects (in meters). The symbol k is a proportionality constant known as the Coulomb's law constant. The value of this constant is dependent upon the medium that the charged objects are immersed in. In the case of air, the value is approximately 9.0 x 10 9 N m 2 / C 2. If the charged objects are present in water, the value of k can be reduced by as much as a factor of 80. It is worthwhile to point out that the units on k are such that when substituted into the equation the units on charge (Coulombs) and the units on distance (meters) will be canceled, leaving a Newton as the unit of force. Electric Field Lines In the previous section of Lesson 4, the vector nature of the electric field strength was discussed. The magnitude or strength of an electric field in the space surrounding a source charge is related directly to the quantity of charge on the source charge and inversely to the distance from the source charge. The direction of the electric field is always directed in the direction that a positive test charge would be pushed or pulled if placed in the space surrounding the source charge. Since electric field is a vector quantity, it can be represented by a vector arrow. For any given location, the arrows point in the direction of the electric field and their length is proportional to the strength of the electric field at that location. Such vector arrows are shown in the diagram below. Note that the lengths of the arrows are longer when closer to the source charge and shorter when further from the source charge. A more useful means of visually representing the vector nature of an electric field is through the use of electric field lines of force. Rather than draw countless vector arrows in the space surrounding a source charge, it is perhaps more useful to draw a pattern of several lines that extend between infinity and the source charge. These pattern of lines, sometimes referred to as electric field lines, point in the direction that a positive test charge would accelerate if placed upon the line. As such, the lines are directed away from positively charged source charges and toward negatively charged source charges. To communicate information about the direction of the field, each line must include an arrowhead that points in the appropriate direction. An electric field line pattern could include an infinite number of lines. Because drawing such large quantities of lines tends to decrease the readability of the patterns, the number of lines is usually limited. The presence of a few lines around a charge is typically sufficient to convey the nature of the electric field in the space surrounding the lines.

39 In each of the above diagrams, the individual source charges in the configuration possess the same amount of charge. Having an identical quantity of charge, each source charge has an equal ability to alter the space surrounding it. Subsequently, the pattern is symmetrical in nature and the number of lines emanating from a source charge or extending towards a source charge is the same. This reinforces a principle discussed earlier that stated that the density of lines surrounding any given source charge is proportional to the quantity of charge on that source charge. If the quantity of charge on a source charge is not identical, the pattern will take on an asymmetric nature, as one of the source charges will have a greater ability to alter the electrical nature of the surrounding space. This is depicted in the electric field line patterns below.

40 The Right Hand Grip Rule gives the direction of the magnetic field around a current carrying conductor. It is in the direction of the fingers, as you grip the conductor with your right hand and your thumb points in the direction of the current. The rule can also be used to indicate the direction of the magnetic field around a solenoid, made up of wire loops. Lenz's Law Emil Lenz gave the following simple rule to find the direction of induced current: The induced current will flow in such a direction so as to oppose the cause that produces it. Let us apply Lenz's law to figure given above. Here the N-pole of the magnet is approaching a coil of several turns. As the N-pole of the magnet moves towards coil, the magnetic flux linking the coil increases. Therefore, an e.m.f and hence current is induced in the coil according to faraday's laws of electromagnetic induction. According to Lenz's law, the direction of the induced current will be such so as to oppose the cause that produces it. In the present case, the cause of the induced current is the increasing magnetic flux linking the coil. Therefore, the induced current will set up magnetic flux that opposes the increase in flux through the coil. Therefore, the induced current will set up magnetic flux that opposes the increase in flux through the coil. This is possible only if the left hand face of the coil becomes N-pole. Once we know the magnetic polarity of the coil face, the direction of the induced current can be easily determined by applying right hand rule for the coil. The Lenz's law can be summed up as under:

