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Connecting Math to Physics A Glencoe Program Student Edition Teacher Wraparound Edition Teacher Chapter Resources Mini Lab Worksheets Physics Lab Worksheets Study Guide Section Quizzes Reinforcement Enrichment Transparency Masters Transparency Worksheets Chapter Assessment Teacher Classroom Resources Teaching Transparencies Laboratory Manual, Student Edition Laboratory Manual, Teacher Edition Probeware Laboratory Manual, Student Edition Probeware Laboratory Manual, Teacher Edition Forensics Laboratory Manual, Student Edition Forensics Laboratory Manual, Teacher Edition Supplemental Problems Additional Challenge Problems Pre-AP/Critical Thinking Problems Physics Test Prep: Studying for the End-of-Course Exam, Student Edition Physics Test Prep: Studying for the End-of-Course Exam, Teacher Edition Connecting Math to Physics Solutions Manual Technology Answer Key Maker ExamView® Pro Interactive Chalkboard McGraw-Hill Learning Network StudentWorks™ CD-ROM TeacherWorks™ CD-ROM physicspp.com Web site Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Physics: Principles and Problems program. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 ISBN 0-07-865894-2 Printed in the United States of America 1 2 3 4 5 6 7 8 9 045 09 08 07 06 05 04 Contents To the Teacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Activity 1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Activity 2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Activity 3 Linear Equations and Graphs . . . . . . . . . . . . . . . . 5 Activity 4 Vectors and Vector Addition . . . . . . . . . . . . . . . . . 7 Activity 5 Vector Addition in Two Dimensions . . . . . . . . . . 9 Activity 6 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . 11 Activity 7 Angular Measurements . . . . . . . . . . . . . . . . . . . . 13 Activity 8 Algebraic Principles . . . . . . . . . . . . . . . . . . . . . . . 15 Activity 9 Exponential Notation . . . . . . . . . . . . . . . . . . . . . 17 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Activity 10 Area and Volume . . . . . . . . . . . . . . . . . . . . . . . . 19 Activity 11 The Area Under a Line . . . . . . . . . . . . . . . . . . . . 21 Activity 12 Reducing Equations. . . . . . . . . . . . . . . . . . . . . . 23 Activity 13 Direct and Inverse Relationships . . . . . . . . . . . 25 Activity 14 Adding and Subtracting Fractions . . . . . . . . . . 27 Activity 15 Order of Operations . . . . . . . . . . . . . . . . . . . . . 29 Activity 16 Graphing the Sine and Cosine Curves . . . . . . . 31 Activity 17 Percentages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Activity 18 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Activity 19 Ratio and Proportion . . . . . . . . . . . . . . . . . . . . 37 Activity 20 The Mass of an Atom . . . . . . . . . . . . . . . . . . . . 39 Activity 21 Exponential Equations and Graphs . . . . . . . . . 41 Teacher Guide Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Physics: Principles and Problems Contents iii To the Teacher Connecting Math to Physics provides activities that help students develop 21 mathematical skills related to the study of physics. These activities provide additional instruction and practice to help students when and where they need it most. Skills range from basic mathematics skills, such as graphing techniques, to slightly more complex skills, such as exponential equations and graphs. CORRELATION TO PHYSICS: PRINCIPLES AND PROBLEMS The activities in Connecting Math to Physics coordinate with the following chapters in Physics: Principles and Problems. Use this chart to help plan the best way to use these activities with your class. Activity Use with: Chapter 1 Activity 2: Graphs Chapter 2 Activity 3: Linear Equations and Graphs Chapter 3 Activity 4: Vectors and Vector Addition Chapters 4 and 5 Activity 5: Vector Addition in Two Dimensions Chapter 5 Activity 6: Scientific Notation Chapter 7 Activity 7: Angular Measurements Chapter 8 Activity 8: Algebraic Principles Chapter 9 Activity 9: Exponential Notation Chapter 11 Activity 10: Area and Volume Chapter 13 Activity 11: The Area Under a Line Chapter 14 Activity 12: Reducing Equations Chapter 15 Activity 13: Direct and Inverse Relationships Chapter 16 Activity 14: Adding and Subtracting Fractions Chapter 17 Activity 15: Order of Operations Chapter 18 Activity 16: Graphing the Sine and Cosine Curves Chapter 19 Activity 17: Percentages Chapter 23 Activity 18: Inequalities Chapter 25 Activity 19: Ratio and Proportion Chapter 26 Activity 20: The Mass of an Atom Chapter 28 Activity 21: Exponential Equations and Graphs Chapter 30 iv To the Teacher Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Activity 1: Measurement Date Period Name ACTIVITY 1 Connecting Math to Physics Recording Measurements Chapter 1 of your textbook discusses how to record measurements using instruments. Practice reading instruments and recording measurements by doing the exercises below. Record the correct reading for each of the instruments shown. 1. Metric ruler Metric Ruler (all measurements are cm.) 2 4 6 8 b a. b. d 10 e a c. c d. e. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2. Graduated cylinder 3 Graduated Cylinder (all quantities are in mL.) 51 20 46 2 45 1 a a. 72 b b. 50 10 44 c c. d 70 e d. e. Significant Digits It is important to record a measurement so that it does not appear to be more precise than the equipment that was used to make the measurement allows. To indicate the precision of a measurement, use the correct number of significant digits. Chapter 1 in your textbook explains how to calculate the number of significant digits. Use your knowledge of significant digits to answer the questions below. Write the following measurements using the correct number of significant digits. 3. 28.25 mL, three significant digits 4. 54.047°C, three significant digits Physics: Principles and Problems Connecting Math to Physics 1 Name 1 Connecting Math to Physics continued 5. 600.006 km, four significant digits 6. 1356 kg 4.2 kg 19.891 kg 7. 10.8 cm 3.06 cm 8. 18.7 g 19.01 g 2.298 g Calculating Relative Uncertainty and Relative Error As you know, there is always uncertainty in any measurement. This uncertainty can affect the final result of an experiment. Therefore, you maybe asked to express uncertainty in terms of relative uncertainty. Relative uncertainty is determined using the following formula. Estimated uncertainty Relative uncertainty (%) 100 Actual Measurement ⏐Accepted value Experimental value⏐ Relative error (%) 100 Accepted value Answer the following questions. 9. The length and width of a room are 12.50 0.01 m and 9.63 0.01 m. Find the perimeter of the room and the uncertainty. 