Physics 20 Lecture Notes

Mr. P. MacDonald Physics 20 Lecture Notes – Spring 2007 “Lousy laws of physics!”

− Bart Simpson In, The Simpsons Hit and Run

TABLE OF CONTENTS Course Outline and Contents...........................................................................iii Formal Lab Write–Up Guide .........................................................................iv Introduction to Physics and Basic Skills .........................................................1 Waves ............................................................................................................21 Sources of Light .............................................................................................40 Mirrors and Reflection ...................................................................................53 Refraction of Light ........................................................................................66 Lenses ............................................................................................................77

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Physics 20 Course Outline “In this room we obey the second law of thermodynamics!” - Homer J. Simpson Mr. Peter MacDonald Room 16 School Contact: 683 – 7700; [email protected] Course Website: http://schools.spsd.sk.ca/evanh/classes/macdonald Classroom Expectations and Details 1. Attendance is mandatory. Unexcused absences and late students do not facilitate learning and lead to poor results. Absences result in missed material and late students disrupt the class. 2. Students are expected to have all of their supplies (pencils, pens, eraser, ruler, etc.) with them at the start of class. Students will be given notice of any new materials (compass, protractor, etc.) at least a few days in advance. 3. Missed Test. a. A zero will be given until the absence has been cleared. b. Students will write a make-up test/quiz (an evaluated work) upon their first day back to class from an absence, or c. Students can choose to omit a missed test and the test weight will be added to the final exam. d. Assignments submitted after the due date will not be penalized. However, no assignments will by accepted after the first group of assignments has been returned to the students. e. There will be the option for extra (bonus) assignments throughout the semester. f. Students with good attendance (no more than three unexcused absences) will be able to replace with worst test and lab with their best. 4. If a student is away, it is up to that student to obtain additional notes and materials. You can get them from a friend, or phone the school and I will assemble the material to be picked up. Material will not be re-taught to students who were absent. a. Students will be provided with a copy of the lecture notes. If lost students can download the course lecture notes from the class website. b. A copy of the problem manual will be given to each student. Any further copies can be downloaded from the class website.

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5. Evaluation Student marks will be determined by a variety of assessment techniques: Final exam, chapter tests, homework checks, a project, and labs. Exam: 30% Tests: 40% Labs: 20% Quizzes: 10% Æ This is a marking scheme only and is subject to change. 6. Supplies • Pencil, Pen, & Eraser • Geometry set • Scientific Calculator • Lecture notes, problem manual

Course Material I Introduction to Physics & Motion 1. 2. 3. 4.

Mathematics Review Dimensional Analysis Significant Figures Scientific Notation

5. 6. 7.

Unit Analysis Proportions Motion

II Waves 1. Wave terminology 2. Universal wave equation 3. Principle of superposition 4. Wave phenomena 5. Sound

III Light A. Characteristics 1. Sources & transmission of light 2. The speed of light 3. Electromagnetic spectrum

C. Refraction 1. Snell’s law 2. Critical angle 3. Total internal reflection

B. Reflection 1. Laws of reflection 2. Planes mirrors 3. Curved mirrors

D. Lenses 1. Thin lens equation 2. Lens magnification equation 3. Lens maker’s equation

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FORMAL LABORATORY REPORT Prelab Before coming to the lab each student must be prepared. It is expected that each student has completed all pre-lab activities such as reading the lab handout and/or relevant material in the textbook or answering assigned questions. Paper 8½" x 11" (21.5 cm x 27.5 cm) white lined paper or letter paper. The report should be single spaced with 12 pt Times Roman font. There should be a 1 inch margin on all sides of the pages. Title Page The title page should include the following items: a title centered 1/3 from the top of the page; an identification containing the student's name, lab partner’s name, course number, due date, and teacher's name located at the bottom right hand corner of the page. Objective The objective is a concise statement outlining the purpose of the experiment. e.g. To determine the boiling point of H2O. Introduction The introduction should contain any prior knowledge on which the experiment is based; including an explanation of principles, definitions, experimental techniques, expected results (hypothesis), theories and laws. Materials The materials section is a list of all equipment, reagents (chemicals), and computer programs that were used to complete the experiment. Drawings of the apparatus setup should be included in this section if needed. Procedure The procedure is a detailed statement (step by step) of how the experiment was performed such that the experiment could be repeated using your report. Safety precautions which were followed should be stated. The procedure must be written in the impersonal (3rd person) past tense: e.g. We are taking the temperature every 2 minutes. 4 The temperature was taken every 2 minutes. 3 Results This section may consist of quantitative and/or qualitative observations of the experiment.

