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Worked Examples from Introductory Physics(Algebra–Based)Vol. I: Basic MechanicsDavid Murdock, TTUOctober 3, 2012

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ContentsPrefacei1 Mathematical Concepts1.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . .1.1.1 Measurement and Units in Physics . . . . . . . . . . . .1.1.2 The Metric System; Converting Units . . . . . . . . . . .1.1.3 Math: You Had This In High School. Oh, Yes You Did. .1.1.4 Math: Trigonometry . . . . . . . . . . . . . . . . . . . .1.1.5 Vectors and Vector Addition . . . . . . . . . . . . . . . .1.1.6 Components of Vectors . . . . . . . . . . . . . . . . . . .1.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.1 Measurement and Units . . . . . . . . . . . . . . . . . .1.2.2 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . .1.2.3 Vectors and Vector Addition . . . . . . . . . . . . . . . .111235568810142 Motion in One Dimension2.1 The Important Stuff . . . . . . . . . . .2.1.1 Displacement . . . . . . . . . . .2.1.2 Speed and Velocity . . . . . . . .2.1.3 Motion With Constant Velocity .2.1.4 Acceleration . . . . . . . . . . . .2.1.5 Motion Where the Acceleration is2.1.6 Free-Fall . . . . . . . . . . . . . .2.2 Worked Examples . . . . . . . . . . . . .2.2.1 Motion Where the Acceleration is2.2.2 Free-Fall . . . . . . . . . . . . . .19191919202021222323243 Motion in Two Dimensions3.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.1 Motion in Two Dimensions, Coordinates and Displacement . . . . . .3333333. . . . . . . . . . . . . . . . . . . . . . . . . .Constant. . . . . . . . . . .Constant. . . . . .

4CONTENTS3.1.2 Velocity and Acceleration . . . . . . . . . . . . .3.1.3 Motion When the Acceleration Is Constant . . . .3.1.4 Free Fall; Projectile Problems . . . . . . . . . . .3.1.5 Ground–To–Ground Projectile: A Long Example3.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . .3.2.1 Velocity and Acceleration . . . . . . . . . . . . .3.2.2 Motion for Constant Acceleration . . . . . . . . .3.2.3 Free–Fall; Projectile Problems . . . . . . . . . . .34353636393940414 Forces I4.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.2 Newton’s 1st Law . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.3 Newton’s 2nd Law . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.4 Units and Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.5 Newton’s 3rd Law . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.6 The Force of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.7 Other Forces Which Appear In Our Problems . . . . . . . . . . . .4.1.8 The Free–Body Diagram: Draw the Damn Picture! . . . . . . . . .4.1.9 Simple Example: What Does the Scale Read? . . . . . . . . . . . .4.1.10 An Important Example: Mass Sliding On a Smooth Inclined Plane4.1.11 Another Important Example: The Attwood Machine . . . . . . . .4.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.1 Newton’s Second Law . . . . . . . . . . . . . . . . . . . . . . . . .4.2.2 The Force of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.3 Applying Newton’s Laws of Motion . . . . . . . . . . . . . . . . . .49494950505151525456565861636365655 Forces II5.1 The Important Stuff . . . . . . . . . . . . . . . . .5.1.1 Introduction . . . . . . . . . . . . . . . . . .5.1.2 Friction Forces . . . . . . . . . . . . . . . .5.1.3 An Important Example: Block Sliding Down5.1.4 Uniform Circular Motion . . . . . . . . . . .5.1.5 Circular Motion and Force . . . . . . . . . .5.1.6 Orbital Motion . . . . . . . . . . . . . . . .5.2 Worked Examples . . . . . . . . . . . . . . . . . . .5.2.1 Friction Forces . . . . . . . . . . . . . . . .5.2.2 Uniform Circular Motion . . . . . . . . . . .5.2.3 Circular Motion and Force . . . . . . . . . .5.2.4 Orbital Motion . . . . . . . . . . . . . . . .69696969707173737575788083. . . . . . . . . .Rough. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Inclined Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5CONTENTS6 Energy6.1 The Important Stuff . . . . . . . . . . . . . . . . .6.1.1 Introduction . . . . . . . . . . . . . . . . . .6.1.2 Kinetic Energy . . . . . . . . . . . . . . . .6.1.3 Work . . . . . . . . . . . . . . . . . . . . . .6.1.4 The Work–Energy Theorem . . . . . . . . .6.1.5 Potential Energy . . . . . . . . . . . . . . .6.1.6 The Spring Force . . . . . . . . . . . . . . .6.1.7 The Principle of Energy Conservation . . . .6.1.8 Solving Problems With Energy Conservation6.1.9 Power . . . . . . . . . . . . . . . . . . . . .6.2 Worked Examples . . . . . . . . . . . . . . . . . . .6.2.1 Kinetic Energy . . . . . . . . . . . . . . . .6.2.2 The Spring Force . . . . . . . . . . . . . . .6.2.3 Solving Problems With Energy Conservation.8787878788898990919292939393947 Momentum7.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . .7.1.1 Momentum; Systems of Particles . . . . . . . . . . . . . .7.1.2 Relation to Force; Impulse . . . . . . . . . . . . . . . . . .7.1.3 The Principle of Momentum Conservation . . . . . . . . .7.1.4 Collisions; Problems Using the Conservation of Momentum7.1.5 Systems of Particles; The Center of Mass . . . . . . . . . .7.1.6 Finding the Center of Mass . . . . . . . . . . . . . . . . .7.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 13113113114114.8 Rotational Kinematics8.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . .8.1.1 Rigid Bodies; Rotating Objects . . . . . . . . . . . . .8.1.2 Angular Displacement . . . . . . . . . . . . . . . . . .8.1.3 Angular Velocity . . . . . . . . . . . . . . . . . . . . .8.1.4 Angular Acceleration . . . . . . . . . . . . . . . . . . .8.1.5 The Case of Constant Angular Acceleration . . . . . .8.1.6 Relation Between Angular and Linear Quantities . . .8.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . .8.2.1 Angular Displacement . . . . . . . . . . . . . . . . . .8.2.2 Angular Velocity and Acceleration . . . . . . . . . . .8.2.3 Rotational Motion with Constant Angular Acceleration8.2.4 Relation Between Angular and Linear Quantities . . .