10. Five measurements were recorded for the mass of a paper clip: 7.998 g, 8.001 g, 8.001 g, 8.003 g, and 7.997 g. What is the best way to report the mass of the paper clip? 2 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Relative error is the ratio of an absolute error to the correct value of the measurement. You can calculate the relative error using the following formula. Date Period Name ACTIVITY 2 Connecting Math to Physics Analyzing Graphs As you have seen in the textbook, the study of physics involves many graphs. It is important to understand graphs and to have the ability to gather information from graphs in order to solve problems. It is easier to understand a graph if you know how to create a graph. Many of the graphs in the textbook are two-dimensional line graphs. Each graph consists of a horizontal axis and a vertical axis, indicating that there are two variables. One variable is represented by the horizontal axis, often called the x-axis, and the other variable is represented by the vertical axis, often called the y-axis. For example, the two-dimensional graph below is a plot of the equation d 8t. This equation has two variables, time, t, and distance, d. Time is on the horizontal axis, and distance is on the vertical axis. Distance (cm) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 25.0 20.0 15.0 10.0 5.00 0.0 1.0 2.0 3.0 4.0 5.0 Time (min) The line in the graph is a visual representation of the algebraic equation. Each point on the line has an x-coordinate and y-coordinate. The points that comprise the line are found by choosing a value for one variable, substituting this value into the equation, and solving for the other variable. Determining points from an equation and plotting these points is one method used to create a graph. This is called the point-plotting method. Some points used to plot the equation d 8t are in the table below. Time, t (min) d 8t Distance, d (cm) Point 0.0 d 8 0.0 0.0 (0.0, 0.0) 1.0 d 8 1.0 8.0 (1.0, 8.0) 2.0 d 8 2.0 16 (2.0, 16) 3.0 d 8 3.0 24 (3.0, 24) To use the point-plotting method, calculate the points, plot them on a graph, and connect the points to form a line or curve. Physics: Principles and Problems Connecting Math to Physics 3 Name Connecting Math to Physics continued Answer the following questions. 1. Plot the equation y x 1. Use the values shown for x to determine y. yx1 x y y Point 0.0 5.0 20.0 10.0 15.0 10.0 0.00 10.0 x 20.0 2. a. Graph the data below for a bicycle ride. t (min) d (km) 0.0 0.0 5.0 1.5 10.0 3.0 15.0 4.5 20.0 6.0 6.0 4.0 3.0 2.0 1.0 0.0 5.0 10 15 20 25 Time (min) b. Use the graph to determine how far the biker has traveled at 12 min. 3. Three friends run a race. The graph below describes their motion. Cecilia a. Who runs at the slowest speed? b. Who passes Aiesha first? Aiesha 75.0 Michelle 50.0 25.0 c. Who travels the farthest after 5 min? 0.0 5.0 10.0 Time (s) 4 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Distance (km) 5.0 Position (km) 2 Date Period Name ACTIVITY 3 Connecting Math to Physics Linear Equations and Graphs Algebraic equations show relationships between variables. Linear equations are a type of algebraic equation commonly used in physics. Examples of linear equations include v at, F ma, and x y 3. When a linear equation is plotted on a coordinate system, the graph is a line. A graph is a useful tool for visualizing an algebraic equation and for analyzing data. The figure below shows a vt graph of the equation v at, where a 2. Velocity (m/s) 8.0 6.0 4.0 2.0 0 1.0 2.0 3.0 4.0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Time (s) An equation is linear when the exponent on each variable is 1. (Note: When the exponent on a variable is 1, the variable may be written as either x or x1.) The equation v 2t is a linear equation and is plotted as a line because both v and t have exponents of one. Determine if each of the following equations is linear. 1. d vt 3. F ma 2. v2 at 4. x2 y3 2 A line has a constant direction, or slope. If the vertical direction of the coordinate system is represented by y and the horizontal direction by x, the slope is defined as the ratio of the change in the vertical direction, y, to the change in the horizontal direction, x. The y slope, or steepness, of a line never changes, and therefore, is the same for all segments x of a line. y2 y1 change in the vertical direction y Slope , where (x1, y1) and (x2, y2) x change in the horizontal direction x2 x1 are two points on the line. If two points on a line are (2,1) and (3,4), y2 y1 3 41 slope 3. When you substitute the (x, y) coordinates, 32 x2 x1 1 maintain the order of the variables according to the subscripts. Physics: Principles and Problems Connecting Math to Physics 5 Name 3 Connecting Math to Physics continued Find the slope of each of the following lines. 5. Two points on the line are (1, 1) and (5, 9). 6. Two points on the line are (0, 0) and (3, 6). 7. Refer to the figure on the previous page. a. Choose two points on the line and calculate the slope. b. Choose two different points on the line and calculate the slope. c. Compare the slope in problems 7a and 7b. Explain the results. Linear equations may be written into a form where the slope of the line and the location of the y-intercept (the point where the line crosses the y-axis) may be determined by simply reviewing the equation. This is called the slope-intercept form of a line. For a linear equation with x and y as the variables, the slope-intercept form of the equation is y mx b, where m is the slope and b is the y-coordinate of the y-intercept. (Note: The x-coordinate will be zero for the y-intercept and any other point on the y-axis.) The linear equation y 3x 2 may be rewritten in slope-intercept form as y 3x 2. From the equation in slope-intercept form, it is evident that the slope of the line is 3 and the y-intercept is (0, 2). If the line crosses the y-axis at the origin, b 0, the slope-intercept form of the equation becomes y mx. Rearrange the linear equation into slope-intercept form and find the slope and y-intercept. 8. y 1 4x 9. x y 8 6 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Slope-Intercept Form of a Line Date Period Name ACTIVITY 4 Connecting Math to Physics Vector Addition in One Dimension When vectors are arranged in a line, they are said to be in one dimension. To add vectors in one dimension, choose a positive direction and assign the vectors a positive or negative magnitude based on the direction in which they point. Then add the magnitudes of the vectors. If the sum of the vectors is positive, the resultant vector points in the positive direction. If the sum is negative, the resultant vector points in the negative direction. For example, to add the vectors shown below, right is chosen as the positive direction. Therefore, the 8.0-m vector is pointing in the positive direction and the 5.2-m vector is pointing in the negative direction. 8.0 m Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5.2 m Arrange the vectors so that the tip of the first vector is connected to the tail of the second. Draw the resultant from the tail of the first vector to the tip of the second. Adding the two vectors, 8.0 m (5.2 m) 2.8 m. The resultant vector’s magnitude is positive and to the right. 8.0 m 5.2 m 2.8 m Draw the resultant of the following vectors, indicating magnitude and direction. 1. 3.0 m 6.1 m 2. 1.5 N 3.0 N 5.0 N Vector Addition in Two Dimensions Using the Pythagorean Theorem Even relatively simple vector problems require addition in more than one dimension. For example, if you walk 0.5 km in one direction, then turn left and walk another 2 km, what is your displacement? To solve this problem, you need to add vectors in two dimensions. Like vectors along the same straight line, vectors in two dimensions can also be added by placing them tip to tail and then drawing the resultant from the tail of the first vector to the tip of the second. As long as the length and direction of each vector remain unchanged, the Physics: Principles and Problems Connecting Math to Physics 7 Name 4 Connecting Math to Physics continued vectors can be moved to achieve the desired tip-to-tail configuration. If the two vectors form a right angle, you create a right triangle by drawing the resultant. nt ta l su Re A triangle is called a right triangle if one of its angles is 90°. The Pythagorean theorem is an equation that can be used to calculate the missing side length in a right triangle. In this type of vector addition, the magnitude of the resultant is the hypotenuse of a right triangle. b R A 90° a B R2 A2 B2 R A2 B2 (3.0 cm )2 ( 4.0 cm )2 5.0 cm Find the missing side length of each of the right triangles described below. 