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Quantitative Results Graphs and Tables – When graphs are required, special attention should be paid to the following items: the type of graph expected (straight line or curve), utilizing the entire graph paper, plotted point size, title of the graph, and axis labels. When numerous measurements have occurred, data is to be placed in a data table whenever possible. Figure headings are placed below the figure and should give a short description of the figure. The figure number should be in bold print. Table headings are found above the table and should also have a brief description. The table number is also in bold print. Calculations – One example of each type of calculation should be included. Results from numerous calculations should be placed in a data table with the proper number of significant figures and correct units. % yield and % error calculations should be included when possible Qualitative Results Observations – This is a qualitative written description and/or sketch of what was seen during the experiment. It may be in the form of a table or simply a written description. Conclusion The conclusion is a concise statement that answers the objective. The result of percent error and/or percent yield should be discussed and compared with known results. A portion of the conclusion should be dedicated to error analysis which discusses any possible sources of error that may have contributed to the percent error or yield. The conclusion should be written in the impersonal past tense. Literature Cited Any information borrowed from another source which is not common knowledge must be cited within the text of the report as outlined in the student handbook. This section should be on a separate final page of the report. Questions Although questions are not part of a formal lab report, they should be answered on a separate sheet of paper and attached to the report where applicable. Important Reminders for a Lab Report 1. Spelling. 2. Significant figures and units regarding measurements and calculations. 3. Avoid personal pronouns. 4. Headings should stand out and each section should be separated by 1 line. 5. Neatness counts Æ use rulers when needed (especially when using tables and graphs), type if possible. ¾ Do not copy verbatim (word for word) from the lab handout or any other source. This is plagiarism and would result in a zero mark and possible further consequences. vi

Introduction & Motion

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Introduction to Physics: Basic Skills & Motion

Introduction & Motion •

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What is physics? o As a science. o In Politics. o In Society o Equations, lab coats, nerds, A-bomb

Ö Study the property, changes, and interactions of matter and energy (nature Æ physica) •

Topics studied within physics: o Geophysics o Bio Physics o Quantum Physics o Quantum Chemistry

•

Evolution of Science o Religious/Political o Always amended/updated o Tentative

• •

Why do different disciplines exist within science? When might a physicist need to know something about the other disciplines within science? Why is observation essential to science? New things are always being learned/discovered; old things always being relearned or re-discovered.

• •

Discovering Physics • • • • •

Many things around the home and elsewhere involve applications of physics. Physics is everywhere. Understanding the principles of physics behind common, everyday things are interesting. By examining typical items found in the home or elsewhere, many ideas in physics can be explored. By applying ideas in physics some types of problems experienced around the home or elsewhere and be solved/prevented. Discuss the difference between the physics of something and knowing how things work/figuring out a problem logically.


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Measurement and Data Analysis •

What is a long way?

•

The Metric System. o Created by French scientist in 1795. o Convenient because different sizes are related by powers of 10. o SI – Système International d’ Unités o Other quantities can be described using the base units: time, length, and mass.

