1 Worked Examples from Introductory Physics Vol. I: Basic Mechanics David Murdock Tenn. Tech. Univ. February 24, 2005
3 Contents To the Student. Yeah, You. i 1 Units and Vectors: Tools for Physics The Important Stuff TheSISystem Changing Units Density Dimensional Analysis Vectors; Vector Addition Multiplying Vectors Worked Examples Changing Units Density Dimensional Analysis Vectors; Vector Addition Multiplying Vectors Motion in One Dimension The Important Stuff Position, Time and Displacement Average Velocity and Average Speed Instantaneous Velocity and Speed Acceleration Constant Acceleration FreeFall Worked Examples Average Velocity and Average Speed Acceleration Constant Acceleration FreeFall Motion in Two and Three Dimensions The Important Stuff Position
4 4 CONTENTS Velocity Acceleration Constant Acceleration in Two Dimensions Projectile Motion Uniform Circular Motion Relative Motion Worked Examples Velocity Acceleration Constant Acceleration in Two Dimensions Projectile Motion Uniform Circular Motion Relative Motion Forces I The Important Stuff Newton s First Law Newton s Second Law Examples of Forces Newton s Third Law Applying Newton s Laws Worked Examples Newton s Second Law Examples of Forces Applying Newton s Laws Forces and Motion II The Important Stuff Friction Forces Uniform Circular Motion Revisited Newton s Law of Gravity (Optional for Calculus Based) Worked Examples Friction Forces Uniform Circular Motion Revisited Newton s Law of Gravity (Optional for Calculus Based) Work, Kinetic Energy and Potential Energy The Important Stuff Kinetic Energy Work SpringForce The Work Kinetic Energy Theorem Power Conservative Forces
5 CONTENTS Potential Energy Conservation of Mechanical Energy Work Done by Non Conservative Forces Relationship Between Conservative Forces and Potential Energy (Optional?) Other Kinds of Energy Worked Examples Kinetic Energy Work The Work Kinetic Energy Theorem Power Conservation of Mechanical Energy Work Done by Non Conservative Forces Relationship Between Conservative Forces and Potential Energy (Optional?) Linear Momentum and Collisions The Important Stuff Linear Momentum Impulse, Average Force Conservation of Linear Momentum Collisions The Center of Mass The Motion of a System of Particles Worked Examples Linear Momentum Impulse, Average Force Collisions Two Dimensional Collisions The Center of Mass The Motion of a System of Particles Appendix A: Conversion Factors 173
6 6 CONTENTS
7 To the Student. Yeah, You. Physics is learned through problem-solving. There is no other way. Problem solving can be very hard to learn, and students often confuse it with the algebra with which one finishes up a problem. But the level of mathematics and calculator skills required in a general physics course is not very great. Any student who has difficulty solving the equations we derive in working these problems really needs to re take some math courses! Physics is all about finding the right equations to solve. The rest of it ought to be easy. My two purposes in composing this book are: (1) To summarize the principles that are absolutely essential in first year physics. This is the material which a student must be familiar with before going in to take an exam. (2) To provide a set of example problems with the most complete, clearest solutions that I know how to give. I hope I ve done something useful in writing this. Of course, nowadays most physics textbooks give lots of example problems (many more than they did in years past) and even some sections on problem solving skills, and there are study guide type books one can buy which have many worked examples in physics. But typically these books don t have enough discussion as to how to set up the problem and why one uses the particular principles to solve them; usually I find that there aren t enough words included between the equations that are written down. Students seem to think so too. Part of the reason for my producing this notebook is the reaction of many students to earlier example notebooks I have written up: You put lots of words between all the equations! At present, most of the problems are taken from the popular calculus based textbooks by Halliday, Resnick and Walker and by Serway. I have copied down the problems nearly verbatim from these books, except possibly to change a number here and there. There s a reason for this: Students will take their exams from individual professors who will state their problems in their own way, and they just have to get used to the professors styles and answer the questions as their teachers have posed them. Style should not get in the way of physics. Organization of the Book: The chapters cover material roughly in the order that it is presented in your physics course, though there may be some differences. The chapters do not correspond to the same chapters in your textbook. Each chapter begins with a summary of the basic principles, where I give the most important equations that we will need in solving the problems. I i
8 ii TO THE STUDENT. YEAH, YOU. have called this The Important Stuff because... well, you get the idea. In general I give no derivations of the equations though learning the derivations is an important part of an education in physics. I refer you to your textbook for those. After that, I give worked examples. The emphasis here is to show how we try to clarify the situation presented in the problem (often with a picture), to show what principles and equations from the chapter are applicable to the situation, and finally to show how to use those equation to solve for the desired quantities. I have not been especially careful about numerical accuracy in solving these problems; oftentimes my results will have more or fewer significant figures than they should, if one takes the data given in the problem literally. But just as often, the textbook author who stated the problem wasn t very careful about accuracy either! Questions about accuracy are very important in lab work, but the focus of this book is problem solving. (After all, the problems are fictional!) I am continually correcting and updating this book, and the date on the title page indicates the version of the copy you are reading. I am sure that no matter what version you are reading there will be some errors and some sections which are incomplete. I apologize in advance. Reactions, please! Please help me with this project: Give me your reaction to this work: Tell me what you liked, what was particularly effective, what was particularly confusing, what you d like to see more of or less of. I can reached at or even at x If this effort is helping you to learn physics, I ll do more of it! DPM