3. If A 6.0 m and B 8.0 m, find R. 4. If Anna walks 12.0 m north and 16 m east, what is her displacement from the starting point? 8 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The hypotenuse, R, of a right triangle is the side of the triangle opposite the 90° angle. The angle opposite side A is a, and the angle opposite side B is b. The Pythagorean theorem states that R2 A2 B2. For example, if A 3.0 cm and B 4.0 cm, the length of R can be found by using the Pythagorean theorem. Date Period Name ACTIVITY 5 Connecting Math to Physics Vector Addition in Two Dimensions Using Trigonometric Ratios For vectors, both the magnitude (corresponding to the side length) and the direction (corresponding to the angle) must be defined. This can be accomplished using the trigonometric ratios of sine (sin) cosine (cos) and tangent (tan). Sin, cos, and tan create relationships between an angle in a right triangle and the side lengths. To find the measure of an angle, use the inverse of the trigonometric ratio, also called the arc. Figure A shows these relationships on the x, y coordinate system. A side opposite a sin a R hypotenuse A a sin1 R y opposite side hypotenuse Ry R sin a R Ay a x y Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. B side adjacent to a cos a R hypotenuse B a cos1 R adjacent side hypotenuse By R cos a R a Bx x y A side opposite a tan a B side adjacent to a A a tan1 B opposite side adjacent side Ay Bx tan a R Ay a Bx Figure A For each right triangle described below, find the missing side length or angle by using one of the trigonometric ratios. 1. If angle a 30.0 and B 4.0 km, find R. 2. If R 13 mm and A 5.0 mm, find angle a. 3. If you walk 60.0 m east and 80.0 m north, what is the displacement from the starting point (magnitude and angle)? 4. If you drive 100.0 km east and 150.0 km south, what is the displacement from the starting point (magnitude and angle)? Physics: Principles and Problems Connecting Math to Physics 9 x Name 5 Connecting Math to Physics continued y Algebraic Addition of Vector Components When more than two vectors are added together, the vectors should be separated into x- and y-components (resolved) based on the coordinate system chosen. In the x, y coordinate system (such as in Figure B) where is measured counterclockwise from the x-axis, Ax A cos and Ay A sin . A x Ax To add multiple vectors, first the x-components of all the vectors are added, forming the resultant vector’s x-component as shown in Figure C. y Ay Figure B y Cy C R B By Ry Ay A Ax Bx x x Rx Figure C Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Rx Ax Bx Cx Cx The same procedure is used for the y-components. Ry Ay By Cy The resultant vector’s magnitude is then found using the Pythagorean theorem, R2 Rx2 Ry2 and the angle of the resultant (the direction) is found using the inverse tangent of the quotient of the y-component divided by the x-component of the resultant, Ry tan1 . Rx Using what you know about trigonometry and vector addition, supply the information requested below. 5. If A 30.0 m and 50.0° ( here is the angle between A and the x-axis), find the magnitude of Ax and Ay using the trigonometric y ratios of sin and cos. C 6. If Ax 36 km and Ay 48 km, find the magnitude of A. 7. Use Figure D to find Ax, Ay, Bx, By, Cx, and Cy, where A 5.0 m, B 13 m, and C 8.0 m. Add the x- and y-components together to find Rx and Ry. Then find the magnitude and direction of R. Hint: create a new right triangle. 45 B 30.0 A 60.0 Figure D 10 Connecting Math to Physics x Physics: Principles and Problems Date Period Name ACTIVITY 6 Connecting Math to Physics Conventional Notation vs. Scientific Notation When numbers have many zeros, it is more convenient to write the numbers in scientific notation. The mass of Earth is approximately 5,970,000,000,000,000,000,000,000 kg. The mass of Earth in scientific notation is 5.971024 kg. A number written in scientific notation is actually two numbers multiplied by each other. The first number is between 1 and 10, and the second number is 10 with an exponent. To determine the first portion of a number in scientific notation, move the decimal point until a number between 1 and 10 is formed. For the mass of Earth, for example, the decimal point is placed between the 5 and the 9, forming 5.97. After the first portion has been determined, count the number of digit spaces the decimal was moved. Use this number as the exponent on the 10. For the mass of Earth, the decimal is moved 24 spaces, so the second portion of the number in scientific form is 1024. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. How do you know if the exponent is positive or negative? If the number between 1 and 10 is less than the original number, the exponent is positive. If the number between 1 and 10 is greater than the original number, the exponent is negative. If the exponent is positive, the number between 1 and 10 is multiplied by 10 raised to the exponent. For the mass of Earth, 5.97 is multiplied by 10 a total of 24 times to yield the original number (5.971024 5.97101010101010101010101010 101010101010101010101010). If the exponent on the 10 is negative, the number between 1 and 10 is divided by 10 raised to the exponent. Any number raised to a negative exponent is equal to the fraction with 1 in the numerator and the number raised to a positive exponent in the denominator. 1 Negative Exponent Rule: 10n 10n The mass of an ant is approximately 0.000004 kg. To form a number between 1 and 10, the decimal point is placed after the 4. The decimal was moved 6 spaces. This number is the exponent on the 10. Is the exponent negative or positive? Since the number between 1 and 10 is greater than the original number, the exponent is negative. For the mass of the ant, 0.000004 4106. To convert the number from scientific notation to conventional notation, 4 is divided by 106: 1 4 4106 4 0.000004. 101010101010 106 Express the following numbers in conventional notation. 1. 103 2. 106 Physics: Principles and Problems Connecting Math to Physics 11 Name 6 Connecting Math to Physics continued 3. 102 4. 105 Express the following numbers in scientific notation. 5. 50,000 6. 3,000,000 7. 850,000,000 8. 0.00009 9. 0.000000075 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 10. 0.0000000001234 Express the following numbers in conventional notation. 11. 2103 12. 7.1106 13. 4.081010 14. 1104 15. 3.5107 16. 6.63951011 12 Connecting Math to Physics Physics: Principles and Problems Date Period Name ACTIVITY 7 Connecting Math to Physics Angular Measurements Angular measurements indicate the amount of rotation. The unit most often used is the 1 degree, where 1° of a circle. In the figure below, represents the rotation, the 360 measurement of an angle. When 360°, a full circle has been measured. 