Worked Examples from Introductory Physics (Algebra–Based ...

1 Worked Examples from Introductory Physics (Algebra Based)Vol. I: Basic MechanicsDavid Murdock, TTUO ctober 3, 20122 ContentsPrefacei1 Mathematical The Important Stuff .. Measurement and Units in Physics .. The Metric System; Converting Units .. Math: You Had This In High School. Oh, Yes You Did.. Math: Trigonometry .. Vectors and Vector Addition .. Components of Vectors .. Worked Examples .. Measurement and Units .. Trigonometry .. Vectors and Vector Addition ..142 Motion in One The Important Stuff .. Displacement .. Speed and Velocity .. Motion With Constant Velocity .. Acceleration .. Motion Where the Acceleration is Constant .. Free-Fall .. Worked Examples .. Motion Where the Acceleration is Constant .. Free-Fall ..243 Motion in Two The Important Stuff .. Motion in Two Dimensions, Coordinates and Displacement.

2 Velocity and Acceleration .. Motion When the Acceleration Is Constant .. Free Fall; Projectile Problems .. Ground To Ground Projectile: A Long Example .. Worked Examples .. Velocity and Acceleration .. Motion for Constant Acceleration .. Free Fall; Projectile Problems ..414 Forces The Important Stuff .. Introduction .. Newton s 1st Law .. Newton s 2nd Law .. Units and Stuff .. Newton s 3rd Law .. The Force of Gravity .. Other Forces Which Appear In Our Problems .. The Free Body Diagram: Draw the Damn Picture! .. Simple Example: What Does the Scale Read? .. An Important Example: Mass Sliding On a Smooth Inclined Plane . Another Important Example: The Attwood Machine .. Worked Examples .. Newton s Second Law .. The Force of Gravity .. Applying Newton s Laws of Motion ..655 Forces The Important Stuff .. Introduction.

3 Friction Forces .. An Important Example: Block Sliding Down Rough Inclined Plane . Uniform Circular Motion .. Circular Motion and Force .. Orbital Motion .. Worked Examples .. Friction Forces .. Uniform Circular Motion .. Circular Motion and Force .. Orbital Motion ..83 CONTENTS56 The Important Stuff .. Introduction .. Kinetic Energy .. Work .. The Work Energy Theorem .. Potential Energy .. The Spring Force .. The Principle of Energy Conservation .. Solving Problems With Energy Conservation .. Power .. Worked Examples .. Kinetic Energy .. The Spring Force .. Solving Problems With Energy Conservation ..947 The Important Stuff .. Momentum; Systems of Particles .. Relation to Force; Impulse .. The Principle of Momentum Conservation .. Collisions; Problems Using the Conservation of Momentum .. Systems of Particles; The Center of Mass.

4 Finding the Center of Mass .. Worked Examples ..1068 Rotational The Important Stuff .. Rigid Bodies; Rotating Objects .. Angular Displacement .. Angular Velocity .. Angular Acceleration .. The Case of Constant Angular Acceleration .. Relation Between Angular and Linear Quantities .. Worked Examples .. Angular Displacement .. Angular Velocity and Acceleration .. Rotational Motion with Constant Angular Acceleration .. Relation Between Angular and Linear Quantities ..1146 CONTENTS9 Rotational The Important Stuff .. Introduction .. Rotational Kinetic Energy .. More on the Moment of Inertia .. Torque .. Another Way to Look at Torque .. Newton s 2nd Law for Rotations .. Solving Problems with Forces, Torques and Rotating Objects .. An Example .. Statics .. Rolling Motion .. Example: Round Object Rolls Down Slope Without Slipping.

5 Angular Momentum .. Worked Examples .. The Moment of Inertia and Rotational Kinetic Energy ..13510 Oscillatory The Important Stuff .. Introduction .. Harmonic Motion .. Displacement, Velocity and Acceleration .. The Reference Circle .. A Real Mass/Spring System .. Energy and the Harmonic Oscillator .. Simple Pendulum .. Physical Pendulum .. Worked Examples .. Harmonic Motion .. Mass Spring System .. Simple Pendulum ..15011 Waves The Important Stuff .. Introduction .. Principle of Superposition .. Harmonic Waves .. Waves on a String .. Sound Waves .. Sound Intensity .. The Doppler Effect .. Worked Examples .. Harmonic Waves .. Waves on a String .. Sound Waves ..1628 CONTENTSP refaceThis booklet can be downloaded free of charge from: date on the cover page serves as an edition number. I m continually tinkering withthese book is: A summary of the material in the first semester of the non calculus Physics course asI teach it at Tennessee Tech.

