Introduction and Mathematical Concepts Chapter 1 Introduction and Mathematical Concepts

Physics has developed out of the efforts 1.1 The Nature of Physics Physics has developed out of the efforts of men and women to explain our physical environment. Physics encompasses a remarkable variety of phenomena: planetary orbits radio and TV waves magnetism lasers many more!

Newton’s Laws → Rocketry Maxwell’s Equations → Telecommunications 1.1 The Nature of Physics Physics predicts how nature will behave in one situation based on the results of experimental data obtained in another situation. Newton’s Laws → Rocketry Maxwell’s Equations → Telecommunications

1.1.1. Which of the following individuals did not make significant contributions in physics? a) Galileo Galilei b) Isaac Newton c) James Clerk Maxwell d) Neville Chamberlain

1.1.1. Which of the following individuals did not make significant contributions in physics? a) Galileo Galilei b) Isaac Newton c) James Clerk Maxwell d) Neville Chamberlain

Physics experiments involve the measurement 1.2 Units Physics experiments involve the measurement of a variety of quantities. These measurements should be accurate and reproducible. The first step in ensuring accuracy and reproducibility is defining the units in which the measurements are made.

SI units meter (m): unit of length kilogram (kg): unit of mass second (s): unit of time

Let’s watch this brief clip on measurement

1.2 Units

1.2 Units

1.2 Units

The units for length, mass, and time (as well as a few others), are regarded as base SI units. These units are used in combination to define additional units for other important physical quantities such as force and energy.

1.1.2. Which of the following statements is not a reason that physics is a required course for students in a wide variety of disciplines? a) There are usually not enough courses for students to take. b) Students can learn to think like physicists. c) Students can learn to apply physics principles to a wide range of problems. d) Physics is both fascinating and fundamental. e) Physics has important things to say about our environment.

1.2.1. The text uses SI units. What do the “S” and the “I” stand for? a) Système International b) Science Institute c) Swiss Institute d) Systematic Information e) Strong Interaction

1.2.1. The text uses SI units. What do the “S” and the “I” stand for? a) Système International b) Science Institute c) Swiss Institute d) Systematic Information e) Strong Interaction

1.2.2. Which of the following units is not an SI base unit? a) slug b) meter c) kilogram d) second

1.2.2. Which of the following units is not an SI base unit? a) slug b) meter c) kilogram d) second

1.2.3. Complete the following statement: The standard meter is defined in terms of the speed of light because a) all scientists have access to sunlight. b) no agreement could be reached on a standard meter stick. c) the yard is defined in terms of the speed of sound in air. d) the normal meter is defined with respect to the circumference of the earth. e) it is a universal constant.

1.3 The Role of Units in Problem Solving THE CONVERSION OF UNITS 1 ft = 0.3048 m 1 mi = 1.609 km 1 hp = 746 W 1 liter = 10-3 m3

1.3 The Role of Units in Problem Solving Example 1 The World’s Highest Waterfall The highest waterfall in the world is Angel Falls in Venezuela, with a total drop of 979.0 m. Express this drop in feet. Since 3.281 feet = 1 meter, it follows that (3.281 feet)/(1 meter) = 1

1.3 The Role of Units in Problem Solving

1.3 The Role of Units in Problem Solving Reasoning Strategy: Converting Between Units 1. In all calculations, write down the units explicitly. 2. Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically. 3. Use the conversion factors located on the page facing the inside cover. Be guided by the fact that multiplying or dividing an equation by a factor of 1 does not alter the equation.

1.3 The Role of Units in Problem Solving Example 2 Interstate Speed Limit Express the speed limit of 65 miles/hour in terms of meters/second. Use 5280 feet = 1 mile and 3600 seconds = 1 hour and 3.281 feet = 1 meter.

1.3 The Role of Units in Problem Solving DIMENSIONAL ANALYSIS [L] = length [M] = mass [T] = time Is the following equation dimensionally correct?

1.3 The Role of Units in Problem Solving Is the following equation dimensionally correct?

1.3.1. Which one of the following statements concerning unit conversion is false? a) Units can be treated as algebraic quantities. b) Units have no numerical significance, so 1.00 kilogram = 1.00 slug. c) Unit conversion factors are given inside the front cover of the text. d) The fact that multiplying an equation by a factor of 1 does not change an equation is important in unit conversion. e) Only quantities with the same units can be added or subtracted.

1.3.1. Which one of the following statements concerning unit conversion is false? a) Units can be treated as algebraic quantities. b) Units have no numerical significance, so 1.00 kilogram = 1.00 slug. c) Unit conversion factors are given inside the front cover of the text. d) The fact that multiplying an equation by a factor of 1 does not change an equation is important in unit conversion. e) Only quantities with the same units can be added or subtracted.