Ö Length: the metre, m. o The metre was first defined at the time of the French Revolution to be one ten-millionth of the distance from the equator to the north pole. o The metre is the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. Ö Time: the second, s. o The division of time into hours, minutes, and seconds dates to the Babylonians. The second was based on the Earth’s rotation, but the Earth’s rotation is not a constant – it can change slightly with the wind! o The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two levels of the ground state of the cesium-133 atom. Ö Mass: the kilogram, kg o It is defined in terms of a particular object, the prototype kilogram kept in a laboratory in France. Ö Charge o Measured in Coulombs, C. The electric charge on an object. Ö Current o Amperes, A. Ö Temperature o Measured in Kelvins, K. Ö Mole, mol. o The amount of a substance. Ö Luminosity, cd, W o Energy per second generated by a light source. Ö All other units (N: NewtonÆforce, J: JoulesÆEnergy, W: WattsÆEnergy per sec, lx: luxÆLuminosity per unit area) are derived the above units. The metre, kilogram, and the second are know as the base units; many other units in mechanics are derived from those three.


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Measurement: The simple rule to remember when measuring accurately is that the last digit of each measured quantity is always estimated. If your measuring device is divided to the nearest ones then you estimate to the nearest tenth. If it is divided to the nearest hundredth then you estimate to the nearest thousandth. Uncertainty in Reading Measurements •

Parallax: The apparent shift in the position of the object and a reference point behind it. Æ Hold thumb out in front of face and switch between eyes. o Reading Gauges

•

Precision: Degree of exactness to which the measurement of a quantity can be produced. o Mass of a piece of chalk: 1.8 g, 1.6 g, and 2.0 g Æ 1.8 ± 0.2 g. The precision is the ± number.

•

Accuracy: Is the extent to which a measured value agrees with the standard value of a quantity. o The actual mass of the chalk is 1.9 g, 3.2 g.

Ö More details about errors-in-measurements to follow later in the section. Scientific Notation This is a format commonly used to express very large or small numbers . It consists of two parts: 1. The numerical part must be an integer (positive or negative) between 1 and 10 (including 1 but not 10) 2. The exponential part is written as a power of 10 (ex. 105 or 10-9) ex. 6.65 x 103 is in scientific notation. ex. 12.09 x 10-3 is not in scientific notation because the numerical part is greater than 10. Every digit in scientific notation is significant so even the zeroes do not need bars The table below gives examples of numbers written in standard and scientific notation. Standard Notation 3429.78 145000000 39800 0.00000076 0.0000000001700

Scientific Notation 3.42978 x 103 1.45 x 108 3.980 x 104 7.6 x 10-7 1.700 x 10-10

If a number is written as a power of 10, but not really in scientific notation because the numerical part is not between 1 and 10, you should change it to scientific notation.


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To change it, an easy rule to remember is if the numerical part must get smaller, the exponent must get larger, and vice versa. ex. 32.7 x 104 becomes 3.27 x 105 (the numerical part got smaller (32.7 to 3.27) by one decimal place so the exponent got larger by one (104 to 105) ex. 155.33 x 102 becomes 1.5533 x 104 (the numerical part got smaller (155.33 to 1.5533) by two decimal places so the exponent got larger by two (102 to x 104) ex. 0.00234 x 105 becomes 2.34 x 102 (the numerical part got larger (0.00234 to 2.34) by three decimal places so the exponent got smaller by three (105 to x 102) ex. 0.000009 x 10-8 becomes 9 x 10-14 (be careful with negative exponents, 10-14 is smaller than 10-8) Scientific Notation Calculations: Æ Use your scientific calculator. A) Addition and Subtraction - all of the numbers must be in the same power of 10. Once they are in the same power of 10, just add/subtract the numerical parts ex. (3.21 × 105) + (1.12 × 105) = 4.33 × 105 ex. (1.281 × 105) + (2.3 × 104) + (1.281 × 105) + (0.23 × 105) = 1.511 × 105 = 1.51 × 105 B) Multiplication - the numbers do not have to be in the same power - multiply the two numerical parts and add the exponents ex. (4.2 × 105) × (9.21 × 102) = 38.682 × 107 = 3.9 × 108 C) Division - the numbers do not have to be in the same power - divide the two numerical parts and subtract the exponents ex. (5.28 × 105) ÷ (1.25 × 107) = 4.224 × 10-2 = 4.22 × 10-2 Significant Figures: Many areas of Physics involve measurements. These measurements are often used in mathematical calculations. Significant figures refers to the digits in a measurement that are deemed to be reliable. The zeros in a number warrant special attention. A zero that is the result of a measurement is significant, but zeros that serve only to mark a decimal point are not significant. If you use measurements, the following rules will help you determine which digits are significant or not. Rules for Significant Figures (answers follow in brackets) 1.