9 Chapter 1 Units and Vectors: Tools for Physics 1.1 The Important Stuff The SI System Physics is based on measurement. Measurements are made by comparisons to well defined standards which define the units for our measurements. The SI system (popularly known as the metric system) is the one used in physics. Its unit of length is the meter, its unit of time is the second and its unit of mass is the kilogram. Other quantities in physics are derived from these. For example the unit of energy is the joule, defined by 1 J = 1 kg m2. s 2 As a convenience in using the SI system we can associate prefixes with the basic units to represent powers of 10. The most commonly used prefixes are given here: Factor Prefix Symbol pico- p 10 9 nano- n 10 6 micro- µ 10 3 milli- m 10 2 centi- c 10 3 kilo- k 10 6 mega- M 10 9 giga- G Other basic units commonly used in physics are: Time : 1 minute = 60 s 1 hour = 60 min etc. Mass : 1 atomic mass unit = 1 u = kg 1
10 2 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS Changing Units In all of our mathematical operations we must always write down the units and we always treat the unit symbols as multiplicative factors. For example, if me multiply 3.0kg by 2.0 m s we get (3.0kg) (2.0 m s )=6.0 kg m s We use the same idea in changing the units in which some physical quantity is expressed. We can multiply the original quantity by a conversion factor, i.e. a ratio of values for which the numerator is the same thing as the denominator. The conversion factor is then equal to 1,and so we do not change the original quantity when we multiply by the conversion factor. Examples of conversion factors are: Density ( 1 min 60 s ) ( ) 100 cm ( 1m 1yr day ) ( 1m ) 3.28 ft A quantity which will be encountered in your study of liquids and solids is the density of a sample. It is usually denoted by ρ and is defined as the ratio of mass to volume: ρ = m V (1.1) The SI units of density are kg m 3 but you often see it expressed in g cm Dimensional Analysis Every equation that we use in physics must have the same type of units on both sides of the equals sign. Our basic unit types (dimensions) are length (L), time (T ) and mass (M). When we do dimensional analysis we focus on the units of a physics equation without worrying about the numerical values Vectors; Vector Addition Many of the quantities we encounter in physics have both magnitude ( how much ) and direction. These are vector quantities. We can represent vectors graphically as arrows and then the sum of two vectors is found (graphically) by joining the head of one to the tail of the other and then connecting head to tail for the combination, as shown in Fig The sum of two (or more) vectors is often called the resultant. We can add vectors in any order we want: A + B = B + A. We say that vector addition is commutative. We express vectors in component form using the unit vectors i, j and k, which each have magnitude 1 and point along the x, y and z axes of the coordinate system, respectively.
11 1.1. THE IMPORTANT STUFF 3 B B A A A+B (a) (b) Figure 1.1: Vector addition. (a) shows the vectors A and B to be summed. (b) shows how to perform the sum graphically. y B y C B C y A y A x A B x x C x Figure 1.2: Addition of vectors by components (in two dimensions). Any vector can be expressed as a sum of multiples of these basic vectors; for example, for the vector A we would write: A = A x i + A y j + A z k. Here we would say that A x is the x component of the vector A; likewise for y and z. In Fig. 1.2 we illustrate how we get the components for a vector which is the sum of two other vectors. If then A = A x i + A y j + A z k and B = B x i + B y j + B z k A + B =(A x + B x )i +(A y + B y )j +(A z + B z )k (1.2) Once we have found the (Cartesian) component of two vectors, addition is simple; just add the corresponding components of the two vectors to get the components of the resultant vector. When we multiply a vector by a scalar, the scalar multiplies each component; If A is a vector and n is a scalar, then ca = ca x i + ca y j + ca z k (1.3)
12 4 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS In terms of its components, the magnitude ( length ) of a vector A (which we write as A) is given by: A = A 2 x + A 2 y + A 2 z (1.4) Many of our physics problems will be in two dimensions (x and y) and then we can also represent it in polar form. If A is a two dimensional vector and θ as the angle that A makes with the +x axis measured counter-clockwise then we can express this vector in terms of components A x and A y or in terms of its magnitude A and the angle θ. These descriptions are related by: A x = A cos θ A y = A sin θ (1.5) A = A 2 x + A 2 y tan θ = A y (1.6) A x When we use Eq. 1.6 to find θ from A x and A y we need to be careful because the inverse tangent operation (as done on a calculator) might give an angle in the wrong quadrant; one must think about the signs of A x and A y Multiplying Vectors There are two ways to multiply two vectors together. The scalar product (or dot product) of the vectors a and b is given by a b = ab cos φ (1.7) where a is the magnitude of a, b is the magnitude of b and φ is the angle between a and b. The scalar product is commutative: a b = b a. One can show that a b is related to the components of a and b by: a b = a x b x + a y b y + a z b z (1.8) If two vectors are perpendicular then their scalar product is zero. The vector product (or cross product) of vectors a and b is a vector c whose magnitude is given by c = ab sin φ (1.9) where φ is the smallest angle between a and b. The direction of c is perpendicular to the plane containing a and b with its orientation given by the right hand rule. One way of using the right hand rule is to let the fingers of the right hand bend (in their natural direction!) from a to b; the direction of the thumb is the direction of c = a b. This is illustrated in Fig The vector product is anti commutative: a b = b a. Relations among the unit vectors for vector products are: i j = k j k = i k i = j (1.10)