1.3.2. Which one of the following pairs of units may not be added together, even after the appropriate unit conversions have been made? a) feet and centimeters b) seconds and slugs c) meters and miles d) grams and kilograms e) hours and years

1.3.2. Which one of the following pairs of units may not be added together, even after the appropriate unit conversions have been made? a) feet and centimeters b) seconds and slugs c) meters and miles d) grams and kilograms e) hours and years

1.3.3. Which one of the following terms is used to refer to the physical nature of a quantity and the type of unit used to specify it? a) scalar b) conversion c) dimension d) vector e) symmetry

1.3.3. Which one of the following terms is used to refer to the physical nature of a quantity and the type of unit used to specify it? a) scalar b) conversion c) dimension d) vector e) symmetry

1.3.4. In dimensional analysis, the dimensions for speed are

1.3.4. In dimensional analysis, the dimensions for speed are

1.4 Trigonometry

1.4 Trigonometry

1.4 Trigonometry

1.4 Trigonometry

1.4 Trigonometry

1.4 Trigonometry Pythagorean theorem:

1.4.1. Which one of the following terms is not a trigonometric function? a) cosine b) tangent c) sine d) hypotenuse e) arc tangent

1.4.1. Which one of the following terms is not a trigonometric function? a) cosine b) tangent c) sine d) hypotenuse e) arc tangent

1.4.2. For a given angle , which one of the following is equal to the ratio of sin /cos ? a) one b) zero c) sin1  d) arc cos  e) tan 

1.4.2. For a given angle , which one of the following is equal to the ratio of sin /cos ? a) one b) zero c) sin1  d) arc cos  e) tan 

1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the sine of the angle ? a)

1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the sine of the angle ? a)

1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the tangent of the angle  ? a)

1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the following ratios is equal to the tangent of the angle  ? a)

1.4.5. Which law, postulate, or theorem states the following: “The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.” a) Snell’s law b) Pythagorean theorem c) Square postulate d) Newton’s first law e) Triangle theorem

1.4.5. Which law, postulate, or theorem states the following: “The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.” a) Snell’s law b) Pythagorean theorem c) Square postulate d) Newton’s first law e) Triangle theorem

A scalar quantity is one that can be described by a single number: 1.5 Scalars and Vectors A scalar quantity is one that can be described by a single number: temperature, speed, mass A vector quantity deals inherently with both magnitude and direction: velocity, force, displacement

By convention, the length of a vector 1.5 Scalars and Vectors Arrows are used to represent vectors. The direction of the arrow gives the direction of the vector. By convention, the length of a vector arrow is proportional to the magnitude of the vector. 8 lb 4 lb

1.5 Scalars and Vectors

1.5.1. Which one of the following statements is true concerning scalar quantities? a) Scalar quantities have both magnitude and direction. b) Scalar quantities must be represented by base units. c) Scalar quantities can be added to vector quantities using rules of trigonometry. d) Scalar quantities can be added to other scalar quantities using rules of ordinary addition. e) Scalar quantities can be added to other scalar quantities using rules of trigonometry.

1.5.1. Which one of the following statements is true concerning scalar quantities? a) Scalar quantities have both magnitude and direction. b) Scalar quantities must be represented by base units. c) Scalar quantities can be added to vector quantities using rules of trigonometry. d) Scalar quantities can be added to other scalar quantities using rules of ordinary addition. e) Scalar quantities can be added to other scalar quantities using rules of trigonometry.

1.5.2. Which one of the following quantities is a vector quantity? a) the age of the pyramids in Egypt b) the mass of a watermelon c) the sun's pull on the earth d) the number of people on board an airplane e) the temperature of molten lava

1.5.2. Which one of the following quantities is a vector quantity? a) the age of the pyramids in Egypt b) the mass of a watermelon c) the sun's pull on the earth d) the number of people on board an airplane e) the temperature of molten lava

1. 5. 3. A vector is represented by an arrow 1.5.3. A vector is represented by an arrow. What is the significance of the length of the arrow? a) Long arrows represent velocities and short arrows represent forces. b) The length of the arrow is proportional to the magnitude of the vector. c) Short arrows represent accelerations and long arrows represent velocities. d) The length of the arrow indicates its direction. e) There is no significance to the length of the arrow.

1. 5. 3. A vector is represented by an arrow 1.5.3. A vector is represented by an arrow. What is the significance of the length of the arrow? a) Long arrows represent velocities and short arrows represent forces. b) The length of the arrow is proportional to the magnitude of the vector. c) Short arrows represent accelerations and long arrows represent velocities. d) The length of the arrow indicates its direction. e) There is no significance to the length of the arrow.

1.5.4. Which one of the following situations involves a vector quantity? a) The mass of the Martian soil probe was 250 kg. b) The overnight low temperature in Toronto was 4.0 C. c) The volume of the soft drink can is 0.360 liters. d) The velocity of the rocket was 325 m/s, due east. e) The light took approximately 500 s to travel from the sun to the earth.

1.5.4. Which one of the following situations involves a vector quantity? a) The mass of the Martian soil probe was 250 kg. b) The overnight low temperature in Toronto was 4.0 C. c) The volume of the soft drink can is 0.360 liters. d) The velocity of the rocket was 325 m/s, due east. e) The light took approximately 500 s to travel from the sun to the earth.