Non-zero digits are always significant. Ex. A) 234.7 L (4) B) 21.921 kg (5)


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2.

A zero between other significant figures is significant. Ex. A) 1.05 m (3) B) 2001 m (4)

3.

Final zeros to the right of the decimal point are significant Ex. A) 6.30 g (3) B) 10.00 ml (4)

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Initial zeros are not significant and serve only to show place of decimal. Ex. A) 0.069 km (2) B) 0.0107 (3)

5. Final zeros in numbers with no decimal point may or may not be significant. To clarify, if they are significant or not, the final significant zero should have a line drawn above it or the number should be written in proper scientific notation (see later section). Only ending zeros that are significant figures to the right of the decimal require a bar. There are some measurements of numbers involved in calculations that are exceptions to these rules. All have an infinite number of significant figures. COUNTS - Ex. 10 marbles, 3 people .... Exact CONSTANTS - Ex. Consider a + 2b = c . The number 2 is a constant. DEFINITIONS - Ex. 1 km = 1000 m, 12 = 1 dozen Significant Figure Calculations As a general rule, the result of any mathematical calculation involving measurements cannot be more precise than the least precise measurement. The rules vary depending on the mathematical operation being performed. 1. Addition and Subtraction: When adding or subtracting measured quantities, the answer should be expressed to the same place value as the measured quantity in the calculation that is the least precise. Steps: 1. Identify the least precise measurement. Identify the least precise place value (furthest to the right). 2. Perform the calculation and the solution must be written to the same place as the place identified in #1 above. Hint: If you line the numbers up on top of each other, taking care to line up the place values (eg. tens above tens, ones above ones, and so on) and then draw a vertical line (called the line of significance) immediately after the least precise measurement ends, your solution should be rounded off so that it, too, ends at this line. A) 94.02 g + 61.1 g + 3.1416 g = 158.2616 g = 158.3 g B) 4.01 m - 2.30642 m = 1.70358 m = 1.70 m


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C) 6500 L + 730 L = 7230 L = 7200 L D) 98 g + 9 g = 107 g = 107 g 2. Multiplication, Division, and Exponents: When multiplying, dividing, etc. measured quantities, the answer should have the same number of significant digits as the measurement with the least number of significant digits (place value is not important). Ex. A) 5.6432 m × 0.020 m = 0.112864 m2 = 0.11 m2 B) 2500 m × 2 m = 5000 m2 = 5 × 103 m2 C) 26.3 mm × 35 mm = 920.5 mm2 = 9.2 × 102 mm2 3. Rounding When completing calculations, do not round any of the intermediate answers on your way to finding the solution to a problem. The only rounding that should occur is the final answer that is being reported. Rule: Equal to or greater than 5 , then round up. Less than 5, then no change. Ex.

51.25 m rounded to 3 sig.figs

Rounds to

51.3 m

51.24 m rounded to 3 sig.figs

Rounds to

51.2 m

Unit Analysis This is the process used to change from one unit to another. It is a three step process outlined below: 1. Write out a pathway. 2. Write out the conversion factors with your desired unit on the left hand side. 3. Write the equation starting with the given and then multiplying by the conversion factors. Be careful to note what side of the fraction the unit is located. 4. Units that are raised to a power other than 1 must be treated differently. The trick is to write the conversion factors (step three) to the same power as the original unit.

