2 Why Study Physics? Mathematics and Physics Speak Scientific Notation and Significant Figures Units Dimensional Analysis Problem Solving Techniques Approximations Graphs Chapter 1: Introduction
3 §1.1 Why Study Physics? Physics is the foundation of every science (astronomy, biology, chemistry…). Many pieces of technology and/or medical equipment and procedures are developed with the help of physicists. Studying physics will help you develop good thinking skills, problem solving skills, and give you the background needed to differentiate between science and pseudoscience.
4 Gain an appreciation for the simplicity of nature. Physics encompasses all natural phenomena.
5 §1.2 Physics Speak Be aware that physicists have their own precise definitions of some words that are different from their common English language definitions. Examples: speed and velocity are no longer synonyms; acceleration is a change of speed or direction.
6 §1.3 Math Galileo Wrote: Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one is wandering in a dark labyrinth. From Opere Il Saggiatore p. 171 by Galileo Galilei ( http://www-gap.dcs.st- and.ac.uk/~history/Mathematicians/Galileo.html) Basically, the language spoken by physicists is mathematics.
7 x is multiplied by the factor m. The terms mx and b are added together. Definitions:
8 x is multiplied by the factor 1/a or x is divided by the factor a. The terms x/a and c are added together. Example:
9 Percentages: Example: You put $10,000 in a CD for one year. The APY is 3.05%. How much interest does the bank pay you at the end of the year? The bank pays you $305 in interest.
10 Example: You have $5,000 invested in stock XYZ. It loses 6.4% of its value today. How much is your investment now worth?
11 The general rule is to multiply by where the (+) is used if the quantity is increasing and (–) is used if the quantity is decreasing.
12 Proportions: A is proportional to B. The value of A is directly dependent on the value of B. A is proportional to 1/B. The value of A is inversely dependent on the value of B.
13 Example: For items at the grocery store: The more you buy, the more you pay. This is just the relationship between cost and weight. To change from to = we need to know the proportionality constant.
14 Example: The area of a circle is The area is proportional to the radius squared. The proportionality constant is .
15 Example: If you have one circle with a radius of 5.0 cm and a second circle with a radius of 3.0 cm, by what factor is the area of the first circle larger than the area of the second circle? The area of a circle is proportional to r 2 : The area of the first circle is 2.8 times larger than the second circle.
16 Example 1.1, p 4 Effect of Increasing Radius on the Volume of a Sphere –The volume of a sphere is given by the equation V = 4/3 r 3, where V is the volume and r is the radius of the sphere. If a basketball has a radius of 12.4 cm and a tennis ball has a radius of 3.20 cm, by what factor is the volume of the basketball larger that the tennis ball?
18 Practice Problem 1.1 – Power Dissipated by a Lightbulb The electric power P dissipated by a lightbulb of resistance R is P = V 2 /R, where V represents the line voltage. During a brownout, the line voltage is 10.0% less than its normal value. How much power is drawn by a lightbulb during the brownout if it normally draws 100.0 W (watts). Assume that the resistance doesn’t change.
20 Follow up problem A spherical balloon is partially blown up and its surface area is measured. More air is then added, increasing the volume of the balloon. If the surface area of the balloon expands be a factor of 2.0 during this procedure, by what factor does the radius of the balloon change?
22 Strategy Relate the surface area S to the radius r using Solution Find the ratio of the new radius to the old. The radius of the balloon increases by a factor of 1.4.
23 Assignment #1 p. 19 – 2, 5, 6, and 9.
24 §1.4 Scientific Notation & Significant Figures This is a shorthand way of writing very large and/or very small numbers.
25 Example: The radius of the sun is 700,000 km. Write as 7.0 10 5 km. Example: The radius of a hydrogen atom is 0.0000000000529 m. This is more easily written as 5.29 10 -11 m. When properly written this number will be between 1.0 and 10.0
26 From the box on page 5: 1.Nonzero digits are always significant. 2.Final ending zeroes written to the right of the decimal are significant. (Example: 7.00.) 3.Zeroes that are placeholders are not significant. (Example: 700,000 versus 700,000.0.) 4.Zeroes written between digits are significant. (Example: 105,000; 150,000.) Significant figures:
27 Example 1.2 For each of these values, identify the number of significant figures and rewrite each in its standard scientific notation. 409.8 s 0.058700 cm 9500 g 950.0 x 10 1 mL
28 Practice Problem 1.2 State the number of significant figures in each of these measurements and rewrite them in standard scientific notation. 0.00010544 kg 0.005800 cm 60200 s
29 Significant Figures in Calculations –When two or more quantities are added or subtracted, the result is as precise as the least precise of the quantities. If the quantities are written in scientific notation with different powers of ten, first rewrite them with the same power of ten. After adding or subtracting, round the result, keeping only as many decimal places as are significant in all of the quantities that were added or subtracted. –When quantities are multiplied or divided, the result has the same number of significant figures as the quantity with the smallest number of significant figures. –In a series of calculations, rounding to the correct number of significant figures should only be done at the end, not at each step. Rounding at each step would increase the chance that roundoff error could snowball and have an adverse effect on the accuracy of the final answer. It’s a good idea to keep at least two extra significant figures in calculations, then round at the end.