41 If the magnetic flux Ф linking a coil will flow in such a direction so as to oppose the increase in flux i.e. the induced current will produce flux as shown in figure given below, If magnetic flux Ф linking a coil is decreasing, the induced current i in the coil will flow in such a direction so as to oppose the decrease in the flux i.e. the induced current will produce flux Ф to aid the flux Ф as shown in figure given below. The Doppler Effect Suppose that there is a happy bug in the center of a circular water puddle. The bug is periodically shaking its legs in order to produce disturbances that travel through the water. If these disturbances originate at a point, then they would travel outward from that point in all directions. Since each disturbance is traveling in the same medium, they would all travel in every direction at the same speed. The pattern produced by the bug's shaking would be a series of concentric circles as shown in the diagram at the right. These circles would reach the edges of the water puddle at the same frequency. An observer at point A (the left edge of the puddle) would observe the disturbances to strike the puddle's edge at the same frequency that would be observed by an observer at point B (at the right edge of the puddle). In fact, the frequency at which disturbances reach the edge of the puddle would be the same as the frequency at which the bug produces the disturbances. If the bug produces disturbances at a frequency of 2 per second, then each observer would observe them approaching at a frequency of 2 per second. Now suppose that our bug is moving to the right across the puddle of water and producing disturbances at the same frequency of 2 disturbances per second. Since the bug is moving towards the right, each consecutive disturbance originates from a position that is closer to observer B and farther from observer A. Subsequently, each consecutive disturbance has a shorter distance to travel before reaching observer B and thus takes less time to reach observer B. Thus, observer B observes that the frequency of arrival of the disturbances is higher than the frequency at which disturbances are produced. On the other hand, each consecutive disturbance has a further distance to travel before reaching observer A. For this reason, observer A observes a frequency of arrival that is less than the frequency at which the disturbances are produced. The net effect of the motion of the bug (the source of waves) is that the observer towards whom the bug is moving observes a frequency that is higher than 2 disturbances/second; and the observer away from whom the bug is moving observes a frequency that is less than 2 disturbances/second. This effect is known as the Doppler effect.

42 The Doppler effect is observed whenever the source of waves is moving with respect to an observer. The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding. It is important to note that the effect does not result because of an actual change in the frequency of the source. Using the example above, the bug is still producing disturbances at a rate of 2 disturbances per second; it just appears to the observer whom the bug is approaching that the disturbances are being produced at a frequency greater than 2 disturbances/second. The effect is only observed because the distance between observer B and the bug is decreasing and the distance between observer A and the bug is increasing. The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc. We are most familiar with the Doppler effect because of our experiences with sound waves. Perhaps you recall an instance in which a police car or emergency vehicle was traveling towards you on the highway. As the car approached with its siren blasting, the pitch of the siren sound (a measure of the siren's frequency) was high; and then suddenly after the car passed by, the pitch of the siren sound was low. That was the Doppler effect - an apparent shift in frequency for a sound wave produced by a moving source. The Doppler effect is of intense interest to astronomers who use the information about the shift in frequency of electromagnetic waves produced by moving stars in our galaxy and beyond in order to derive information about those stars and galaxies. The belief that the universe is expanding is based in part upon observations of electromagnetic waves emitted by stars in distant galaxies. Furthermore, specific information about stars within galaxies can be determined by application of the Doppler effect. Galaxies are clusters of stars that typically rotate about some center of mass point. Electromagnetic radiation emitted by such stars in a distant galaxy would appear to be shifted downward in frequency (a red shift) if the star is rotating in its cluster in a direction that is away from the Earth. On the other hand, there is an upward shift in frequency (a blue shift) of such observed radiation if the star is rotating in a direction that is towards the Earth. Interference of Waves What happens when two waves meet while they travel through the same medium? What effect will the meeting of the waves have upon the appearance of the medium? Will the two waves bounce off each other upon meeting (much like two billiard balls would) or will the two waves pass through each other? These questions involving the meeting of two or more waves along the same medium pertain to the topic of wave interference. Wave interference is the phenomenon that occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium. To begin our exploration of wave interference, consider two pulses of the same amplitude traveling in different directions along the same medium. Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped. At that moment, the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units. The diagrams below depict the before and during interference snapshots of the medium for two such pulses. The individual sine pulses are drawn in red and blue and the resulting displacement of the medium is drawn in green.

43 This type of interference is sometimes called constructive interference. Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement that is greater than the displacement of the two interfering pulses. Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering waves are displaced downward. This is shown in the diagram below for two downward displaced pulses. In this case, a sine pulse with a maximum displacement of -1 unit (negative means a downward displacement) interferes with a sine pulse with a maximum displacement of -1 unit. These two pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units. Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the opposite direction. For instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive interference occurs. This is depicted in the diagram below. In the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting displacement of the particles of the medium. This "destruction" is not a permanent condition. In fact, to say that the two waves destroy each other can be partially misleading. When it is said that the two pulses destroy each other, what is meant is that when overlapped, the effect of one of the pulses on the displacement of a given particle of the medium is destroyed or canceled by the effect of the other pulse. Recall from Lesson 1 that waves transport energy through a medium by means of each individual particle pulling upon its nearest neighbor. When two pulses with opposite displacements (i.e., one pulse displaced up and the other down) meet at a given location, the upward pull of one pulse is balanced (canceled or destroyed) by the downward pull of the other pulse. Once the two pulses pass through each other, there is still an upward displaced pulse and a downward displaced pulse heading in the same direction that they were heading before the interference. Destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the largest-amplitude wave. Reflection, Refraction, and Diffraction The wave doesn't just stop when it reaches the end of the medium. Rather, a wave will undergo certain behaviors when it encounters the end of the medium. Specifically, there will be some reflection off the boundary and some transmission into the new medium. But what if the wave is traveling in a two-dimensional medium such as a water wave traveling through ocean water? Or what if the wave is traveling in a three-dimensional medium such as a sound wave or a light wave traveling through air? What types of behaviors can be expected of such two- and threedimensional waves?