Examples: 1. 2.0 km = _______ m.


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2. 3.75 h = ______ s.

3. 25.5 km/h = ______ m/s.

4. 1.5 m2 = _______ cm2.

5. 9.81 m/s2 = _______ ft/min2.

6. 8.54 g/cm3 = ________ kg/m3.

This process is also called times-sign-line by many teachers. It can be used for any conversions as long as you know the conversion factors (see conversion sheets at the end of the section).

Proportionality: Ö Physics equations often involve many variables. It is useful to be able to determine how a change to one variable would impact upon the other variables, assuming that all other variable remain constant. Ö In an equation, two variables can be either directly or indirectly proportional to each other. If they are directly proportional, a change to one variable has the


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same effect on a second variable. For example, in the equation for velocity (v = d/t) if the displacement d doubles, then the velocity also doubles (assuming that all other variables remain constant). Ö To clarify, just think of a car that travels twice as far in the same amount of time as a slower car. It did so by doubling its velocity. On the other hand, if time doubles, velocity is reduced by half. Again, think of a car that takes twice as long to travel a certain trip. This would be because its velocity was half as much as usual. Therefore, time and velocity are said to be indirectly proportional. Examples: 1. A car is traveling at 75 km/h a. What is the car’s new velocity if it travels twice the distance in the same period of time?

b. What is the car’s new velocity if it travels the same distance in twice the time?

c. What is the car’s new velocity if it travels the same distance in half the time?

d. What is the car’s new velocity if it travels three times the distance in twothirds the time?

2. The force of gravity between two masses a distance r apart is given by the formula (G is the universal gravitational constant)

F =G

M 1M 2 r2

a. Suppose during an experiment F measures 25 N. What will F measure if the second mass is tripled?


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b. What will F measure if M1 is halved and M2 is quartered?

c. What will F measure if r is doubled?

d. What will F measure if M1 is doubled, M2, is quartered, and r is tripled?

Motion Average Velocity: A measure of the displacement divided by the total time and is related to the direction of travel. Represented by the formula v =

∆d , the ∆ symbol is the Greek ∆t

letter capital delta and usually used in math and science to represent the words “change in”. Speed is change in total distance over time, but is independent of direction. Examples: Take into account significant figures. 1. What is the velocity of someone walking 120.5 m in 33.5 s?

2. What distance is traveled by a car driving 105 km/h in 2.75 hours?

3. How long does it take to drive to Drumheller Alberta, which is 5.0 × 102 km from Saskatoon? Assume an average velocity of v = 1.10 × 102 km/h.

4. How many meters does a car traveled in 6.75 min? The velocity of the car is 95 km/h.


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Circular Motion Velocity for objects traveling in a circle uses the same concepts for motion in a straight line. The difference is that in a circle an object is constantly changing direction. To simplify we say an object is rotating/revolving in a clockwise or counter-clockwise manner. The formula for circular velocity is: v =

∆d 2πr . Where r is the radius of the circle and = ∆t T

T is the time period for one complete rotation. We will take all rotating objects as having a positive velocity. Examples: Take into account significant figures. 1. How fast is the edge of a 4.0 cm radius CD traveling if its period is 1.57 × 10-3 s?

a. Change that to km/h.

2. What is the radius of a car tire that travels at 75 km/h and rotates 5.0 times per second?

Displacement – Time Graphs: D-T Graphs The displacement – time relationship is represented graphically by a straight line. Many quantities are measured and analyzed with the aid of graphs. Graphs provide a reference point from which to compare data. The slope of the line is the average velocity. Consider the three graphs below. Each line makes the same angle with the x-axis. You have to look carefully at the scale to infer which graphs represent the fastest and slowest speeds.