3 4 2 3 rad 120 rad 135 2 rad 90 3 rad 60 4 rad 45 rad 180 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5 4 6 rad 30 0 rad 0 2 rad 360 rad 225 3 2 7 4 rad 315 rad 270 Another type of angular measurement is the radian. The radian measurement is equal to the arc length, d, of a circle with a unit radius (radius of 1). An arc is any portion of the circle’s perimeter, or circumference. The figure above shows the angular measurements of a circle in both degrees and radians. How is the length of an arc measured? Recall that the circumference of a circle is 2r. If r 1, C 2. For a full revolution, 2 rad 360°. For half of a revolution, C 2 rad rad rad 180°. To convert between rads and degrees, use the 2 2 equality rad 180° to form a conversion factor. Convert from radians to degrees. 3 1. rad 2 2. rad 6 Physics: Principles and Problems Connecting Math to Physics 13 Name 7 Connecting Math to Physics continued 3. rad 8 3 4. rad 4 Convert from degrees to radians. Use in your answers. (Hint: use the conversion factor to create a fraction, and then reduce the fraction.) 5. 360° 6. 45° 7. 135° 8. 60° Find the missing value. (Note: d refers to the arc length, not the diameter.) 9. When rad and r 2.0 m, find d. 10. When rad and r 9.0 mm, find d. 3 11. When 2 rad and d 2.0 cm, find r. 2 12. When rad and d 10.0 m, find r. 3 13. When d 10.0 mm and r 2.0 mm, find . 14. When d 90.0 km and r 30.0 km, find . 14 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. When the radius is not equal to one, the angular measurement in rads is not the same as the arc length, d. The relationship between d, r, and is as follows: d r, where is in radians (not degrees). When r 1, d . Date Period Name ACTIVITY 8 Connecting Math to Physics Properties of Real Numbers and Algebraic Terms To understand physics and the mathematics required when solving physics problems, it is necessary to know the basic properties of real numbers and algebraic terms. Real numbers include all of the numbers on a number line, as in the x-axis of a graph. The terms of an algebraic expression are the parts of the expression added together. For example, the terms of the algebraic expression 3x 4y 1 are 3x, 4y, and 1. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The basic properties of real numbers and algebraic terms include the commutative property, associative property, distributive property, and the property of equality. These properties allow equations to be rearranged and solved. The commutative property states that two numbers or algebraic terms can be added or multiplied in any order. For example, 3 2 2 3, x y y x, 3 2 2 3, and xy yx. The commutative property allows an equation to be rearranged. Therefore, it is proper to write F ma as F am. The answer will not change. The associative property states that when three or more terms are added or multiplied together, it does not matter which terms are added or multiplied together first. For example, (4 3) 2 4 (3 2) and (xy)z x(yz). The distributive property states that multiplication may be disbursed over addition: z(x y) zx zy. When two terms are added together and then multiplied by another term, the answer is equal to multiplying the added terms by the term outside the parentheses and then adding the two new terms together. For example, 2(3x 4) (2 3x) (2 4) 6x 8. Identify the property used. 1. tf ti ti tf 3. 3x 3y 3(x y) 2. 2(RB)cos 2R(B cos ) 4. A (B C) (A B) C When solving an algebraic equation, the goal is to isolate the variable on one side of the equals sign and to have all other terms on the opposite side of the equals sign. The equation must be rearranged to solve for the variable. A property of equality is used to move terms from one side of the equation to the other. This property states that any mathematical operation can be performed on one side of the equation as long as it is also performed on the other side of the equation. For example, 1 1. If 3 is added to both sides, 1 3 1 3 resulting in 4 4. Because 3 is added to both sides, the left and the right side remain equal. If x 1 2, x can be isolated on the left side of the equation if 1 is subtracted from both sides. x 1 1 2 1 and x 1. Because 1 is subtracted from both sides of the equation, the equality remains true. The opposite operation must be performed to move a term from one side of the equation to the other. The opposite operation of subtraction is addition and the opposite operation d of division is multiplication. To solve for d in the equation v , the t must be moved to t the left side to isolate d on the right side. If d is divided by t, t must be multiplied to cancel Physics: Principles and Problems Connecting Math to Physics 15 Name 8 Connecting Math to Physics continued it on the right side since multiplication is the opposite of division. To maintain equality, if d t is multiplied on the right side, it also must be multiplied on the left side. v ; t d t v t; tv d. t Use the commutative, associative, distributive, and equality properties to solve the equations. 5. x 5 8, solve for x. d 7. v , solve for t. t 6. 2z 4, solve for z. 8. I m(vf vi), solve for vf. Algebraic Substitution Derive the indicated equation using substitution and the commutative, associative, distributive, and equality properties. vf vi v 9. Change a to t using v vf vi. t a 10. Change F ma to I using a r, Fr, and I mr2. 16 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. In the textbook, the equation F ma was transformed into the equation Ft p. This conversion is possible using the properties described above and the concept of substitution. Whenever a variable is replaced with a number or algebraic expression, substitution has occurred. For example, if a 9.80 m/s2 in the equation F ma, the numerical value for a v may be substituted in for the variable. F m 9.80 m/s2. If a instead of a number, t v v F m. Because the expression is equal to a, it may be substituted into any equation t t for a. This is the first step in transforming F ma into Ft p. The complete steps are as v v follows: (1) F ma; (2) F m Substitution of a ; (3) Ft mv (Property of t t Equality, multiply both sides of the equation by t); (4) Ft m(vf vi) (Substitution of v vf vi); (5) Ft mvf mvi (Distributive Property); (6) Ft pf pi (Substitution of the momentum equation p mv) (7) Ft p (Substitution of p pf pi). Throughout the study of physics, equations will be manipulated to derive new equations. Date Period Name ACTIVITY 9 Connecting Math to Physics Exponential Notation 1 In the equation for kinetic energy, KE mv2, the variable v has an exponent of two. The 2 exponent is the small number to the upper right of the variable. The general form of exponential notation is b x, where b is the base and x is the exponent. Exponential notation is applicable to many situations where it is necessary to multiply a number or variable by itself several times. Rather than writing the base multiple times, it is written once with an exponent. The exponent is equal to the number of times the base is multiplied by itself. For example, v v v2 and 8 8 8 8 8 8 86. If a number or a variable has an exponent of one, the exponent is generally omitted: m1 m and 31 3. In the equation d a 2, the exponent is in the denominator. This equation may be rewritten as a dt2, t 1 where t has a negative exponent. The negative exponent rule states that x bx. For b 1 1 93 and 7 6(7) 67. An exponent also may be a fraction. Square example, 3 6 9 and cubed roots are examples of numbers written with exponents as fractions. The square 1 1 root sign, b, may be written as b 2 and the cubed root b may be written as b 3 . 