30 Example and Practice Problem 1.3 Significant Figures in Addition. –Calculate the sum 44.56005 s + 0.0698 s + 1103.2 s. Significant Figures in Subtraction. –Calculate the difference 568.42 m – 3.924 m and write the result in scientific notation. How many significant figures are in the result?
32 Example and Practice Problem 1.4 Significant Figures in Multiplication –Find the product of 45.26 m/s and 2.41 s. How many significant figures does the product have? Significant Figures in Division –Write the solution to 28.84 m divided by 6.2 s with the correct number of significant figures.
33 Follow-up Problems Perform these operations with the appropriate number of significant figures. 3.783 x 10 6 kg + 1.25 x 10 8 kg (3.783 x 10 6 m) / (3.0 x 10 -2 s)
35 (a)Strategy Rewrite the numbers so that the power of 10 is the same for each. Then add and give the answer with the number of significant figures determined by the less precise of the two numbers. Solution Perform the operation with the appropriate number of significant figures. (b)Strategy Find the quotient and give the answer with the number of significant figures determined by the number with the fewest significant figures. Solution Perform the operation with the appropriate number of significant figures.
36 Follow-up Problem Solve the following problem and express the answer in meters with the appropriate number of significant figures and in scientific notation: 3.08 x 10 -1 km + 2.00 x 10 3 cm
38 Strategy Convert each length to meters. Then, rewrite the numbers so that the power of 10 is the same for each. Finally, add and give the answer with the number of significant figures determined by the less precise of the two numbers. Solution Solve the problem.
39 Assignment #2 p. 19 – 14, 16, and 18
40 §1.5 Units Some of the standard SI unit prefixes and their respective powers of 10. PrefixPower of 10PrefixPower of 10 tera (T)10 12 centi (c)10 -2 giga (G)10 9 milli (m)10 -3 mega (M)10 6 micro ( ) 10 -6 kilo (k)10 3 nano (n)10 -9
41 Dimensions are basic types of quantities that can be measured or computed. Examples are length, time, mass, electric current, and temperature. A unit is a standard amount of a dimensional quantity. There is a need for a system of units. SI units will be used throughout this class.
42 The quantities in this column are based on an agreed upon standard.
43 Example: The SI unit of energy is the joule. 1 joule = 1 kg m 2 /sec 2 Derived unit Base units A derived unit is composed of combinations of base units.
44 Units can be freely converted from one to another. Examples: 12 inches = 1 foot 1 inch = 2.54 cm
45 Example: The density of air is 1.3 kg/m 3. Change the units to slugs/ft 3. 1 slug = 14.59 kg 1 m = 3.28 feet
46 Example 1.5 – Buying Clothes in a Foreign Country Michel, an exchange student from France, is studying in the United States. He wishes to buy a new pair of jeans, but the sizes are all in inches. He does remember that 1 m = 3.28 ft and that 1 ft = 12 in. If his waist is 82 cm, what is his waist size in inches?
48 Practice Problem 1.5 – Driving on the Autobahn A BMW convertible travels on the German autobahn at a speed of 128 km/h. What is the speed of the car (a) in meters per second? (b) in miles per hour?
50 Example 1.6 – Conversion of Volume A beaker of water contains 255 mL of water. (1 mL – 1 milliliter; 1 L = 1000 cm 3.) What is the volume of the water in (a) cubic centimeters? (b) cubic meters?
52 Practice Problem 1.6 – Surface Area of Earth The radius of Earth is 6.4 x 10 3 km. Find the surface area of Earth in square meters and in square miles. (Surface are of a sphere = 4 r 2.)
54 Follow-up Problem A snail crawls at a pace of 5.0 cm/min. Express the snail’s speed in (a) ft/s and (b) mi/h.
56 (a)Strategy There are 12 inches in one foot, 2.54 centimeters in one inch, and 60 seconds in one minute. Solution Express the snail’s speed in feet per second. (b)Strategy There are 5280 feet in one mile, 12 inches in one foot, 2.54 centimeters in one inch, and 60 minutes in one hour. Solution Express the snail’s speed in miles per hour.