13 1.2. WORKED EXAMPLES 5 C A C A f B B (a) (b) Figure 1.3: (a) Finding the direction of A B. Fingers of the right hand sweep from A to B in the shortest and least painful way. The extended thumb points in the direction of C. (b) Vectors A, B and C. The magnitude of C is C = AB sin φ. The vector product of a and b can be computed from the components of these vectors by: a b =(a y b z a z b y )i +(a z b x a x b z )j +(a x b y a y b x )k (1.11) which can be abbreviated by the notation of the determinant: i j k a b = a x a y a z (1.12) b x b y b z 1.2 Worked Examples Changing Units 1. The Empire State Building is 1472 ft high. Express this height in both meters and centimeters. To do the first unit conversion (feet to meters), we can use the relation (see the Conversion Factors in the back of this book): 1m = ft We set up the conversion factor so that ft cancels and leaves meters: ( ) 1m 1472 ft = (1472 ft) = 448.6m ft So the height can be expressed as m. To convert this to centimeters, use: 1 m = 100 cm
14 6 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS and get: ( ) 100 cm m = (448.6m) = cm 1m The Empire State Building is cm high! 2. A rectangular building lot is 100.0ft by 150.0ft. Determine the area of this lot in m 2. is The area of a rectangle is just the product of its length and width so the area of the lot A = (100.0 ft)(150.0ft)= ft 2 To convert this to units of m 2 we can use the relation 1 m = ft but the conversion factor needs to be applied twice so as to cancel ft 2 and get m 2. We write: ( ) 1m ft 2 =( ft 2 ) = m ft The area of the lot is m The Earth is approximately a sphere of radius m. (a) What is its circumference in kilometers? (b) What is its surface area in square kilometers? (c) What is its volume in cubic kilometers? (a) The circumference of the sphere of radius R, i.e. the distance around any great circle is C =2πR. Using the given value of R we find: C =2πR =2π( m) = m. To convert this to kilometers, use the relation 1 km = 10 3 m in a conversion factor: ( ) 1km C = m=( m) = km 10 3 m The circumference of the Earth is km. (b) The surface area of a sphere of radius R is A =4πR 2. So we get A =4πR 2 =4π( m) 2 = m 2 Again, use 1 km = 10 3 m but to cancel out the units m 2 and replace them with km 2 it must be applied twice: ( ) 2 1km A = m 2 =( m 2 ) = km m
15 1.2. WORKED EXAMPLES 7 The surface area of the Earth is km 2. (c) The volume of a sphere of radius R is V = 4 3 πr3. So we get V = 4 3 πr3 = 4 3 π( m) 3 = m 3 Again, use 1 km = 10 3 m but to cancel out the units m 3 and replace them with km 3 it must be applied three times: ( ) 3 1km V = m 3 =( m 3 ) = km m The volume of the Earth is km Calculate the number of kilometers in 20.0mi using only the following conversion factors: 1 mi = 5280 ft, 1 ft = 12 in, 1in=2.54 cm, 1 m = 100 cm, 1 km = 1000 m. Set up the factors of 1 as follows: 20.0 mi = ( ) 5280 ft (20.0 mi) 1mi = 32.2km ( ) 12 in 1ft ( 2.54 cm 1in ) ( ) ( ) 1m 1km 100 cm 1000 m Setting up the factors of 1 in this way, all of the unit symbols cancel except for km (kilometers) which we keep as the units of the answer. 5. One gallon of paint (volume = m 3 ) covers an area of 25.0m 3. What is the thickness of the paint on the wall? We will assume that the volume which the paint occupies while it s covering the wall is the same as it has when it is in the can. (There are reasons why this may not be true, but let s just do this and proceed.) The paint on the wall covers an area A and has a thickness τ; the volume occupied is the area time the thickness: V = Aτ. We have V and A; we just need to solve for τ: τ = V A = m 3 = m. 25.0m 2 The thickness is m. This quantity can also be expressed as mm. 6. A certain brand of house paint claims a coverage of 460 ft2 gal. (a) Express this quantity in square meters per liter. (b) Express this quantity in SI base units. (c) What is the inverse of the original quantity, and what is its physical significance?
16 8 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS (a) Use the following relations in forming the conversion factors: 1 m = 3.28 ft and 1000 liter = 264 gal. To get proper cancellation of the units we set it up as: ( ) ( ) 460 ft2 ft2 1m gal = (460 ) =11.3 m2 gal gal L 3.28 ft 1000 L (b) Even though the units of the answer to part (a) are based on the metric system, they are not made from the base units of the SI system, which are m, s, and kg. To make the complete conversion to SI units we need to use the relation 1 m 3 = 1000 L. Then we get: ( ) 11.3 m2 m L = (11.3 ) = m 1 L L 1m 3 So the coverage can also be expressed (not so meaningfully, perhaps) as m 1. (c) The inverse (reciprocal) of the quantity as it was originally expressed is ( ) 460 ft 2 1 gal = gal. ft 2 Of course when we take the reciprocal the units in the numerator and denominator also switch places! Now, the first expression of the quantity tells us that 460 ft 2 are associated with every gallon, that is, each gallon will provide 460 ft 2 of coverage. The new expression tells us that gal are associated with every ft 2, that is, to cover one square foot of surface with paint, one needs gallons of it. 7. Express the speed of light, m s in (a) feet per nanosecond and (b) millimeters per picosecond. (a) For this conversion we can use the following facts: to get: 1m=3.28 ft and 1 ns = 10 9 s ( ) 3.28 ft m = ( m) s s 1m = 0.98 ft ns In these new units, the speed of light is 0.98 ft ns. (b) For this conversion we can use: 1 mm = 10 3 m and 1 ps = s ( 10 9 ) s 1ns and set up the factors as follows: ( ) ( 1mm m = ( m 12 ) ) s s s 10 3 m 1ps = mm ps
17 1.2. WORKED EXAMPLES 9 1 mm In these new units, the speed of light is ps 8. One molecule of water (H 2 O) contains two atoms of hydrogen and one atom of oxygen. A hydrogen atom has a mass of 1.0u and an atom of oxygen has a mass of 16 u, approximately. (a) What is the mass in kilograms of one molecule of water? (b) How many molecules of water are in the world s oceans, which have an estimated total mass of kg? (a) We are given the masses of the atoms of H and O in atomic mass units; using these values, one molecule of H 2 O has a mass of m H2 O = 2(1.0 u) + 16 u = 18 u Use the relation between u (atomic mass units) and kilograms to convert this to kg: ( ) kg m H2 O = (18 u) = kg 1u One water molecule has a mass of kg. (b) To get the number of molecules in all the oceans, divide the mass of all the oceans water by the mass of one molecule:... a large number of molecules! Density N = kg kg = Calculate the density of a solid cube that measures 5.00 cm on each side and has a mass of 350 g. The volume of this cube is So from Eq. 1.1 the density of the cube is V =(5.00 cm) (5.00 cm) (5.00 cm) = 125 cm 3 ρ = m V = 350 g 125 cm 3 =2.80 g cm The mass of the planet Saturn is kg and its radius is m. Calculate its density.