3 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Rewrite the expression in exponential notation with a positive exponent. 2. 2 2 2 2 1 7. 34 1 8. zx 3. 1 9. 16 4. a a 10. c2 5. 75 11. 8 6. v8 12. x3 1. 9 9 9 3 3 Exponent product rule: When multiplying two quantities with the same base, the product may be written as the base with the exponents added together: b x by b xy . For example, 3233 323 35 243. The product rule may be illustrated by expanding the exponents: 3233 (3 3) (3 3 3) 35 243. Exponent quotient rule: When dividing two quantities with the same base, subtract 1 bx 43 1 b xy . For example, 435 42 . The quotient rule the exponents: y 2 5 16 b 4 4 may be illustrated by expanding the exponents: 1 43 444 1 1 . 44444 44 16 45 42 Physics: Principles and Problems Connecting Math to Physics 17 Name 9 Connecting Math to Physics continued Find the decimal value of the expression. 13. 104 6.04 19. 6.02 14. 2.003 39 20. 38 15. 8.001 8.002 2.02 21. 2.04 16. 5.04 5.06 42 22. 43 17. 7.06 7.05 52 23. 51 18. 9.01 9.01 Find the decimal value of the expression. 24. (2.0104) (3.0105) 25. (1.0106) (5.0103) 26. (4.2102) (1.0103) 6.0103 27. 2.0102 9.9104 28. 3.0102 18 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The exponent product and quotient rules are applicable to numbers written in scientific notation. Numbers in scientific notation have a number multiplied by an exponent with a base of 10. For example, 4.0103 and 2.1102 are numbers written in scientific notation. To find the product of these numbers, multiply the standard number portions and the base 10 portions separately. To multiply the base 10 portions, use the exponent product rule. (3.0103) (2.0102) (3.0 2.0) (103 102) 6.01032 6.0105. This same procedure may be used to divide the numbers in scientific notation, only the decimal numbers are divided and the exponent quotient rule is used. 4.0103 4.0 103 (4.0103) (2.1102) 1.91032 1.9101 2 2.110 2.1 102 Date Period Name ACTIVITY 10 Connecting Math to Physics The geometric concepts of area and volume are used when studying the properties of fluids. For example, pressure is the force on a surface divided by the area of the surface and for an ideal gas the pressure times the volume is proportional to the temperature. Area Area, A, is the number of square units needed to cover a surface. Some common shapes and the formulas for calculating the area of each shape are shown below: r Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. w h l Rectangle b Triangle A lw A 2 bh Circle 1 A r 2 To find the area of a shape, use the applicable formula(s). For example, the area of a rectangle with a length of 8.0 m and a width of 5.0 m is found using the formula A lw. Thus, A lw 8.0 m 5.0 m 4.0101 m2. To find the area of a circle with a radius of 3.00 km, use the formula A r2. The area of the circle, A (3.00 km)2 28.3 km2. Areas of shapes are additive. To find the area of a shape that consists of a rectangle and a triangle, add the area of the rectangle to the area of the triangle. Find the area of each of the following shapes described below. 1. A rectangular driveway that is 3.05 m wide and 64.0 m long 3. Circle with r 8.00 cm 4. A shape formed by the figure below 2. A flower garden in the shape of an equilateral triangle whose side is 6.25 m (Hint: The base is the length 3 of a side and the height is () 2 times the length of a side.) 4.80 m 4.80 m Physics: Principles and Problems Connecting Math to Physics 19 Name 10 Connecting Math to Physics continued Volume You have learned that the volume of water displaced by an object is proportional to the buoyant force. The volume, V, of a three-dimensional object is the amount of space it occupies. The units for volume are length units cubed, such as m3 or km3. Some common formulas for volume are shown below: r h h r w l Rectangular solid Right circular cylinder V lwh V r 2 h Sphere 4 V 3 r 3 4 (3.00 km)2 3 113 km3 Find the volume of the shape. 5. A physics laboratory workbook with l 27.7 cm, w 21.6 cm, and h 3.7 cm 6. A plastic jewel case for a computer CD-ROM with l 14.1 cm, w 12.4 cm, and h 1.0 mm 7. A salad crouton cube whose side measures 7.00 mm 8. A cylindrical juice glass with diameter 6.5 cm and h 11.0 cm 9. A basketball with diameter 22 cm 20 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Using the formula V lhw, the volume of a rectangular solid with a length of 8.0 m, a height of 5.0 m, and a width of 4.0 m is: V lhw 8.0 m5.0 m4.0 m 160 m3. 4 To find the volume of a sphere with a radius of 3.00 km, use the formula V r3. 3 4 V 3 Date Period Name ACTIVITY 11 Connecting Math to Physics The Area Under a Graph Graphs are used throughout your textbook to visualize the relationships between variables and to gain information. When a linear equation is plotted on a coordinate system, the graph is a line. Two important attributes of a graphed line are the slope and the area under the line. You have learned about the slope of a line, m. If the vertical direction of the coordinate system is represented by y and the horizontal direction is represented by x, the slope is defined as the ratio of the change in the vertical direction, y, to the change in the y horizontal direction, x: m . In the v-t graph below, the equation of the line is x v at, where a 2. The slope of the graph is equal to the constant acceleration, v2 v1 6.0 m/s 2.0 m/s a 2.0 m/s2. t2 t1 3.0 s 1.0 s Velocity (m/s) 6.0 4.0 2.0 0.0 m 1.0 2.0 3.0 y x 4.0 Time (s) The area under the graph is equal to the area of the shape that is formed by the axes and 1 1 the line. In the figure above, the shape is a triangle where A bh 4.0 s 8.0 m/s 2 2 16 m. Notice that the unit for vt is the meter. Therefore, the area under a v-t graph is always is equal to the displacement, d. The unit that results from the multiplication of the y-axis value and the x-axis value indicates the variable the area under the graph is equal to. In the figure below showing an F-d graph, the area under the graph is the area of the rectangle A bh 1.5 m20 N 30 Nm 30 J. The unit for Fd is the joule. Therefore, the area under the F-d graph is equal to the amount of work done. 30.0 Force (N) Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 8.0 20.0 10.0 0.00 0.50 1.00 1.50 Displacement (m) Physics: Principles and Problems Connecting Math to Physics 21 Name Connecting Math to Physics 11 continued For each of the following exercises, find the area under the graph and indicate the variable it is equal to based on the units. Acceleration (m/s2) 1. 6.0 5.0 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 5.0 6.0 7.0 Time (s) 2. Force (N) 4.0 2.0 0.0 2.0 4.0 6.0 8.0 3.0 4.0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Time (s) 3. Velocity (m/s) 4.0 3.0 2.0 1.0 0.0 1.0 2.0 Time (s) Force (N) 4. 3.0 0.0 2.0 Displacement (m) 22 Connecting Math to Physics Physics: Principles and Problems Date Period Name ACTIVITY 12 Connecting Math to Physics Reducing Equations An algebraic equation shows a relationship between variables. The equation may have one, two, three, or more variables. Whenever a variable in an equation is equal to zero, the equation may be reduced to a form that is easier to use. Recall the Doppler effect equation v vd that you learned in Chapter 15. This equation, fd fs , may be reduced to v vs v fd fs if the stationary detector velocity, vd, is equal to zero. How is the equation v vs reduced if both vd and vs are equal to zero? v f v v0 fd fs v0 s fs Reduce the following equations by replacing the indicated variable(s) with zero. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2 1. y x 8, where x 0. 5 2 2. y x 8, where y 0. 5 df di 3. Average velocity: v , where di 0. tf ti df di 4. Average velocity: v , where df 0. tf ti vf vi 5. Average acceleration: a , where vi 0 and ti 0. tf ti Physics: Principles and Problems Connecting Math to Physics 23 Name 12 Connecting Math to Physics continued vf vi 6. Average acceleration: a , where a 0. tf ti 7. Velocity with constant acceleration: vf 2 vi2 2a(df di), where di 0. 8. Velocity with constant acceleration: vf 2 vi2 2a(df di), where a 0. 9. Work: W Fd cos , where 0. 10. Momentum: m1v1i m2v2i m1v1f m2v2f , 11. Mechanical energy: hi 1 1 mvi2 mghi mvf 2 mghf , where vi 0 and hf 2 2 2 12. Heat energy: Q mC(Tf Ti), where Ti 0. 13. Velocity, v, of a mass at the end of a pendulum, length, l, after being released at rest from an angle, , with the vertical: cos ) v 2gl(1 , where 90° 24 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. where v2i v1f 0 Date Period Name ACTIVITY 13 Connecting Math to Physics Direct and Inverse Relationships Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. In Chapter 7, direct and inverse relationships of variables were discussed. When considering the relationship between the variables on opposite sides of the equals sign, assume all of the other variables and constants are equal to one and replace the equals sign with the d proportion sign to see if the relationship is direct or inverse. For Fg mg and v , Fg is t 1 directly proportional to m, Fg m, and v is inversely proportional to t, v . To determine t v2 the relationship between ac and v2 in the equation ac , assume r is one and replace the r equals sign with the proportion sign to get ac v2. ac is directly proportional to v2. This is a direct relationship because as a is increased, v2 is also increased. What is the relationship between illumination, I, and the distance from an illumination source, d, in the equation k I ? If k is constant, as the illumination increases the distance decreases. If an increase d2 in I results in a decrease in d, the variables are said to be in an inverse relationship. I is 1 inversely proportional to d2: I . d2 Express the relationship between the indicated variables, and determine whether the relationship is direct or inverse. 42r 1. For equation ac , what is the relationship between ac and T2? T2 42 2. For equation T2 r3, what is the relationship between T2 and r3? Gms 42 3. For equation T2 r3, what is the relationship between T2 and ms? Gms 4. If the value of ms is increased in problem 3, is T2 increased or decreased? 5. For equation S 4r2, what is the relationship between S and r2? Physics: Principles and Problems Connecting Math to Physics 25 Name 13 Connecting Math to Physics continued 4 6. In the equation V r3, if v increased by 4 times by how much would r increase or 3 decrease? 1 2 k 7. For equation n , what is the relationship between n and l ? If the left side of the 1 l2 1 equation were changed to 2n, how would l 2 be affected? 8. For equation z x, what is the relationship between z andx? 9. If the value of x is increased in problem 8, is z increased or decreased? Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 26 Connecting Math to Physics Physics: Principles and Problems Date Period Name ACTIVITY 14 Connecting Math to Physics Adding and Subtracting Fractions The basic operation of division may be illustrated in the form of a fraction. A fraction is a a number with the general form , where a is the numerator and b is the denominator. In b this form, the numerator, a, is divided by the denominator, b. Examples of fractions include 1 3 v C , , , and . 2 4 t 2xy Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. When a number is between zero and one, a fraction may be used instead of a decimal. The numerator is the number of equal parts in the number out of the total number of parts that 1 make up the whole, the denominator. For example, the fraction indicates that the 2 3 number has one part out of two parts total. The fraction indicates that a number has 4 three parts out of four parts. Any integer that is equal to or greater than 1 also may be 3 written as a fraction. The number 3 may be written as because 3 1 3. When the 1 numerator and the denominator are the same number, the value of the fraction is one. This 4 is because a number divided by itself is equal to one; for example, 1. 4 Like all numbers and variables, fractions may be added together. The number of parts that the fraction is divided into, the denominator, must be the same when fractions are added. If the denominators of the fractions are the same, add the numerators and keep the common denominator. 2 1 4 5 1 1 For example, and 1. 6 6 6 2 2 2 To subtract fractions with common denominators, subtract the numerators and keep the common denominator. 4 3 1 4 2 2 For example, and . 6 6 6 5 5 5 If the denominators are not the same, the fractions must be converted to have the same denominator prior to addition or subtraction. To do this, multiply the first number by a fraction that consists of the second fraction’s denominator, and multiply the second number by a fraction that consists of the first fraction’s denominator. For example, 2 13 1 2 1 3 1 12 23 32 2 3 2 3 3 2 3 2 5 . 6 6 6 For example, remember, if a number is multiplied by 1, its value does not change. When fractions contain variables, the same procedures used above apply. For example, 2x x 2x 5 x 3y 10x 3xy 3y 5 3y 5 5 3y 15y 15y 10x 3xy . 15y Physics: Principles and Problems Connecting Math to Physics 27 Name 14 Connecting Math to Physics continued Form a new fraction using addition or subtraction. 1 3 1. 5 5 3 2 8. 4 3 3 1 2. 4 4 3 2 9. 10 5 3 6 10. 4 7 7 5 4. 8 8 5 1 11. z 2z 5 3 5. xy xy 3 5 12. 2y 3y x y 6. 2 2 6 2 13. y x 1 4 7. 2 3 2 4 14. m n 28 Connecting Math to Physics Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 2 3 3. 6 6 Physics: Principles and Problems Date Period Name ACTIVITY 15 Connecting Math to Physics Order of Operations Equations are used throughout the study of physics. These equations consist of operations such as addition, subtraction, multiplication, division, and trigonometric functions. When solving equations, it is important to follow an order in which the operations are performed. For example, what number is equal to 3 4 2 5? The correct answer is 6. Depending on which operation is performed first, however, other answers to this problem include 9 and 21. The correct answer of 6 is found using the proper order of operations. The order of operations is a set of rules that ensures the correct answer is calculated. Order of Operations: 1. Perform all operations within grouping symbols, such as parentheses. 2. Evaluate all exponential expressions. 3. Evaluate all trigonometric functions. 4. Perform all multiplication and division in the order they occur from left to right. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 5. Perform all addition and subtraction in the order they occur from left to right. For the example 3 4 2 5 there are no grouping symbols, exponents, or trigonometric functions. Begin with the multiplication of 4 and 2 to form 8. Then add 3 and 8 to form 11, and subtract 5 from 11 to get 6. To evaluate sin 30° (24 8.0) 4.0 2.0 3.02 20.5, use the following procedure: sin 30° 32 4.0 2.0 3.02 20.5 sin 30° 32 4.0 2.0 9.0 20.5 0.500 32 4.0 2.0 9.0 20.5 0.500 8.0 2.0 9.0 20.5 0.500 16 9.0 20.5 17 9.0 20.5 26 20.5 5 If a number is in the form of a fraction with operations in the numerator and the denominator, evaluate the numerator and the denominator separately, and then divide. Parentheses may be added for clarity. 18 4.0 3.0 18 4.0 3.0 18 4.0 3.0 22 3.0 19 Evaluate . 0.95. 2.0 10 2.0 10 20 20 20 Find the value of each of the following expressions. 1. 8.0 4.0 9.0 2. 100.0 10.0 25.0 Physics: Principles and Problems Connecting Math to Physics 29 Name 15 Connecting Math to Physics continued 3. 8.0 2.0 4.0 4. 75 25 5.0 10.0 52 5. 81 sin 90.0° 6. 30.0 3.0 8.0 7. 2(5 3) 8 8. 48 (4.0 6.0) 10.0 9. 6.0 cos 60.0° 5.02 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 10. 50.000 (sin 30.0°)3 2.0 3.0 11. 12 6.0 8.0 30 Connecting Math to Physics Physics: Principles and Problems Date Period Name ACTIVITY 16 Connecting Math to Physics Graphing the Sine and Cosine Curves The trigonometric functions of sine and cosine are used in many physical applications. The sine and cosine functions may be graphed on a coordinate system in a manner similar to a line. The sine and cosine graphs are not lines, however, but harmonic waves. The equation for the trigonometric function of sine may be written as y sin . To graph this equation, use the point-plotting method. Place on the horizontal axis and y on the vertical axis. Enter a value for and use your calculator to find y. This forms one point on the graph. Recall that may be in degrees or radians. The following is a table with points on the y sin graph: Degrees 0 45° 90° 135° 180° 225° 270° 315° 360° Radians 0 4 2 3 4 5 4 3 2 7 4 2 0 0.707 1.000 0.707 0 0.707 1.000 0.707 0 360 y sin Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Plotting the points above results in the graph of y sin . y 1.0 0.5 0 45 4 90 2 135 3 4 180 225 270 315 5 4 3 2 7 4 2 0.5 1.0 The graph of y sin has a repetitive pattern. This is a periodic function. The equation also may be written as y A sin , where A is the amplitude, or height, of the crest above or below the horizontal axis. For the equation y sin , A is equal to one. The equation y 4 sin has an amplitude of 4. When and A 4, y A sin 4 sin 2 2 4 1 4. The coordinate for this point is , 4 . As you will see below, the graph of the 2 cosine function is similar to the sine function. Physics: Principles and Problems Connecting Math to Physics 31 Name Connecting Math to Physics 16 continued 1. Fill in the table below: Degrees 0 45° 90° 135° 180° 225° 270° 315° 360° Radians y cos 2. Graph the equation y cos . 4. How does the graph of y cos differ from the graph of y sin ? 32 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 3. Is y cos a periodic function? Date Period Name ACTIVITY 17 Connecting Math to Physics Percentages Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Percentages are useful when comparing quantities. The word percent means “per hundred.” A percent is the parts per 100. Any percent may be written as a number over 100. For 50 example, 50% is equal to . Recall that a fraction is one way of representing the 100 operation of division. To convert a percent to a decimal, change the percent to a fraction with a denominator of 100 and then divide the numerator by the denominator. For 50 example, 50% 0.5. To convert a decimal to a percent, multiply the decimal by 100 100 and add the percent sign: 0.183 0.183 100 18.3% When the resistance, R, in a circuit with a constant voltage source is increased, the current 1 will decrease because these variables have an inverse relationship: I . If, in a certain R circuit, R is increased from 50 to 100 , the current decreases from 0.500 A to 0.250 A. What is the percent decrease in the current? To determine the percent decrease or increase, (Final value) (Initial value) use the following equation: % Change 100. If the result (Initial value) is positive, the change is a percent increase. If the result is negative, the change is a percent decrease. (Note: If the initial value is zero, the percent change is undefined because zero is in the denominator.) When the current in the above scenario is decreased from 0.500 A to 0.250 A, the 0.250 A 0.500 A % Change 100 50.0%. The result is negative so there is a 0.500 A (100 ) (50 ) percent decrease. The percent change in the resistance is 100 100%. (100 ) The result is positive so there is a percent increase. Therefore, the current, I, is decreased by 50% when the resistance, R, is increased by 100%. Convert the percent to a decimal. 1. 25% 5. 0.25% 2. 40% 1 6. % 4 3. 100% 1 7. % 8 4. 0.5% Physics: Principles and Problems Connecting Math to Physics 33 Name 17 Connecting Math to Physics continued Convert the decimal to a percent. 8. 0.1 11. 0.68 9. 0.25 12. 0.99 10. 1.0 13. 1.1 Calculate the percent increase or decrease. 15. If the current in a circuit is increased from 0.75 A to 0.80 A, what is the percent increase in the current? 16. If the velocity of a car is reduced from 90.0 km/h to 50.0 km/h, what is the percent decrease in the car’s velocity? 34 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 14. If the price of a shirt is reduced from $25 to $20, what is the percent decrease in the price? Date Period Name ACTIVITY 18 Connecting Math to Physics Inequalities Inequalities are used to help understand the relationships between variables such as voltage, current, and the number of coils in step-up and step-down transformers. An inequality is a mathematical statement that indicates that one number, variable, or algebraic expression has a greater or lesser value than another. The general forms of inequalities are the following: a b a is less than b a b a is greater than b To use the inequality signs, determine if one number, variable, or algebraic expression is greater than the other. The inequality sign points toward the smaller value. For example, because the number 3 is less than the number 5, the inequality points toward the 3: 3 5 or 5 3. Insert or to make the inequality true. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. 1 2. 10 5 20 5. 10 20 6. 13 2 200 3. 100 68 7. 0 4. 1 5 8. 9 7 If the inequality consists of a variable on one side of the inequality sign and a number on the other side, the inequality has many solutions. For example, if x 2, then the solution for x is all numbers greater than 2. An inequality in this form may be graphed on a number line. A number line is the same as the horizontal axis on a graph. Zero separates the positive and negative numbers on the number line. The positive numbers are located on the right of the zero and negative numbers are on the left. 3 2 1 0 1 2 3 To graph an inequality with a variable on one side and a number on the other, a circle is placed on the number line and the weight of the line is increased either to the left or to the right of the circle depending on the inequality sign. For example, to graph x 2, place a circle on the number line at 2 and increase the thickness of the number line from the circle to the right, indicating that x is any number greater than 2. 2 1 Physics: Principles and Problems 0 1 2 3 4 Connecting Math to Physics 35 Name Connecting Math to Physics 18 continued The open circle indicates that x is not equal to the number. If you were to graph x 2 where x is greater than or equal to 2, the circle would be filled in to form a dot, as shown below. 