Mathematical Concepts: A Comprehensive Guide with Worked Examples, Summaries of Physics

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Worked Examples from Introductory Physics

(Algebra–Based)

Vol. I: Basic Mechanics

David Murdock, TTU

October 3, 2012

• 1 Mathematical Concepts Preface i
  • 1.1 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
• 2 Motion in One Dimension
  • 2.1 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 is Constant
    • 2.1.6 Free-Fall
  • 2.2 Worked Examples
    • 2.2.1 Motion Where the Acceleration is Constant
    • 2.2.2 Free-Fall
• 3 Motion in Two Dimensions
  • 3.1 The Important Stuff
    • 3.1.1 Motion in Two Dimensions, Coordinates and Displacement
    • 3.1.2 Velocity and Acceleration 4 CONTENTS
    • 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 Example
  • 3.2 Worked Examples
    • 3.2.1 Velocity and Acceleration
    • 3.2.2 Motion for Constant Acceleration
    • 3.2.3 Free–Fall; Projectile Problems
• 4 Forces I
  • 4.1 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 Plane
    • 4.1.11 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
• 5 Forces II
  • 5.1 The Important Stuff
    • 5.1.1 Introduction
    • 5.1.2 Friction Forces
    • 5.1.3 An Important Example: Block Sliding Down Rough Inclined Plane
    • 5.1.4 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
• CONTENTS
• 6 Energy
  • 6.1 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 Conservation
    • 6.1.9 Power
  • 6.2 Worked Examples
    • 6.2.1 Kinetic Energy
    • 6.2.2 The Spring Force
    • 6.2.3 Solving Problems With Energy Conservation
• 7 Momentum
  • 7.1 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 Momentum
    • 7.1.5 Systems of Particles; The Center of Mass
    • 7.1.6 Finding the Center of Mass
  • 7.2 Worked Examples
• 8 Rotational Kinematics
  • 8.1 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 Acceleration
    • 8.2.4 Relation Between Angular and Linear Quantities
• 9 Rotational Dynamics 6 CONTENTS
  • 9.1 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
• 10 Oscillatory Motion
  • 10.1 The Important Stuff
    • 10.1.1 Introduction
    • 10.1.2 Harmonic Motion
    • 10.1.3 Displacement, Velocity and Acceleration
    • 10.1.4 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
• 11 Waves I
  • 11.1 The Important Stuff
    • 11.1.1 Introduction
    • 11.1.2 Principle of Superposition
    • 11.1.3 Harmonic Waves
    • 11.1.4 Waves on a String
    • 11.1.5 Sound Waves
    • 11.1.6 Sound Intensity
• CONTENTS - 11.1.7 The Doppler Effect
  • 11.2 Worked Examples
    • 11.2.1 Harmonic Waves
    • 11.2.2 Waves on a String
    • 11.2.3 Sound Waves

8 CONTENTS

ii PREFACE

Chapter 1

Mathematical Concepts

1.1 The Important Stuff

1.1.1 Measurement and Units in Physics

Physics is concerned with the relations between measured quantities in the natural world. We make 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 MKS system, 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 the exact relations:

1 cm = 10−^2 m and 1 in = 2.54 cm

From this we can get:

1 m = 3.281 ft and 1 km = 0.6214 mi

Everyone knows the (exact) relations between the common units of time:

1 minute = 60 sec 1 hour = 60 min 1 day = 24 h

and we also have the (pretty accurate) relation:

1 year = 365.24 days

Finally, the unit of mass is the kilogram. The meaning of mass is not so clear unless you have already studies physics. For now, suffice it to say that a mass of 1 kilogram has a weight of — pounds. Later on we will make the distinction between “mass” and “weight”.

1.1. THE IMPORTANT STUFF 3

x

y

x (^) y

z

(a) (^) (b)

Figure 1.1: (a) Rectangle with sides x and y. Area is A = xy. I hope you knew that. (b) Rectangular box with 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 seconds in the denominator (to cancel what we’ve got already; the conversion factor is still equal to 1). So:

3. 68 × 104 s = (3. 68 × 104 s)

( (^) 1 min

60 s

) = 613 min

1.1.3 Math: You Had This In High School. Oh, Yes You Did.

The mathematical demands of a “non–calculus” physics course are not extensive, but you do have to be proficient with the little bit of mathematics that we will use! It’s just the stuff you 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 us to express large and small numbers conveniently (and also express the precision of those numbers). We will need the basic algebra operations of powers and roots and we will solve equations to find the “unknowns”.

Usually the algebra will be very simple. But if we are ever faced with an equation that looks like ax^2 + bx + c = 0 (1.1)

where x is the unknown and a, b and c are given numbers (constants) then there are two possible answers for x which you can find from the quadratic formula:

x =

−b ±

b^2 − 4 ac 2 a

On occasion you will need to know some facts from geometry. Starting simple and working upwards, the simplest shapes are the rectangle and rectangular box, shown in Fig. 1.1. If

4 CHAPTER 1. MATHEMATICAL CONCEPTS

R R

(a) (b)

D

Figure 1.2: (a) Circle; C = πD = 2πR; A = πR^2. (b) Sphere; A = 4πR^2 ; V = 43 πR^3. You’ve seen these formulae before. Oh, yes you have.