18 10 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS r 2 r 1 Figure 1.4: Cross section of copper shell in Example 11. The planet Saturn is roughly a sphere. (But only roughly! Actually its shape is rather distorted.) Using the formula for the volume of a sphere, we find the volume of Saturn: V = 4 3 πr3 = 4 3 π( m) 3 = m 3 Now using the definition of density we find: ρ = m V = kg kg = m3 m 3 g While this answer is correct, it is useful to express the result in units of. Using our cm 3 conversion factors in the usual way, we get: ( kg =( kg 3 ) ( ) g 1m 3 ) =0.623 g m 3 m 3 cm 1kg 100 cm 3 The average density of Saturn is g cm 3. Interestingly, this is less than the density of water. 11. How many grams of copper are required to make a hollow spherical shell with an inner radius of 5.70 cm and an outer radius of 5.75 cm? The density of copper is 8.93 g/ cm 3. A cross section of the copper sphere is shown in Fig The outer and inner radii are noted as r 2 and r 1, respectively. We must find the volume of space occupied by the copper metal; this volume is the difference in the volumes of the two spherical surfaces: With the given values of the radii, we find: V copper = V 2 V 1 = 4 3 πr πr3 1 = 4 3 π(r3 2 r 3 1) V copper = 4 3 π((5.75 cm)3 (5.70 cm) 3 )=20.6cm 3 Now use the definition of density to find the mass of the copper contained in the shell: ρ = m copper V copper = m copper = ρv copper = ( 8.93 g cm 3 ) (20.6cm 3 )=184g
19 1.2. WORKED EXAMPLES grams of copper are required to make the spherical shell of the given dimensions. 12. One cubic meter (1.00 m 3 ) of aluminum has a mass of kg, and 1.00 m 3 of iron has a mass of kg. Find the radius of a solid aluminum sphere that will balance a solid iron sphere of radius 2.00 cm on an equal arm balance. In the statement of the problem, we are given the densities of aluminum and iron: ρ Al = kg m 3 and ρ Fe = kg m 3. A solid iron sphere of radius R =2.00 cm = m has a volume V Fe = 4 3 πr3 = 4 3 π( m) 3 = m 3 so that from M Fe = ρ Fe V Fe we find the mass of the iron sphere: M Fe = ρ Fe V Fe = ( kg m 3 ) ( m 3 )= kg If this sphere balances one made from aluminum in an equal arm balance, then they have the same mass. So M Al = kg is the mass of the aluminum sphere. From M Al = ρ Al V Al we can find its volume: V Al = M Al ρ Al = kg kg m 3 = m 3 Having the volume of the sphere, we can find its radius: ( ) 1 3VAl 3 V Al = 4 3 πr3 = R = 4π This gives: ( 3( m 3 ) 1 3 ) R = = m=2.86 cm 4π The aluminum sphere must have a radius of 2.86 cm to balance the iron sphere Dimensional Analysis 13. The period T of a simple pendulum is measured in time units and is l T =2π g. where l is the length of the pendulum and g is the free fall acceleration in units of length divided by the square of time. Show that this equation is dimensionally correct.
20 12 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS The period (T ) of a pendulum is the amount of time it takes to makes one full swing back and forth. It is measured in units of time so its dimensions are represented by T. On the right side of the equation we have the length l, whose dimensions are represented by L. We are told that g is a length divided by the square of a time so its dimensions must be L/T 2. There is a factor of 2π on the right side, but this is a pure number and has no units. So the dimensions of the right side are: ( L ) L = T 2 = T T 2 so that the right hand side must also have units of time. Both sides of the equation agree in their units, which must be true for it to be a valid equation! 14. The volume of an object as a function of time is calculated by V = At 3 + B/t, where t is time measured in seconds and V is in cubic meters. Determine the dimension of the constants A and B. Both sides of the equation for volume must have the same dimensions, and those must be the dimensions of volume where are L 3 (SI units of m 3 ). Since we can only add terms with the same dimensions, each of the terms on right side of the equation (At 3 and B/t) must have the same dimensions, namely L 3. Suppose we denote the units of A by [A]. Then our comment about the dimensions of the first term gives us: [A]T 3 = L 3 = [A]= L3 T 3 so A has dimensions L 3 /T 3. In the SI system, it would have units of m 3 / s 3. Suppose we denote the units of B by [B]. Then our comment about the dimensions of the second term gives us: [B] T = L3 = [B] =L 3 T so B has dimensions L 3 T. In the SI system, it would have units of m 3 s. 15. Newton s law of universal gravitation is F = G Mm r 2 Here F is the force of gravity, M and m are masses, and r is a length. Force has the SI units of kg m/s 2. What are the SI units of the constant G? If we denote the dimensions of F by [F ] (and the same for the other quantities) then then dimensions of the quantities in Newton s Law are: [M] =M (mass) [m] =M [r] =L [F ]: ML T 2
21 1.2. WORKED EXAMPLES 13 What we don t know (yet) is [G], the dimensions of G. Putting the known dimensions into Newton s Law, we must have: ML T =[G]M M 2 L 2 since the dimensions must be the same on both sides. Doing some algebra with the dimensions, this gives: ( ) ML L 2 [G] = T 2 M = L3 2 MT 2 so the dimensions of G are L 3 /(MT 2 ). In the SI system, G has units of m 3 kg s In quantum mechanics, the fundamental constant called Planck s constant, h, has dimensions of [ML 2 T 1 ]. Construct a quantity with the dimensions of length from h, a mass m, and c, the speed of light. The problem suggests that there is some product of powers of h, m and c which has dimensions of length. If these powers are r, s and t, respectively, then we are looking for values of r, s and t such that h r m s c t has dimensions of length. What are the dimensions of this product, as written? We were given the dimensions of h, namely [ML 2 T 1 ]; the dimensions of m are M, and the dimensions of c are L = LT 1 (it T is a speed). So the dimensions of h r m s c t are: [ML 2 T 1 ] r [M] s [LT 1 ] t = M r+s L 2r+t T r t where we have used the laws of combining exponents which we all remember from algebra. Now, since this is supposed to have dimensions of length, the power of L must be 1 but the other powers are zero. This gives the equations: r + s = 0 2r + t = 1 r t = 0 which is a set of three equations for three unknowns. Easy to solve! The last of them gives r = t. Substituting this into the second equation gives 2r + t =2( t)+t = t =1 = t = 1 Then r = +1 and the first equation gives us s = 1. With these values, we can confidently say that h r m s c t = h 1 m 1 c 1 = h mc has units of length.