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Driving Around Town 4000

3500

Displacement (km)

3000

2500

2000

Graph #1

1500

1000

500

0 0

5

10

15

20

25

30

35

40

35

40

Time (h)


700

Displacement (km)

600

500

400

Graph #2

300

200

100

0 0

5

10

15

20

25

30

Time (h)


1400

Displacement (km)

1200

1000

Graph #3

800

600

400

200

0 0

5

10

15

20

Time (h)

25

30

35

40


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Placing all of the above lines within one graph yields an easy-to-see relationship: Driving Around Town 4000

3500

#1

Displacement (km)

3000

2500

2000

#3 1500

1000

#2

500

0 0

5

10

15

20

25

30

35

Time (h)

Recall, to make things easier: Ö Objects always start traveling to the right; the positive direction. Ö Objects traveling to the left are moving in the negative direction. Ö The steeper the slope, the greater the velocity. Ö A positive slope is a positive velocity. Ö A negative slope means the object is traveling in the opposite direction. Ö Velocity is a measure of speed and direction. Speed is independent of direction. Ö Distance is a measure of the total distance traveled. Ö Displacement is a measure of the net distance from a starting (reference) point. Ö Instantaneous velocity is the slope at a specific time, t.

40


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D-T graphs relay much information of an objects trip: D-T Graph Analysis Displacement = 20 m Velocity = 0 m/s Distance = 20 m

Displacement from origin is increasing. Velocity = +2 m/s.

25

20

Displacement from origin is decreasing. Velocity = -4 m/s.

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Displacement = 0 m Velocity = +1 m/s Distance = 80 m

Displacement (m)

10

Displacement = 0 m Velocity = -4 m/s Distance = 40 m

5

0 0 -5

-10

5

10

15

20

25

30

35

45

Displacement from origin is increasing. Velocity = -4 m/s Displacement from origin is decreasing. Velocity = +1 m/s

-15

-20

40

Displacement = -20 m Velocity = 0 m/s Distance = 60 m

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Time (s)

Acceleration and V-T Graphs Average acceleration is the change in velocity over change in time:

a=

∆v v f − vi = ∆t t f − t i

Examples: Take into account significant figures. 1. Starting from rest, an object accelerations to a velocity of 175 m/s in 6.5s. What was the average acceleration?

2. How long does it take a bus to accelerate from 62 km/h to 95 km/h if the average acceleration is 5.8 m/s2?

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Velocity – Time Graphs Recall: Ö Objects always start traveling to the right; the positive direction. Ö Objects traveling to the left are moving in the negative direction. Ö The steeper the slope, the greater the acceleration. Ö A positive slope is a positive acceleration Æ velocity ⇑. Ö A negative slope means negative acceleration Æ velocity ⇓. Ö An object changes direction when the graph crosses the x-axis. Ö Displacement is a measured by calculating the area under the graph. o The area above the x-axis is taken to be positive displacement (distance from starting position is increasing). o The area below the x-axis is taken to be negative displacement (changing direction, distance from starting position is decreasing). Ö Distance is the total area contained between the graph and the x-axis (the area contained below the x-axis is taken as an absolute value). Ö Instantaneous acceleration is the slope at a specific time, t. V-T graphs contain much information about the path of an accelerating object. V-T Graph Analysis Acceleration = 0 m/s2 Velocity & Speed = 1000 m/s Displacement ⇑ Distance ⇑

1500

1000

Acceleration = + 200 m/s2 Velocity ⇑ Speed ⇑ Displacement ⇑ Distance ⇑

Acceleration = -200 m/s2 Velocity ⇓ Speed ⇓ Displacement ⇑ Distance ⇑

Velocity (m/s)

500

Acceleration = +100 m/s2 Velocity & Speed = 0 m/s Displacement = -17500 m Distance = 37500 m

0 -20

-500

-1000

-15

-10

-5

0

5

10

15

20

25

30

35

40

45

50

2

Acceleration = 0 m/s Velocity = -1000 m/s, Speed = +1000 m/s Displacement = 0 m Distance = 20000 m Back to starting position.