6CONTENTS9 Rotational Dynamics9.1 The Important Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.2 Rotational Kinetic Energy . . . . . . . . . . . . . . . . . . . .9.1.3 More on the Moment of Inertia . . . . . . . . . . . . . . . . .9.1.4 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.5 Another Way to Look at Torque . . . . . . . . . . . . . . . . .9.1.6 Newton’s 2nd Law for Rotations . . . . . . . . . . . . . . . . .9.1.7 Solving Problems with Forces, Torques and Rotating Objects .9.1.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.9 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.10 Rolling Motion . . . . . . . . . . . . . . . . . . . . . . . . . .9.1.11 Example: Round Object Rolls Down Slope Without Slipping .9.1.12 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . .9.2 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.2.1 The Moment of Inertia and Rotational Kinetic Energy . . . 0 Oscillatory Motion10.1 The Important Stuff . . . . . . . . . . . . . . .10.1.1 Introduction . . . . . . . . . . . . . . . .10.1.2 Harmonic Motion . . . . . . . . . . . . .10.1.3 Displacement, Velocity and Acceleration10.1.4 The Reference Circle . . . . . . . . . . .10.1.5 A Real Mass/Spring System . . . . . . .10.1.6 Energy and the Harmonic Oscillator . .10.1.7 Simple Pendulum . . . . . . . . . . . . .10.1.8 Physical Pendulum . . . . . . . . . . . .10.2 Worked Examples . . . . . . . . . . . . . . . . .10.2.1 Harmonic Motion . . . . . . . . . . . . .10.2.2 Mass–Spring System . . . . . . . . . . .10.2.3 Simple Pendulum . . . . . . . . . . . . .13713713713714014114414514614814914914915011 Waves I11.1 The Important Stuff . . . . . . .11.1.1 Introduction . . . . . . . .11.1.2 Principle of Superposition11.1.3 Harmonic Waves . . . . .11.1.4 Waves on a String . . . . .11.1.5 Sound Waves . . . . . . .11.1.6 Sound Intensity . . . . . .151151151152154157157158.