22 14 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS c y x Figure 1.5: Vector c, found in Example 17. With c x = 9.0 and c y = +10.0, the direction of c is in the second quadrant Vectors; Vector Addition 17. (a) What is the sum in unit vector notation of the two vectors a =4.0i +3.0j and b = 13.0i +7.0j? (b) What are the magnitude and direction of a + b? (a) Summing the corresponding components of vectors a and b we find: a + b = ( )i +( )j = 9.0i +10.0j This is the sum of the two vectors is unit vector form. (b) Using our results from (a), the magnitude of a + b is a + b = ( 9.0) 2 + (10.0) 2 =13.4 and if c = a + b points in a direction θ as measured from the positive x axis, then the tangent of θ is found from ( ) cy tan θ = = 1.11 If we naively take the arctangent using a calculator, we are told: c x θ = tan 1 ( 1.11) = 48.0 which is not correct because (as shown in Fig. 1.5), with c x negative, and c y positive, the correct angle must be in the second quadrant. The calculator was fooled because angles which differ by multiples of 180 have the same tangent. The direction we really want is θ = = Vector a has magnitude 5.0m and is directed east. Vector b has magnitude 4.0m and is directed 35 west of north. What are (a) the magnitude and (b) the
23 1.2. WORKED EXAMPLES 15 y N b 4.0 m 35 o W 5.0 m a E x S Figure 1.6: Vectors a and b as given in Example 18. direction of a + b? What are (c) the magnitude and (d) the direction of b a? Draw a vector diagram for each combination. (a) The vectors are shown in Fig (On the axes are shown the common directions N, S, E, W and also the x and y axes; North is the positive y direction, East is the positive x direction, etc.) Expressing the vectors in i, j notation, we have: and a =(5.00 m)i b = (4.00 m) sin 35 +(4.00 m) cos 35 c irc = ( 2.29 m)i +(3.28 m)j So if vector c is the sum of vectors a and b then: The magnitude of c is c x = a x + b x =(5.00 m) + ( 2.29 m) = 2.71 m c y = a y + b y =(0.00 m) + (3.28 m) = 3.28 m c = c 2 x + c 2 y = (2.71 m) 2 +(3.28 m) 2 =4.25 m (b) If the direction of c, as measured counterclockwise from the +x axis is θ then tan θ = c y c x = 3.28 m 2.71 m =1.211 then the tan 1 operation on a calculator gives θ = tan 1 (1.211) = 50.4 and since vector c must lie in the first quadrant this angle is correct. We note that this angle is =39.6
24 16 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS y N c b d -a b y N W a E x W E x S S (a) (b) Figure 1.7: (a) Vector diagram showing the addition a + b. (b) Vector diagram showing b a. just shy of the +y axis (the North direction). So we can also express the direction by saying it is 39.6 East of North. A vector diagram showing a, b and c is given in Fig. 1.7(a). (c) If the vector d is given by d = b a then the components of d are given by The magnitude of c is d x = b x a x =( 2.29 m) (5.00 m) = 7.29 m c y = a y + b y =(3.28 m) (0.00 m) + (3.28 m) = 3.28 m d = d 2 x + d 2 y = ( 7.29 m) 2 +(3.28 m) 2 =8.00 m (d) If the direction of d, as measured counterclockwise from the +x axis is θ then tan θ = d y = 3.28 m d x 7.29 m = Naively pushing buttons on the calculator gives θ = tan 1 ( 0.450) = 24.2 which can t be right because from the signs of its components we know that d must lie in the second quadrant. We need to add 180 to get the correct answer for the tan 1 operation: θ = = 156 But we note that this angle is =24 shy of the y axis, so the direction can also be expressed as 24 North of West. A vector diagram showing a, b and d is given in Fig. 1.7(b). 19. The two vectors a and b in Fig. 1.8 have equal magnitudes of 10.0m. Find
25 1.2. WORKED EXAMPLES 17 y b 105 o a 30 o x Figure 1.8: Vectors for Example 19. (a) the x component and (b) the y component of their vector sum r, (c) the magnitude of r and (d) the angle r makes with the positive direction of the x axis. (a) First, find the x and y components of the vectors a and b. The vector a makes an angle of 30 with the +x axis, so its components are a x = a cos 30 = (10.0 m) cos 30 =8.66 m a y = a sin 30 = (10.0 m) sin 30 =5.00 m The vector b makes an angle of 135 with the +x axis (30 plus 105 more) so its components are b x = b cos 135 = (10.0 m) cos 135 = 7.07 m b y = b sin 135 = (10.0 m) sin 135 =7.07 m Then if r = a + b, the x and y components of the vector r are: r x = a x + b x =8.66 m 7.07m=1.59 m r y = a y + b y =5.00m+7.07 m = m So the x component of the sum is r x =1.59 m, and... (b)...the y component of the sum is r y =12.07 m. (c) The magnitude of the vector r is r = r 2 x + r2 y = (1.59 m) 2 + (12.07 m) 2 =12.18 m (d) To get the direction of the vector r expressed as an angle θ measured from the +x axis, we note: tan θ = r y r x =7.59 and then take the inverse tangent of 7.59: θ = tan 1 (7.59) = 82.5