Acceleration = -200 m/s2 Velocity & Speed = 0 m/s Displacement = 10000 m Distance = 10000 m Direction has changed. Acceleration = -200 m/s2 Velocity ⇓ Speed ⇑ Displacement ⇓ Distance ⇑

-1500 Time (s)

Acceleration = +100 m/s2 Velocity = ⇑ Speed ⇓ Displacement = ⇓ (farther from starting position, but in opposite direction) Distance = ⇑

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Example: Refer to the Changing Speed Limits graph for the following questions. Changing Speed Limits 200

150

Velocity (km/h)

100

50

0 0

10

20

30

-50

-100

Time (h)

1. What is the acceleration at 5.0 h?

2. What is the acceleration at 33.75 h?

3. What distance was traveled after 15 h? 12h

40

50

60


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4. What was the displacement after 40.0 h?

5. What was the total displacement traveled? Distance traveled?

6. How long did it take to drive a distance of 75.8 km from the starting position?

Introduction & Motion: Conversion Sheets

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Prefix

Symbol

10α

exa

E

1018

1 000 000 000 000 000 000

(Em, Eg, El, Es)

peta

P

1015

1 000 000 000 000 000

(Pm, Pg, Pl, Ps)

tera

T

1012

1 000 000 000 000

(Tm, Tg, Tl, Ts)

giga

G

109

1 000 000 000

mega

M

106

1 000 000

kilo

k

103

1 000

(km, kg, kl, ks)

hecto

h

102

100

(hm, hg, hl, hs)

deca

da

101

10

Factor by Which Base Unit is Multiplied

d

10–1

centi

c

–2

milli micro

(Mm, Mg, Ml, Ms)

(dam, dag, dal, das)

1 (metre – m, gram – g, litre – l, second – s)

100 deci

(Gm, Gg, Gl, Gs)

(dm, dg, dl, ds)

0.1

10

(cm, cg, cl, cs)

0.01

m

10–3

(mm, mg, ml, ms)

0.001

µ

10–6

(µm, µg, µl, µs)

0.000 001

nano

n

10

–9

(nm, ng, nl, ns)

0.000 000 001

pico

p

10–12

(pm, pg, pl, ps)

0.000 000 000 001

femto

f

10–15

(fm, fg, fl, fs)

0.000 000 000 000 001

atto

a

10–18

(am, ag, al, as)

0.000 000 000 000 000 001

Numbers differ by multiplying (up) or dividing (down) by 1000. Æ Move 3 decimal places

Numbers differ by multiplying (up) or dividing (down) by 10. Æ Move 1 decimal place.

Numbers differ by multiplying (up) or dividing (down) by 1000. Æ Move 3 decimal places

¾ The number of decimal places to move is the same as the difference of the exponents. ¾ The above prefixes can be used for any unit by placing the symbol for the prefix in front of the unit’s symbol.

Decimal place moves to the left.

Decimal place moves to the right.


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Converting Between Time Units

Time Period

Divide

Years 365.25

Weeks 7

Days 24

Hours 60

Minutes

Multiply 60

Seconds


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Converting between Metric and English System Mass 1 lb = 454.5 g 1 oz = 28.4 g 1 kg = 2.2 lb 1 lb = 16 oz