7CONTENTS11.1.7 The Doppler Effect11.2 Worked Examples . . . . .11.2.1 Harmonic Waves .11.2.2 Waves on a String .11.2.3 Sound Waves . . .160161161162162

8CONTENTS

PrefaceThis booklet can be downloaded free of charge from:http://iweb.tntech.edu/murdock/books.htmlThe date on the cover page serves as an edition number. I’m continually tinkering withthese booklets.This book is: A summary of the material in the first semester of the non–calculus physics course asI teach it at Tennessee Tech. A set of example problems typical of those given in non–calculus physics courses solvedand explained as well as I know how.It is not intended as a substitute for any textbook suggested by a professor. . . at least notyet! It’s just here to help you with the physics course you’re taking. Read it alongside thetext they told you to buy. The subjects should be in the rough order that they’re coveredin class, though the chapter numbers won’t exactly match those in your textbook.Feedback and errata will be appreciated. Send mail to me at:murdock@tntech.edui

iiPREFACE

Chapter 1Mathematical Concepts1.11.1.1The Important StuffMeasurement and Units in PhysicsPhysics is concerned with the relations between measured quantities in the natural world. Wemake measurements (length, time, etc) in terms of various standards for these quantities.In physics we generally use the “metric system”, or more precisely, the SI or MKSsystem, so called because it is based on the meter, the second and the kilogram.The meter is related to basic length unit of the “English” system —the inch— by theexact relations:1 cm 10 2 mand1 in 2.54 cmFrom this we can get:1 m 3.281 ftand1 km 0.6214 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 365.24 daysFinally, the unit of mass is the kilogram. The meaning of mass is not so clear unlessyou have already studies physics. For now, suffice it to say that a mass of 1 kilogram has aweight of — pounds. Later on we will make the distinction between “mass” and “weight”.1

2CHAPTER 1. MATHEMATICAL CONCEPTS1.1.2The 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:Factor10 1210 910 610 310 2103106109Prefix Some examples:1 ms 1 millisecond 10 3 s1 µm 1 micrometer 10 6 sOftentimes in science we need to change the units in which a quantity is expressed.We might want to change a length expressed in feet to one expressed in meters, or a timeexpressed in days to one expressed in seconds.First, 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. In any case we can’t simply ignoreor erase a unit symbol.With this in mind we can set up conversion factors, which contain the same quantity onthe top and bottom (and so are equal to 1) which will cancel the old units and give newones.For example, 60 seconds is equal to one minute. Then we have 60 s 11 min so we can multiply by this factor without changing the value of a number. But it can giveus new units for the number. To convert 8.44 min to seconds, use this factor and cancel thesymbol “min”: 60 s8.44 min (8.44 min) 506 s1 min

31.1. THE IMPORTANT STUFFyzxx(a)y(b)Figure 1.1: (a) Rectangle with sides x and y. Area is A xy. I hope you knew that. (b) Rectangular boxwith sides x, y and z. Volume is V xyz. I hope you knew that too.If we have to convert 3.68 104 s 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). So: 1 min443.68 10 s (3.68 10 s) 613 min60 s1.1.3Math: You Had This In High School. Oh, Yes You Did.The mathematical demands of a “non–calculus” physics course are not extensive, but youdo have to be proficient with the little bit of mathematics that we will use! It’s just the stuffyou had in high school. Oh, yes you did. Don’t tell me you didn’t.We 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(1.1)where x is the unknown and a, b and c are given numbers (constants) then there are twopossible answers for x which you can find from the quadratic formula: b b2 4acx (1.2)2aOn 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. 1.1. If

4CHAPTER 1. MATHEMATICAL CONCEPTSRRD(b)(a)Figure 1.2: (a) Circle; C πD 2πR; A πR2 . (b) Sphere; A 4πR2; V 43 πR3. You’ve seen theseformulae before. Oh, yes you have.ARh(a)h(b)Figure 1.3: (a) Circular cylinder of radius R and height h. Volume is V πR2h. (b) Right cylinder ofarbitrary shape. If the area of the cross section is A, the volume is V Ah.the rectangle has sides x and y its area is A xy. Since it is the product of two lengths,the units of area in the SI system are m2 . For the rectangular box with sides x, y and z, thevolume is V xyz. A volume is the product of three lengths so its units are m3.Other formulae worth mentioning here are for the circle and the sphere; see Fig. 1.2.A circle is specified by its radius R (or its diameter D, which is twice the radius). Thedistance around the circle is the circumference, C. The circumference and area A of thecircle are given byC πD 2πRA πR2(1.3)A sphere is specified by its radius R. The surface area A and volume V of a sphere aregiven byA 4πR2V 34 πR3(1.4)Another simple shape is the (right) circular cylinder, shown in Fig. 1.3(a). If the cylinderhas radius R and height h, its volume is V πR2h. This is a special case of the generalright cylinder (see Fig. 1.3(b)) where if the area of the cross section is A and the height ish, the volume is V Ah.