26 18 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS y A 12.0 m C 20.0 o 15.0 m 40.0 o x Figure 1.9: Vectors A and C as described in Example 20. Since the components of r are both positive, the vector does lie in the first quadrant so that the inverse tangent operation has (this time) given the correct answer. So the direction of r is given by θ = In the sum A + B = C, vector A has a magnitude of 12.0m and is angled 40.0 counterclockwise from the +x direction, and vector C has magnitude of 15.0m and is angled 20.0 counterclockwise from the x direction. What are (a) the magnitude and (b) the angle (relative to +x) of B? (a) Vectors A and C are diagrammed in Fig From these we can get the components of A and C (watch the signs on vector C from the odd way that its angle is given!): A x = (12.0 m) cos(40.0 )=9.19 m C x = (15.0 m) cos(20.0 )= 14.1m A y = (12.0 m) sin(40.0 )=7.71 m C y = (15.0 m) sin(20.0 )= 5.13 m (Note, the vectors in this problem have units to go along with their magnitudes, namely m (meters).) Then from the relation A + B = C it follows that B = C A, and from this we find the components of B: B x = C x A x = 14.1m 9.19 m = 23.3m B y = C y A y = 5.13 m 7.71 m = 12.8m Then we find the magnitude of vector B: B = Bx 2 + By 2 = ( 23.3) 2 +( 12.8) 2 m=26.6m (b) We find the direction of B from: tan θ = ( ) By B x =0.551 If we naively press the atan button on our calculators to get θ, we are told: θ = tan 1 (0.551) = 28.9 (?)
27 1.2. WORKED EXAMPLES 19 which cannot be correct because from the components of B (both negative) we know that vector B lies in the third quadrant. So we need to ad 180 to the naive result to get the correct answer: θ = = This is the angle of B, measured counterclockwise from the +x axis. 21. If a b =2c, a + b =4c and c =3i +4j, then what are a and b? We notice that if we add the first two relations together, the vector b will cancel: (a b)+(a + b) =(2c)+(4c) which gives: 2a =6c = a =3c and we can use the last of the given equations to substitute for c; weget a =3c = 3(3i +4j) =9i +12j Then we can rearrange the first of the equations to solve for b: b = a 2c =(9i +12j) 2(3i +4j) = (9 6)i + (12 8)j = 3i +4j So we have found: a =9i +12j and b =3i +4j 22. If A =(6.0i 8.0j) units, B =( 8.0i +3.0j) units, and C = (26.0i +19.0j) units, determine a and b so that aa + bb + C =0. The condition on the vectors given in the problem: aa + bb + C =0 is a condition on the individual components of the vectors. It implies: So that we have the equations: aa x + bb x + C x = 0 and aa y + bb y + C y =0. 6.0a 8.0b = 0 8.0a +3.0b = = 0
28 20 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS y A 45 0 B 45 0 x C Figure 1.10: Vectors for Example 23 We have two equations for two unknowns so we can find a and b. The are lots of ways to do this; one could multiply the first equation by 4 and the second equation by 3 to get: Adding these gives 24.0a 32.0b = a +9.0b = = b = 0 = b = =7.0 and then the first of the original equations gives us a: 6.0a =8.0b 26.0 =8.0(7.0) 26.0 =30.0 = a = =5.0 and our solution is a =7.0 b = Three vectors are oriented as shown in Fig. 1.10, where A = 20.0 units, B = 40.0 units, and C = 30.0 units. Find (a) the x and y components of the resultant vector and (b) the magnitude and direction of the resultant vector. (a) Let s first put these vectors into unit vector notation : A = 20.0j B = (40.0 cos 45 )i + (40.0 sin 45 )j =28.3i +28.3j C = (30.0 cos( 45 ))i + (30.0 sin( 45 ))j =21.2i 21.2j Adding the components together, the resultant (total) vector is: Resultant = A + B + C = ( )i + ( )j = 49.5i +27.1j
29 1.2. WORKED EXAMPLES 21 So the x component of the resultant vector is 49.5 and the y component of the resultant is (b) If we call the resultant vector R, then the magnitude of R is given by R = R 2 x + R 2 y = (49.5) 2 + (27.1) 2 =56.4 To find its direction (given by θ, measured counterclockwise from the x axis), we find: tan θ = R y = 27.1 R x 49.5 =0.547 and then taking the inverse tangent gives a possible answer for θ: θ = tan 1 (0.547) = Is this the right answer for θ? Since both components of R are positive, it must lie in the first quadrant and so θ must be between 0 and 90. So the direction of R is given by A vector B, when added to the vector C =3.0i+4.0j, yields a resultant vector that is in the positive y direction and has a magnitude equal to that of C. What is the magnitude of B? If the vector B is denoted by B = B x i + B y j then the resultant of B and C is B + C =(B x +3.0)i +(B y +4.0)j. We are told that the resultant points in the positive y direction, so its x component must be zero. Then: B x +3.0 =0 = B x = 3.0. Now, the magnitude of C is C = C 2 x + C2 y = (3.0) 2 +(4.0) 2 =5.0 so that if the magnitude of B + C is also 5.0 then we get B + C = (0) 2 +(B y +4.0) 2 =5.0 = (B y +4.0) 2 =25.0. The last equation gives (B y +4.0) = ±5.0 and apparently there are two possible answers B y =+1.0 and B y = 9.0 but the second case gives a resultant vector B + C which points in the negative y direction so we omit it. Then with B y =1.0 we find the magnitude of B: B = (B x ) 2 +(B y ) 2 = ( 3.0) 2 +(1.0) 2 =3.2 The magnitude of vector B is 3.2.