R

h (^) h

A

(a) (^) (b)

Figure 1.3: (a) Circular cylinder of radius R and height h. Volume is V = πR^2 h. (b) Right cylinder of arbitrary 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 m^2. For the rectangular box with sides x, y and z, the volume is V = xyz. A volume is the product of three lengths so its units are m^3. 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). The distance around the circle is the circumference, C. The circumference and area A of the circle are given by C = πD = 2πR A = πR^2 (1.3) A sphere is specified by its radius R. The surface area A and volume V of a sphere are given by A = 4πR^2 V = 43 πR^3 (1.4) Another simple shape is the (right) circular cylinder, shown in Fig. 1.3(a). If the cylinder has radius R and height h, its volume is V = πR^2 h. This is a special case of the general right cylinder (see Fig. 1.3(b)) where if the area of the cross section is A and the height is h, the volume is V = Ah.

6 CHAPTER 1. MATHEMATICAL CONCEPTS

A

B

C

A

B

Figure 1.5: Vectors A and B are added to give the vector C = A + B.

A x

A y

A

y

x

Figure 1.6: Vector A is split up into components.

Vectors are represented by arrows which show their magnitude and direction. The laws of physics will require us to add vectors, and to represent this operation on paper, we add the arrows. The way to add arrows, say to add arrow A to arrow B we join the tail of B to the head of A and then draw a new arrow from the tail of A to the head of B. The result is A + B. This is shown in Figure 1.5. Vectors can be multiplied by ordinary numbers (called scalars), giving new vectors, as shown in Fig. 1.5.

1.1.6 Components of Vectors

Addition of vectors would be rather messy if we didn’t have an easy technique for handling the trigonometry. Vector addition is made much easier when we split the vectors into parts that run along the x axis and parts that run along the y axis. These are called the x and y components of the vector. In Figure 1.6A vector split up into components: One component is a vector that runs along 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–

1.1. THE IMPORTANT STUFF 7

y

x

y

x

A

A

(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 length Ax, where the relation between the two is:

Ax = A cos θ (1.8)

Likewise, the length of the component that runs along y is

Ay = 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, then one of its components will be negative. For example, if the vector’s direction is in quadrant II 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 by the following relations:

A =

√ A^2 x + A^2 y tan θ =

Ay Ax

where θ is the angle which gives the direction of A, measured counterclockwise from the +x axis.

Once we have the x and y components of two vectors it is easy to add the vectors since the x components of the individual vectors add to give the x component of the sum, and the y components of the individual vectors add to give the y component of the sum. This is illustrated in Figure 1.8. Expressing this with math, if we say that A + B = C, we mean

Ax + Bx = Cx and Ay + By = Cy (1.11)

One we have the x and y components of the total vector C, we can get the magnitude and direction of C with

C =

√ C x^2 + C y^2 and tan θC =

Cy Cx

1.2. WORKED EXAMPLES 9

(b) Using the fact that a milligram is a thousandth of a gram: 1 mg = 10−^3 g, and our answer from (a), we find

m = 5 × 10 −^3 g = (5 × 10 −^3 g)

( 1 mg 10 −^3 g

) = 5 mg

(c) Using the fact that a microgram is 10−^6 (one millionth) of a gram: 1 μg = 10−^6 g

m = 5 × 10 −^3 g = (5 × 10 −^3 g)

( 1 μg 10 −^6 g

) = 5 × 103 μg

2. Vesna Vulovic survived the longest fall on record without a parachute when her plane exploded and she fell 6 miles, 551 yards. What is the distance in meters? [CJ6 1-2]

Convert the two lengths (i.e. 6 miles and 551 yards) to meters and then find the sum. Use the fact that 1 mile equals 1.6093 km to get:

6 mile = (6 mile)

( 1 .6093 km 1 mile

) ( 103 m 1 km

) = 9656.1 m

and we can use the exact relation 1 in = 2.54 cm to get

551 yd = (551 yd)

( 36 in 1 yd

) ( 2 .54 cm 1 in

) ( 1 m 102 cm

)

= 503 .8 m

Add the two lengths:

LTotal = 9656.1 m + 503.8 m = 1. 0160 × 104 m

3. How many seconds are there in (a) one hour and thirty–five minutes and (b) one day? [CJ6 1-3]

(a) Change one hour to seconds using the unit–factor method:

1 h = (1 h)

( 60 min 1 h

) ( 60 s 1 min

) = 3600 s

10 CHAPTER 1. MATHEMATICAL CONCEPTS

Likewise change 35 min to seconds:

35 min = (35 min)

( (^) 60 s

1 min

) = 2100 s

The total is 1 h + 35 min = 3600 s + 2100s = 5700 s

(b) Change one day to seconds; use the unit factors:

1 day = (1 day)

( 24 h 1 day

) ( 60 min 1 h

) ( (^) 60 s

1 min

)

= 86 , 400 s

4. Bicyclists in the Tour de France reach speeds of 34. 0 miles per hour (mi/h) on flat sections of the road. What is this speed in (a) kilometers per hour (km/h) and (b) meters per second (m/s)? [CJ6 1-4]

(a) Use the relation between miles and kilometers:

1 mi = 1.609 km

to get

v = 34. 0 mi h = (34. 0 mi h )

( 1 .609 km 1 mi

) = 54. 7 km h

(b) Using our answer from (a) along with the relations

1 km = 10^3 m and 1 hr = (60 min)

( (^) 60 s

1 min

) = 3600 s

to get

v = (54. 7 km h )

( 1 h 3600 s

) ( 103 m 1 km

) = 15. 2 m s

1.2.2 Trigonometry

5. For the right triangle with sides as shown in Figure 1.9, find side x and the angle θ.