1.6 Vector Addition and Subtraction Often it is necessary to add one vector to another.

1.6 Vector Addition and Subtraction 3 m 5 m 8 m

1.6 Vector Addition and Subtraction

1.6 Vector Addition and Subtraction 2.00 m 6.00 m

1.6 Vector Addition and Subtraction 2.00 m 6.00 m

1.6 Vector Addition and Subtraction 6.32 m 2.00 m 6.00 m

1.6 Vector Addition and Subtraction When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.

1.6 Vector Addition and Subtraction

1.6.1. A and B are vectors. Vector A is directed due west and vector B is directed due north. Which of the following choices correctly indicates the directions of vectors A and B? a) A is directed due west and B is directed due north b) A is directed due west and B is directed due south c) A is directed due east and B is directed due south d) A is directed due east and B is directed due north e) A is directed due north and B is directed due west

1.6.1. A and B are vectors. Vector A is directed due west and vector B is directed due north. Which of the following choices correctly indicates the directions of vectors A and B? a) A is directed due west and B is directed due north b) A is directed due west and B is directed due south c) A is directed due east and B is directed due south d) A is directed due east and B is directed due north e) A is directed due north and B is directed due west

1.6.2. Which one of the following statements concerning vectors and scalars is false? a) In calculations, the vector components of a vector may be used in place of the vector itself. b) It is possible to use vector components that are not perpendicular. c) A scalar component may be either positive or negative. d) A vector that is zero may have components other than zero. e) Two vectors are equal only if they have the same magnitude and direction.

1.6.2. Which one of the following statements concerning vectors and scalars is false? a) In calculations, the vector components of a vector may be used in place of the vector itself. b) It is possible to use vector components that are not perpendicular. c) A scalar component may be either positive or negative. d) A vector that is zero may have components other than zero. e) Two vectors are equal only if they have the same magnitude and direction.

1.7 The Components of a Vector

1.7 The Components of a Vector

1.7 The Components of a Vector It is often easier to work with the scalar components rather than the vector components.

1.7 The Components of a Vector Example A displacement vector has a magnitude of 175 m and points at an angle of 50.0 degrees relative to the x axis. Find the x and y components of this vector.

1. 7. 1. A, B, and, C are three vectors 1.7.1. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A. In mathematical form, A = B + C. Which one of the following statements concerning the components of vectors B and C must be true if Ay = 0? a) The y components of vectors B and C are both equal to zero. b) The y components of vectors B and C when added together equal zero. c) By  Cy = 0 or Cy  By = 0 d) Either answer a or answer b is correct, but never both. e) Either answer a or answer b is correct. It is also possible that both are correct.

1. 7. 1. A, B, and, C are three vectors 1.7.1. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A. In mathematical form, A = B + C. Which one of the following statements concerning the components of vectors B and C must be true if Ay = 0? a) The y components of vectors B and C are both equal to zero. b) The y components of vectors B and C when added together equal zero. c) By  Cy = 0 or Cy  By = 0 d) Either answer a or answer b is correct, but never both. e) Either answer a or answer b is correct. It is also possible that both are correct.

1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25 relative to the x axis. Which of the following choices indicates the horizontal and vertical components of vector r? rx ry a) +22 km/h +66 km/h b) +39 km/h +79 km/h c) +79 km/h +39 km/h d) +66 km/h +22 km/h e) +72 km/h +48 km/h

1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25 relative to the x axis. Which of the following choices indicates the horizontal and vertical components of vector r? rx ry a) +22 km/h +66 km/h b) +39 km/h +79 km/h c) +79 km/h +39 km/h d) +66 km/h +22 km/h e) +72 km/h +48 km/h

1. 7. 3. A, B, and, C are three vectors 1.7.3. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A. Vector A has a magnitude of 88 units and it is directed at an angle of 44 relative to the x axis as shown. Find the scalar components of vectors B and C. Bx By Cx Cy a) 63 0 0 61 b) 0 61 63 0 c) 63 0 61 0 d) 0 63 0 61 e) 61 0 63 0

1. 7. 3. A, B, and, C are three vectors 1.7.3. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A. Vector A has a magnitude of 88 units and it is directed at an angle of 44 relative to the x axis as shown. Find the scalar components of vectors B and C. Bx By Cx Cy a) 63 0 0 61 b) 0 61 63 0 c) 63 0 61 0 d) 0 63 0 61 e) 61 0 63 0

1.8 Addition of Vectors by Means of Components

1.8 Addition of Vectors by Means of Components

1. 8. 1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s 1.8.1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s. Vector B has scalar components Bx = 22 m/s and By = 18 m/s. Determine the scalar components of vector C = A  B. Cx Cy a) 13 m/s 3 m/s b) 57 m/s 33 m/s c) 13 m/s 33 m/s d) 57 m/s 3 m/s e) 57 m/s 3 m/s

1. 8. 1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s 1.8.1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s. Vector B has scalar components Bx = 22 m/s and By = 18 m/s. Determine the scalar components of vector C = A  B. Cx Cy a) 13 m/s 3 m/s b) 57 m/s 33 m/s c) 13 m/s 33 m/s d) 57 m/s 3 m/s e) 57 m/s 3 m/s