30 22 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS Multiplying Vectors 25. Vector A extends from the origin to a point having polar coordinates (7, 70 ) and vector B extends from the origin to a point having polar coordinates (4, 130 ). Find A B. We can use Eq. 1.7 to find A B. We have the magnitudes of the two vectors (namely A = 7 and B = 4) and the angle φ between the two is φ = =60. Then we get: A B = AB cos φ = (7)(4) cos 60 = Find the angle between A = 5i 3j +2kandB= 2j 2k. Eq. 1.7 allows us to find the cosine of the angle between two vectors as long as we know their magnitudes and their dot product. The magnitudes of the vectors A and B are: A = A 2 x + A 2 y + A 2 z = B = Bx 2 + By 2 + Bz 2 = and their dot product is: ( 5) 2 +( 3) 2 + (2) 2 =6.164 (0) 2 +( 2) 2 +( 2) 2 =2.828 A B = A x B x + A y B y + A z B z =( 5)(0) + ( 3)( 2) + (2)( 2)=2 Then from Eq. 1.7, if φ is the angle between A and B, wehave cos φ = A B AB = 2 (6.164)(2.828) =0.114 which then gives φ = Two vectors a and b have the components, in arbitrary units, a x = 3.2, a y =1.6, b x =0.50, b y =4.5. (a) Find the angle between the directions of a and b. (b) Find the components of a vector c that is perpendicular to a, is in the xy plane and has a magnitude of 5.0 units. (a) The scalar product has something to do with the angle between two vectors... if the angle between a and b is φ then from Eq. 1.7 we have: cos φ = a b. ab
31 1.2. WORKED EXAMPLES 23 We can compute the right hand side of this equation since we know the components of a and b. First, find a b. Using Eq. 1.8 we find: Now find the magnitudes of a and b: This gives us: From which we get φ by: a b = a x b x + a y b y = (3.2)(0.50) + (1.6)(4.5) = 8.8 a = a 2 x + a 2 y = b = b 2 x + b 2 y = (3.2) 2 +(1.6) 2 =3.6 (0.50) 2 +(4.5) 2 =4.5 cos φ = a b ab = 8.8 (3.6)(4.5) =0.54 φ = cos 1 (0.54) = 57 (b) Let the components of the vector c be c x and c y (we are told that it lies in the xy plane). If c is perpendicular to a then the dot product of the two vectors must give zero. This tells us: a c = a x c x + a y c y =(3.2)c x +(1.6)c y =0 This equation doesn t allow us to solve for the components of c but it does give us: c x = c y = 0.50c y Since the vector c has magnitude 5.0, we know that c = c 2 x + c 2 y =5.0 Using the previous equation to substitute for c x gives: c = c 2 x + c 2 y = ( 0.50 c y ) 2 + c 2 y = 1.25 c 2 y =5.0 Squaring the last line gives 1.25c 2 y =25 = c 2 y =20. = c y = ±4.5 One must be careful... there are two possible solutions for c y here. If c y =4.5 then we have c x = 0.50 c y =( 0.50)(4.5) = 2.2
32 24 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS But if c y = 4.5 then we have c x = 0.50 c y =( 0.50)( 4.5)=2.2 So the two possibilities for the vector c are c x = 2.2 c y =4.5 and c x =2.2 c y = Two vectors are given by A = 3i +4jandB=2i +3j. Find (a) A B and (b) the angle between A and B. (a) Setting up the determinant in Eq (or just using Eq for the cross product) we find: i j k A B = =(0 0)i +(0 0)j +(( 9) (8))k = 17k (b) To get the angle between A and B it is easiest to use the dot product and Eq The magnitudes of A and B are: A = A 2 x + A 2 y = and the dot product of the two vectors is ( 3) 2 + (4) 2 =5 B = Bx 2 + By 2 = (2) 2 + (3) 2 =3.61 A B = A x B x + A y B y + A z B z =( 3)(2) + (4)(3) = 6 so then if φ is the angle between A and B we get: cos φ = A B AB = 6 (5)(3.61) =0.333 which gives φ = Prove that two vectors must have equal magnitudes if their sum is perpendicular to their difference. Suppose the condition stated in this problem holds for the two vectors a and b. If the sum a + b is perpendicular to the difference a b then the dot product of these two vectors is zero: (a + b) (a b) =0
33 1.2. WORKED EXAMPLES 25 Use the distributive property of the dot product to expand the left side of this equation. We get: a a a b + b a b b But the dot product of a vector with itself gives the magnitude squared: a a = a 2 x + a 2 y + a 2 z = a 2 (likewise b b = b 2 ) and the dot product is commutative: a b = b a. Using these facts, we then have a 2 a b + a b + b 2 =0, which gives: a 2 b 2 =0 = a 2 = b 2 Since the magnitude of a vector must be a positive number, this implies a = b and so vectors a and b have the same magnitude. 30. For the following three vectors, what is 3C (2A B)? A =2.00i +3.00j 4.00k B = 3.00i j k C = 7.00i 8.00j Actually, from the properties of scalar multiplication we can combine the factors in the desired vector product to give: 3C (2A B) =6C (A B). Evaluate A B first: i j k A B = =( )i + ( )j +( )k =22.0i +8.0j +17.0k Then: C (A B) =(7.0)(22.0) (8.0)(8.0) + (0.0)(17.0) = 90 So the answer we want is: 6C (A B) = (6)(90.0) = A student claims to have found a vector A such that (2i 3j +4k) A =(4i +3j k).