6 A set of example problems typical of those given in non calculus Physics courses solvedand explained as well as I know is not intended as a substitute for any textbook suggestedby a professor.. at least notyet! It s just here to help you with the Physics course you retaking. Read it alongside thetext they told you to buy. The subjects should be in theroughorder that they re coveredin class, though the chapter numbers won t exactly match those in your and errata will be appreciated. Send mail to me 1 Mathematical The Important Measurement and Units in PhysicsPhysics is concerned with the relations between measured quantities in the natural world. Wemake measurements (length, time, etc) in terms of variousstandardsfor these Physics we generally use the metric system , or more precisely, theSIorMKSsystem, so called because it is based on themeter, thesecondand meter is related to basic length unit of the English system the inch by theexactrelations:1 cm = 10 2mand1 in = cmFrom this we can get:1 m = ftand1 km = miEveryone knows the (exact) relations between the common units of time:1 minute = 60 sec1 hour = 60 min1 day = 24 hand we also have the (pretty accurate) relation:1 year = daysFinally, the unit ofmassis the kilogram.

7 Themeaningof mass is not so clear unlessyou have already studies Physics . For now, suffice it to say that amassof 1 kilogram has aweightof pounds. Later on we will make the distinction between mass and weight .12 CHAPTER 1. MATHEMATICAL The Metric System; Converting UnitsTo make the SI system more convenient we can associate prefixes with the basic units torepresent powers of 10. The most commonly used prefixes are given here:FactorPrefixSymbol10 12pico-p10 9nano-n10 6micro- 10 3milli-m10 2centi-c103kilo-k106mega-M109giga-GSome Examples :1 ms = 1 millisecond = 10 3s1 m = 1 micrometer = 10 6sOftentimes in science we need to change the units in which a quantity is might want to change a length expressed in feet to one expressed in meters, or a timeexpressed in days to one expressed in , be aware that in the math we do for Physics problems a unit symbol like cm (centimeter) or yr (year) is treated as a multiplicative factor which we can cancel if thesame factor occurs in the numerator and denominator.

8 In any case we can t simply ignoreor erase a unit this in mind we can set up conversion factors, which contain the samequantityonthe top and bottom (and so are equal to 1) which will cancel theold units and give example, 60 seconds is equal to one minute. Then we have(60 s1 min)= 1so we can multiply by this factor without changing thevalueof a number. But it can giveus newunitsfor the number. To convert min to seconds, use this factor and cancel thesymbol min min = ( min)(60 s1 min)= 506 THE IMPORTANT STUFF3xyxyz(a)(b)Figure :(a) Rectangle with sidesxandy. Area isA=xy. I hope you knew that. (b) Rectangular boxwith sidesx,yandz. Volume isV=xyz. I hope you knew that we have to convert 104s to minutes, we would use a conversion factor with secondsin the denominator (to cancel what we ve got already; the conversion factor is still equal to1). 104s = ( 104s)(1 min60 s)= 613 Math: You Had This In High School.

9 Oh, Yes You mathematical demands of a non calculus Physics course are not extensive, but youdo have to be proficient with the little bit of mathematics that wewilluse! It s just the stuffyou had in high school. Oh, yes you did. Don t tell me you didn will often use scientific notation to express our numbers,because this allows usto express large and small numbers conveniently (and also express the precision of thosenumbers). We will need the basic algebra operations of powers and roots and we will solveequations to find the unknowns .Usually the algebra will be very simple. But if we are ever faced with an equation thatlooks likeax2+bx+c= 0( )wherexis the unknown anda,bandcare given numbers (constants) then there are twopossible answers forxwhich you can find from thequadratic formula:x= b b2 4ac2a( )On occasion you will need to know some facts from geometry. Starting simple and workingupwards, the simplest shapes are the rectangle and rectangular box, shown in Fig.

10 If4 CHAPTER 1. MATHEMATICAL CONCEPTSRR(a)(b)DFigure :(a) Circle;C= D= 2 R;A= R2. (b) Sphere;A= 4 R2;V=43 R3. You ve seen theseformulae before. Oh, yes you (a)(b)Figure :(a) Circular cylinder of radiusRand heighth. Volume isV= R2h. (b) Right cylinder ofarbitrary shape. If the area of the cross section isA, the volume isV= rectangle has sidesxandyits area isA=xy. Since it is the product of twolengths,the units of area in the SI system are m2. For the rectangular box with sidesx,yandz, thevolume isV=xyz. A volume is the product ofthreelengths so its units are formulae worth mentioning here are for the circle and the sphere; see Fig. circle is specified by its radiusR(or its diameterD, which is twice the radius). Thedistance around the circle is the circumference,C. The circumference and areaAof thecircle are given byC= D= 2 RA= R2( )A sphere is specified by its radiusR.