44 The study of waves in two dimensions is often done using a ripple tank. A ripple tank is a large glass-bottomed tank of water that is used to study the behavior of water waves. A light typically shines upon the water from above and illuminates a white sheet of paper placed directly below the tank. A portion of light is absorbed by the water as it passes through the tank. A crest of water will absorb more light than a trough. So the bright spots represent wave troughs and the dark spots represent wave crests. As the water waves move through the ripple tank, the dark and bright spots move as well. As the waves encounter obstacles in their path, their behavior can be observed by watching the movement of the dark and bright spots on the sheet of paper. Ripple tank demonstrations are commonly done in a Physics class in order to discuss the principles underlying the reflection, refraction, and diffraction of waves. If a linear object attached to an oscillator bobs back and forth within the water, it becomes a source of straight waves. These straight waves have alternating crests and troughs. As viewed on the sheet of paper below the tank, the crests are the dark lines stretching across the paper and the troughs are the bright lines. These waves will travel through the water until they encounter an obstacle - such as the wall of the tank or an object placed within the water. The diagram at the right depicts a series of straight waves approaching a long barrier extending at an angle across the tank of water. The direction that these wavefronts (straight-line crests) are traveling through the water is represented by the blue arrow. The blue arrow is called a ray and is drawn perpendicular to the wavefronts. Upon reaching the barrier placed within the water, these waves bounce off the water and head in a different direction. The diagram below shows the reflected wavefronts and the reflected ray. Regardless of the angle at which the wavefronts approach the barrier, one general law of reflection holds true: the waves will always reflect in such a way that the angle at which they approach the barrier equals the angle at which they reflect off the barrier. This is known as the law of reflection. The discussion above pertains to the reflection of waves off of straight surfaces. But what if the surface is curved, perhaps in the shape of a parabola? What generalizations can be made for the reflection of water waves off parabolic surfaces? Suppose that a rubber tube having the shape of a parabola is placed within the water. The diagram at the right depicts such a parabolic barrier in the ripple tank. Several wavefronts are approaching the barrier; the ray is drawn for these wavefronts. Upon reflection off the parabolic barrier, the water waves will change direction and head towards a point. This is depicted in the diagram below. It is as though all the energy being carried by the water waves is converged at a single point - the point is known as the focal point. After passing through the focal point, the waves spread out through the water.

45 Reflection involves a change in direction of waves when they bounce off a barrier. Refraction of waves involves a change in the direction of waves as they pass from one medium to another. Refraction, or the bending of the path of the waves, is accompanied by a change in speed and wavelength of the waves. It was mentioned that the speed of a wave is dependent upon the properties of the medium through which the waves travel. So if the medium (and its properties) is changed, the speed of the waves is changed. The most significant property of water that would affect the speed of waves traveling on its surface is the depth of the water. Water waves travel fastest when the medium is the deepest. Thus, if water waves are passing from deep water into shallow water, they will slow down. This decrease in speed will also be accompanied by a decrease in wavelength. So as water waves are transmitted from deep water into shallow water, the speed decreases, the wavelength decreases, and the direction changes. This boundary behavior of water waves can be observed in a ripple tank if the tank is partitioned into a deep and a shallow section. If a pane of glass is placed in the bottom of the tank, one part of the tank will be deep and the other part of the tank will be shallow. Waves traveling from the deep end to the shallow end can be seen to refract (i.e., bend), decrease wavelength (the wavefronts get closer together), and slow down (they take a longer time to travel the same distance). When traveling from deep water to shallow water, the waves are seen to bend in such a manner that they seem to be traveling more perpendicular to the surface. If traveling from shallow water to deep water, the waves bend in the opposite direction. Ohm's Law There are certain formulas in Physics that are so powerful and so pervasive that they reach the state of popular knowledge. A student of Physics has written such formulas down so many times that they have memorized it without trying to. Certainly to the professionals in the field, such formulas are so central that they become engraved in their minds. In the field of Modern Physics, there is E = m c 2. In the field of Newtonian Mechanics, there is F net = m a. In the field of Wave Mechanics, there is v = f. And in the field of current electricity, there is V = I R. The predominant equation which pervades the study of electric circuits is the equation V = I R In words, the electric potential difference between two points on a circuit ( V) is equivalent to the product of the current between those two points (I) and the total resistance of all electrical devices present between those two points (R). Through the rest of this unit of The Physics Classroom, this equation will become the most common equation which we see. Often referred to as the Ohm's law equation, this equation is a powerful predictor of the relationship between potential difference, current and resistance. Parallel Circuits As mentioned in a previous section of Lesson 4, two or more electrical devices in a circuit can be connected by series connections or by parallel connections. When all the devices are connected using parallel connections, the circuit is referred to as a parallel circuit. In a parallel circuit, each device is placed in its own separate branch. The presence of branch lines means that there are multiple pathways by which charge can traverse the external circuit. Each charge passing through the loop of the external circuit will pass through a single resistor present in a single branch. When arriving at the branching location or node, a charge makes a choice as to which branch to travel through on its journey back to the low potential terminal.