34 26 CHAPTER 1. UNITS AND VECTORS: TOOLS FOR PHYSICS Do you believe this claim? Explain. Frankly, I ve been in this teaching business so long and I ve grown so cynical that I don t believe anything any student claims anymore, and this case is no exception. But enough about me; let s see if we can provide a mathematical answer. We might try to work out a solution for A, but let s think about some of the basic properties of the cross product. We know that the cross product of two vectors must be perpendicular to each of the multiplied vectors. So if the student is telling the truth, it must be true that (4i +3j k) is perpendicular to (2i 3j +4k). Is it? We can test this by taking the dot product of the two vectors: (4i +3j k) (2i 3j +4k) = (4)(2) + (3)( 3) + ( 1)(4) = 5. The dot product does not give zero as it must if the two vectors are perpendicular. So we have a contradiction. There can t be any vector A for which the relation is true.
35 Chapter 2 Motion in One Dimension 2.1 The Important Stuff Position, Time and Displacement We begin our study of motion by considering objects which are very small in comparison to the size of their movement through space. When we can deal with an object in this way we refer to it as a particle. In this chapter we deal with the case where a particle moves along a straight line. The particle s location is specified by its coordinate, which will be denoted by x or y. As the particle moves, its coordinate changes with the time, t. The change in position from x 1 to x 2 of the particle is the displacement x, with x = x 2 x Average Velocity and Average Speed When a particle has a displacement x in a change of time t, its average velocity for that time interval is by v = x t = x 2 x 1 (2.1) t 2 t 1 The average speed of the particle is absolute value of the average velocity and is given Distance travelled s = (2.2) t In general, the value of the average velocity for a moving particle depends on the initial and final times for which we have found the displacements Instantaneous Velocity and Speed We can answer the question how fast is a particle moving at a particular time t? by finding the instantaneous velocity. This is the limiting case of the average velocity when the time 27
36 28 CHAPTER 2. MOTION IN ONE DIMENSION interval t include the time t and is as small as we can imagine: x v = lim t 0 t = dx dt (2.3) The instantaneous speed is the absolute value (magnitude) of the instantaneous velocity. If we make a plot of x vs. t for a moving particle the instantaneous velocity is the slope of the tangent to the curve at any point Acceleration When a particle s velocity changes, then we way that the particle undergoes an acceleration. If a particle s velocity changes from v 1 to v 2 during the time interval t 1 to t 2 then we define the average acceleration as v = x t = x 2 x 1 t 2 t 1 (2.4) As with velocity it is usually more important to think about the instantaneous acceleration, given by v a = lim t 0 t = dv (2.5) dt If the acceleration a is positive it means that the velocity is instantaneously increasing; if a is negative, then v is instantaneously decreasing. Oftentimes we will encounter the word deceleration in a problem. This word is used when the sense of the acceleration is opposite that of the instantaneous velocity (the motion). Then the magnitude of acceleration is given, with its direction being understood Constant Acceleration A very useful special case of accelerated motion is the one where the acceleration a is constant. For this case, one can show that the following are true: v = v 0 + at (2.6) x = x 0 + v 0 t at2 (2.7) v 2 = v0 2 +2a(x x 0 ) (2.8) x = x (v v)t (2.9) In these equations, we mean that the particle has position x 0 and velocity v 0 at time t =0; it has position x and velocity v at time t. These equations are valid only for the case of constant acceleration.
37 2.2. WORKED EXAMPLES Free Fall An object tossed up or down near the surface of the earth has a constant downward acceleration of magnitude 9.80 m. This number is always denoted by g. Be very careful about s 2 the sign; in a coordinate system where the y axis points straight up, the acceleration of a freely falling object is a y = 9.80 m = g (2.10) s 2 Here we are assuming that the air has no effect on the motion of the falling object. For an object which falls for a long distance this can be a bad assumption. Remember that an object in free fall has an acceleration equal to 9.80 m while it is s 2 moving up, while it is moving down, while it is at maximum height... always! 2.2 Worked Examples Average Velocity and Average Speed 1. Boston Red Sox pitcher Roger Clemens could routinely throw a fastball at a horizontal speed of 160 km. How long did the ball take to reach home plate 18.4m hr away? We assume that the ball moves in a horizontal straight line with an average speed of 160 km/hr. Of course, in reality this is not quite true for a thrown baseball. We are given the average velocity of the ball s motion and also a particular displacement, namely x =18.4 m. Equation 2.1 gives us: v = x t = t = x v But before using it, it might be convenient to change the units of v. We have: v = 160 km hr ( ) ( ) 1000 m 1hr =44.4 m s 1km 3600 s Then we find: t = x v = 18.4m 44.4 m s =0.414 s The ball takes seconds to reach home plate.
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Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
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This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed
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Introduction to Vectors A vector is a physical quantity that has both magnitude and direction. An example is a plane flying NE at 200 km/hr. This vector is written as 200 Km/hr at 45. Another example is