2 1 0 1 2 3 4 Graph the following inequalities on a number line. 9. x 3 10. x 2 11. x 5 Determine the equation of the inequality from the graphs below. 13. 14. 7 6 5 4 3 2 1 0 2 3 4 5 6 7 36 Connecting Math to Physics 8 9 Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 12. x 1 Date Period Name ACTIVITY 19 Connecting Math to Physics Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Ratio and Proportion a A ratio is a comparison of two quantities by division. The general form of a ratio is . For b example, if 1.0 aspirin tablet should be taken for every 50.0 kg of body mass, the ratio of 1.0 aspirin tablets to mass is . When an equation states that two ratios are equal to one 50.0 kg c a another, it is called a proportion. So is a proportion. Because a proportion is a d b statement of equality, if three of the quantities in a proportion are known, the fourth quantity may be calculated by solving for the unknown. If a person’s mass is 100.0 kg instead of 50.0 kg, the number of aspirin tablets required may be calculated using a proportion. Setting the known ratio of 1.0 tablet for every 50.0 kg equal to the ratio with 1.0 x an unknown forms the proportion: . Both ratios must maintain the 50.0 kg 100.0 kg same form, with the number of tablets in the numerator and the body mass in the denominator. To solve for x, isolate the variable on one side of the equation and move the 1.0 numbers to the other side: x 100.0 kg 2.0. Thus, 2.0 tablets are needed for a 50.0 kg mass of 100.0 kg. The cross product principle is a quick method that may be used to solve a proportion for an unknown. c a Cross Product Principle If , then ad bc. b d The numerator of the first ratio is multiplied by the denominator of the second ratio, and the numerator of the second ratio is multiplied by the denominator of the first ratio. For any proportion, ad must equal bc. In the aspirin-to-mass example above, the proportion is x 1.0 written as . Therefore, 1.0 100.0 kg x 50.0 kg. 50.0 kg 100.0 kg Solve the following proportions for the unknown variable. 12 4 1. 3 x 3 x 2. 4 8 Physics: Principles and Problems Connecting Math to Physics 37 Name 19 Connecting Math to Physics continued x 2 3. 10 20 1 3x 4. 9 12 6 4x 2 5. 4 8 12 6x 3 6. 2x 3 If q 1.601019 C, V 60.0 V, B 0.050 T, and r 0.080 m, find m. 8. Ibuprofen is given to a patient to reduce pain. In a child six months to 12 years of age, the dose is based on body weight. If the dose is 5 mg per kg of body weight, how much ibuprofen should be given to a child that weighs 10 kg? Hint: set up a proportion. 38 Connecting Math to Physics Physics: Principles and Problems Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 7. The mass spectrometer discussed in Chapter 26 is an important scientific instrument. q The mass spectrometer measures an ion’s charge-to-mass ratio, . The charge-to-mass m ratio is equal to the ratio of twice the potential difference divided by the product of square of the magnetic field strength and the square of the radius of the ion’s circular 2V q 2V path, . This is expressed by the equation . m B2r2 B2r2 Date Period Name ACTIVITY 20 Connecting Math to Physics The Mass of an Atom Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Like other measurements, the mass of an atom may be written in a variety of units. The atomic mass unit is one way to measure the mass of a single atom. The carbon-12 atom is 1 used as the standard for the atomic mass unit, u. One u is equal to the mass of a 12 carbon-12 atom. The atomic mass of a carbon-12 atom is, therefore, 12 u. Each element’s 1 atomic mass is based on a scale that uses of the mass of a carbon-12 atom as 1. You 12 can find the atomic mass units of other elements in the periodic table. For example, the atomic mass of one hydrogen atom, H, is 1.0 u and the atomic mass of one sodium atom, Na, is 23 u. These atomic masses indicate that the Na atom is approximately 23 times the mass of the H atom and the Na atom is almost two times the mass of the carbon, C, atom. To convert from atomic mass units to grams, use the following relationship: 1 u 1.661024 g. The quantity of atoms in many physics applications is much greater than a single atom and the unit mole is used instead. The unit of the mole, abbreviated mol, is a unit that is used to quantify a large number of atoms. One mole is defined as the number of carbon-12 atoms that are in 12 g of pure carbon. This quantity is equivalent to 6.0221023 atoms. atoms Stated in another way, 1 mol 6.0221023 atoms. 6.0221023 is also known as mol Avogadro’s number. The units of an atom (or molecule) may be provided in grams, atomic mass units, or moles. Using the equivalents above, a mass may be converted between these units using dimensional analysis. 1u 3.01024 u. For example, to convert 5.0 g of oxygen, O, to u: (5.0 g O) 1.661024 g To convert 3.01024 u of O to moles: 1 atom 1 mole 0.31 mol. (3.01024 u O) 16 u 6.0221023 atoms Convert to the indicated mass unit. Use Appendix D in your textbook to find the atomic mass of each element. 1. 8.0 g O to u 2. 20.0 g C to u 3. 5.0 g Na to u Physics: Principles and Problems Connecting Math to Physics 39 Name 20 Connecting Math to Physics continued 4. 10.0 g K to u 5. 6.01026 u of Cd to moles 6. 9.01024 u of Ag to moles 7. 4.01028 u of Cl to moles Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 8. 251030 u of Au to moles 9. 8.0 g O to moles 10. 20.0 g C to moles 11. 5.0 g Na to moles 40 Connecting Math to Physics Physics: Principles and Problems Date Period Name ACTIVITY 21 Connecting Math to Physics Exponential Equations and Graphs Algebraic equations show relationships between variables. There are many types of equations, including the linear equations that were discussed previously. An exponential equation, or function, is another type of algebraic equation. An exponential function 1 contains a variable in the exponent. y 4x and y xt are examples of exponential 2 functions. Determine whether the following equations are exponential. 1. y 8x 2. d vt Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1 3. v at 2 4. F ma 5. y 5t3 Exponential functions are used in the study of radioactive decay. The amount of a radioactive isotope remaining in a sample is expressed by the equation: remaining 1 t original , where t is the number of half-lives that have passed. When an exponential 2 function is plotted on a coordinate system, the graph is curved. The figure below is a graph 1 t of the equation y . 2 y 9 8 7 6 5 4 3 2 1 4 3 2 Physics: Principles and Problems 1 0 1 2 3 4 5 6 t Connecting Math to Physics 41 Name 21 Connecting Math to Physics continued Like other algebraic equations, an exponential function may be graphed using ordered pairs, or points. To graph an equation using this method, a series of ordered pairs is calculated. Each ordered pair is found by selecting a numerical value for the variable on the horizontal axis and substituting that number into the equation to solve for the variable on the vertical axis. This procedure is repeated several times until enough ordered pairs have been found to establish the shape of the curve. After the ordered pairs are plotted, they are connected to form the graph. Plot the following exponential equations using the point-plotting method. 6. y 2x Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 7. y 2x1 1 x 8. y 4 42 Connecting Math to Physics Physics: Principles and Problems