46 A short comparison and contrast between series and parallel circuits was made in an earlier section of Lesson 4. In that section, it was emphasized that the act of adding more resistors to a parallel circuit results in the rather unexpected result of having less overall resistance. Since there are multiple pathways by which charge can flow, adding another resistor in a separate branch provides another pathway by which to direct charge through the main area of resistance within the circuit. This decreased resistance resulting from increasing the number of branches will have the effect of increasing the rate at which charge flows (also known as the current). In an effort to make this rather unexpected result more reasonable, a tollway analogy was introduced. A tollbooth is the main location of resistance to car flow on a tollway. Adding additional tollbooths within their own branch on a tollway will provide more pathways for cars to flow through the toll station. These additional tollbooths will decrease the overall resistance to car flow and increase the rate at which they flow. Current The rate at which charge flows through a circuit is known as the current. Charge does NOT pile up and begin to accumulate at any given location such that the current at one location is more than at other locations. Charge does NOT become used up by resistors in such a manner that there is less current at one location compared to another. In a parallel circuit, charge divides up into separate branches such that there can be more current in one branch than there is in another. Nonetheless, when taken as a whole, the total amount of current in all the branches when added together is the same as the amount of current at locations outside the branches. The rule that current is everywhere the same still works, only with a twist. The current outside the branches is the same as the sum of the current in the individual branches. It is still the same amount of current, only split up into more than one pathway. In equation form, this principle can be written as I total = I 1 + I 2 + I where I total is the total amount of current outside the branches (and in the battery) and I 1, I 2, and I 3 represent the current in the individual branches of the circuit. Throughout this unit, there has been an extensive reliance upon the analogy between charge flow and water flow. Once more, we will return to the analogy to illustrate how the sum of the current values in the branches is equal to the amount outside of the branches. The flow of charge in wires is analogous to the flow of water in pipes. Consider the diagrams below in which the flow of water in pipes becomes divided into separate branches. At each node (branching location), the water takes two or more separate pathways. The rate at which water flows into the node (measured in gallons per minute) will be equal to the sum of the flow rates in the individual branches beyond the node. Similarly, when two or more branches feed into a node, the rate at which water flows out of the node will be equal to the sum of the flow rates in the individual branches that feed into the node.

47 The same principle of flow division applies to electric circuits. The rate at which charge flows into a node is equal to the sum of the flow rates in the individual branches beyond the node. This is illustrated in the examples shown below. In the examples a new circuit symbol is introduced - the letter A enclosed within a circle. This is the symbol for an ammeter - a device used to measure the current at a specific point. An ammeter is capable of measuring the current while offering negligible resistance to the flow of charge. Diagram A displays two resistors in parallel with nodes at point A and point B. Charge flows into point A at a rate of 6 amps and divides into two pathways - one through resistor 1 and the other through resistor 2. The current in the branch with resistor 1 is 2 amps and the current in the branch with resistor 2 is 4 amps. After these two branches meet again at point B to form a single line, the current again becomes 6 amps. Thus we see the principle that the current outside the branches is equal to the sum of the current in the individual branches holds true. I total = I 1 + I 2 6 amps = 2 amps + 4 amps Diagram B above may be slightly more complicated with its three resistors placed in parallel. Four nodes are identified on the diagram and labeled A, B, C and D. Charge flows into point A at a rate of 12 amps and divides into two pathways - one passing through resistor 1 and the other heading towards point B (and resistors 2 and 3). The 12 amps of current is divided into a 2 amp pathway (through resistor 1) and a 10 amp pathway (heading toward point B). At point B, there is further division of the flow into two pathways - one through resistor 2 and the other through resistor 3. The current of 10 amps approaching point B is divided into a 6-amp pathway (through resistor 2) and a 4- amp pathway (through resistor 3). Thus, it is seen that the current values in the three branches are 2 amps, 6 amps and 4 amps and that the sum of the current values in the individual branches is equal to the current outside the branches. I total = I 1 + I 2 + I 3 12 amps = 2 amps + 6 amps + 4 amps A flow analysis at points C and D can also be conducted and it is observed that the sum of the flow rates heading into these points is equal to the flow rate that is found immediately beyond these points.