51.1. THE IMPORTANT STUFFfcaqbFigure 1.4: Right triangle with sides a, b and c.1.1.4Math: TrigonometryYou will also need some simple trigonometry. This won’t amount to much more than relatingthe sides of a right triangle, that is, a triangle with two sides joined at 90 .Such a triangle is shown in Fig. 1.4. The sides a, b and c are related by the PythagoreanTheorem: a 2 b 2 c2 c a 2 b2(1.5)We only need the angle θ to determine the shape of the triangle and this gives the ratiosof the sides of the triangle. The ratios are given by:sin θ accos θ bctan θ ab(1.6)Or you can remember these ratios in term of their positions with respect to the angle θ. Ifthe sides area oppositeb adjacentc hypothenusethen the ratios aresin θ opphypcos θ adjhyptan θ oppadj(1.7)If you pick out the first letters of the “words” in Eq. 1.7 in order, they spell out SOHCAHTOA. If you want to remember the trig ratios by intoning “SOHCAHT OA”, be my guest,but don’t do it near me.1.1.5Vectors and Vector AdditionThroughout our study of physics we will discuss quantities which have a size (that is, amagnitude) as well as a direction These quantities are called vectors. Examples of vectorsare velocity, acceleration, force, and the electric field.

6CHAPTER 1. MATHEMATICAL CONCEPTSBCBAAFigure 1.5: Vectors A and B are added to give the vector C A B.yAAyAxxFigure 1.6: Vector A is split up into components.Vectors are represented by arrows which show their magnitude and direction. The lawsof physics will require us to add vectors, and to represent this operation on paper, we addthe arrows. The way to add arrows, say to add arrow A to arrow B we join the tail of B tothe head of A and then draw a new arrow from the tail of A to the head of B. The resultis A B. This is shown in Figure 1.5.Vectors can be multiplied by ordinary numbers (called scalars), giving new vectors, asshown in Fig. 1.5.1.1.6Components of VectorsAddition of vectors would be rather messy if we didn’t have an easy technique for handlingthe trigonometry. Vector addition is made much easier when we split the vectors into partsthat run along the x axis and parts that run along the y axis. These are called the x and ycomponents of the vector.In Figure 1.6A vector split up into components: One component is a vector that runsalong the x axis; the other is one running along the y axis.If we let A be the magnitude of vector A and θ is its direction as measured counter–

71.1. THE IMPORTANT STUFFyAyxxA(a)(b)Figure 1.7: Vectors can have negative components when they’re in the other quadrants.clockwise from the x axis, then the component of this vector that runs along x has lengthAx , where the relation between the two is:Ax A cos θ(1.8)Likewise, the length of the component that runs along y isAy A sin θ(1.9)Actually, we don’t literally mean “length” here since that implies a positive number.When the vector A has a direction lying in quadrants II, III or IV (as in Figure 1.7, thenone of its components will be negative. For example, if the vector’s direction is in quadrantII as in Fig. 1.7(a), its x component is negative while its y component is positive.Now if we have the components of a vector we can find its magnitude and direction bythe following relations:qAyA A2x A2y tan θ (1.10)Axwhere θ is the angle which gives the direction of A, measured counterclockwise from the xaxis.Once we have the x and y components of two vectors it is easy to add the vectors sincethe x components of the individual vectors add to give the x component of the sum, andthe y components of the individual vectors add to give the y component of the sum. This isillustrated in Figure 1.8. Expressing this with math, if we say that A B

Worked Examples from Introductory Physics (Algebra–Based) Vol. I: Basic Mechanics David Murdock, TTU October 3, 2012

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