57 Assignment #3 p. 20 – 23, 25, 26, 28, 29, and 31.
58 §1.6 Dimensional Analysis Dimensions are basic types of quantities such as length [L]; time [T]; or mass [M]. The square brackets are referring to dimensions not units.
59 Example (text problem 1.84): Use dimensional analysis to determine how the period of a pendulum depends on mass, the length of the pendulum, and the acceleration due to gravity (here the units are distance/time 2 ). Mass of the pendulum [M] Length of the pendulum [L] Acceleration of gravity [L/T 2 ] The period of a pendulum is how long it takes to complete 1 swing; the dimensions are time [T].
60 §1.7 Problem Solving Techniques Summary of the list on page 13: Read the problem thoroughly. Draw a picture. Label the picture with the given information. What is unknown? What physical principles apply? Are their multiple steps needed? Work symbolically! It is easier to catch mistakes. Calculate the end result. Don’t forget units! Check your answer for reasonableness.
61 §1.8 Approximations All of the problems that we will do this semester will be an approximation of reality. We will use models of how things work to compute our desired results. The more effects we include, the more correct our results will be.
62 Example (text problem 1.44): Estimate the number of times a human heart beats during its lifetime. Estimate that a typical heart beats ~60 times per minute:
63 Example 1.9 – Number of Cells in the Human Body Averaged-sized cells in the human body are about 10 m in length. How many cells are in the human body? Make an order-of-magnitude estimate.
65 Practice Problem 1.9 – Drinking Water Consumed in the United States How many liters of water are swallowed by the people living in the United States in one year? This is a type of problem made famous by Enrico Fermi (1901-1954), who was a master at this sort of back-of-the- envelope calculation. Such problems are often called Fermi problems in his honor. (1 liter = 10 -3 m 3 ≈ 1 quart.)
67 Follow-up Problem What is the order of magnitude of the number of seconds in a year?
69 Strategy Estimate the appropriate orders of magnitude. Solution Find the order of magnitude of the number of seconds in one year. seconds/minute ~ 10 2, minutes/hour ~ 10 2, hours/day ~ 10 1, days/year ~ 10 2
70 §1.9 Graphs Experimenters vary a quantity (the independent variable) and measure another quantity (the dependent variable). Dependent variable here Independent variable here
71 Be sure to label the axes with both the quantity and its unit. For example: Position (meters) Time (seconds)
72 Example (text problem 1.49): A nurse recorded the values shown in the table for a patient’s temperature. Plot a graph of temperature versus time and find (a) the patient’s temperature at noon, (b) the slope of the graph, and (c) if you would expect the graph to follow the same trend over the next 12 hours? Explain. TimeDecimal timeTemp (°F) 10:00 AM10.0100.00 10:30 AM10.5100.45 11:00 AM11.0100.90 11:30 AM11.5101.35 12:45 PM12.75102.48 The given data:
74 (a) (b) (c) Reading from the graph: 101.8 F. No.
75 Example 1.10 – Length of a Spring In an introductory physics laboratory experiment, students are investigating how the length of a spring varies with the weight hanging from it. Various weights (accurately calibrated to 0.01 N) ranging up to 6.00 N can be hung from the spring; then the length of the spring is measured with a meterstick. The goal is to see if the weight F and length L are related by F = kx where x = (L – L o ), L o is the length of the spring when no weight in hanging from it, and k is called the spring constant of the spring. Graph the data in the table and calculate k for this spring. F (N)00.501.002.503.003.504.005.006.00 L (cm)9.410.212.517.919.722.523.028.829.5
76 Fig. 01.04
79 Practice Problem 1.10 – Another Weight of the Spring What is the length of the spring of Example 1.10 when a weight of 8.00 N is suspended? Assume that the relationship found in Example 1.10 still holds for this weight.
81 Follow-up Problem An object is moving in the x-direction. A graph of the distance it has moved as a function of time is shown. (a) What are the slope and vertical axis intercept? (b) What physical significance do the slope and intercept on the vertical axis have for this graph?
83 (a)Strategy Refer to the figure. Use the definition of the slope of a line and the fact that the vertical axis intercept is the x-value corresponding to t = 0. Solution Compute the slope. When t = 0, x = 3.0 km; therefore, the vertical axis intercept is 3.0 km. (b)Strategy and Solution The physical significance of the slope of the graph is that it represents the speed of the object. The physical significance of the vertical axis intercept is that it represents the starting position of the object (position at time zero).
84 Assignment #4 p. 21 – 52 and 54 p. 22 – 62, 68, 69 and 70
85 Summary Math Skills The SI System of Units Dimensional Analysis