Length 1 in = 2.25 cm 1 mi = 1.6 km 1 yd = 0.91 m 1 m = 3.3 ft 1 ft = 12 in

Volume 1 qt = 0.946 l 1 gal = 3.784 l 2.1 pt = 1 l 1 gal = 4 qt 2 pt = 1 qt

lb Æ pounds oz Æ ounce

in Æ inch mi Æ mile yd Æ yard ft Æ feet

qt Æ quart gal Æ gallon pt Æ pint

Waves & Sound

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Waves

Mr. MacDonald

Waves & Sound

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Mr. MacDonald

Waves Ö A wave is a transfer of energy, in a form of a disturbance usually through a material substance, or medium. Ö Electromagnetic Waves Ö Sound waves Ö Water waves Ö Pressure waves Ö Gravity waves Ö Matter waves Ö When objects repeat a pattern of motion (e.g. a pendulum), we say that object is vibrating or oscillating. Ö The oscillation is repeated over and over with the same time interval each time. Ö One complete oscillation is called a cycle. Ö The number of cycles per second is called the frequency, f. The frequency is measured in Hertz (Hz). Ö The period , T, usually measured in seconds, is the time required for one cycle. The frequency and period are reciprocals of each other. frequency =

period =

cycles 1 = time T

time 1 = cycle f

Examples 1. A pendulum completes 30 cycles in 15 seconds. Calculate its frequency and period.

2. What is the period of a pendulum that has a frequency of 10 Hz?

3. What is the frequency, in Hz, of the Earth’s orbit about the Sun?

Waves & Sound

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Mr. MacDonald

Wave Motion & Terminology Ö Waves transmit energy, not matter. Ö It is a disturbance from some normal value of the medium that is transmitted, not the medium itself. Ö These types of waves are called periodic waves; where the motions are repeated at regular time intervals. Ö A single disturbance is called a pulse, or shock wave. Creating half of a cycle results in a pulse.

Transverse Waves Ö The particles in the medium vibrate at right angles to the direction in which the wave travels. Ö The high section is called the crest, and the low section is called the trough. Ö The height of the crest, or depth of the trough, from the equilibrium position is called the amplitude. Ö For periodic waves, the distance between successive crests and troughs is equal and is called the wavelength. The symbol for the wavelength is the Greek letter lambda, λ. Ö The period of a transverse wave is the time it takes for one wavelength (one cycle) to pass a fixed point. Ö The frequency is the number of wavelengths that passed a fixed point in one second. Ö Examples include water waves and making vibrations on a rope.

library.thinkquest.org/. ../Waves/basic.htm

Waves & Sound

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Mr. MacDonald

Longitudinal Waves Ö The vibrations of the particles are parallel to the direction of motion. Ö There are a compressions and rarefactions created in longitudinal waves. Ö One wavelength is the distance between the midpoints of successive compressions or rarefactions. Ö The amplitude is the maximum displacement of the particles from their rest position. Amplitude is a measure of the wave’s energy. Ö The period of a longitudinal wave is the time it takes for one wavelength (one cycle) to pass a fixed point. Ö The frequency is the number of wavelengths that passed a fixed point in one second. Ö Sound waves, pressure waves are examples.

www.christian81.free-online.co.uk

Transmission of Waves Ö When a wave is generated in a spring or a rope, the wave travels a distance of one wavelength, λ, along the rope in the time required for one complete vibration of the source (the period). We can use the formula for velocity to derive the wave equation:

Waves & Sound

Mr. MacDonald

25

velocity, v = and therefore but Therefore

change in distance, ∆d change in time, ∆t ∆d = λ, and ∆t = T λ T 1 f = T v = fλ v=

Ö The wave equation, v = fλ, applies to all waves, visible and invisible. Examples 1. The wavelength of a water wave in a ripple tank is 0.080 m. If the frequency of the wave is 2.5 Hz, what is its speed?

2. The distance between successive crests in a series of water waves is 4.0 m, and the crests travel 9.0 m in 4.5 s. What is the frequency of the waves?

3. The period of a sound wave from a piano is 1.18 × 10-2 s. If the speed of the wave in air is 3.4 × 102 m/s, what is its wavelength?

Transmission and Reflection •

Waves travel at uniform speed as long as the medium they are in does not change. (Note: If the tension changes, then that is a change in medium.)

•

When waves propagate into a different medium, the frequency stays the same. The wave velocity changes.

•

Thus, the wavelength must change as well. v is directly proportional to λ.

Waves & Sound

Mr. MacDonald

26

•

Assuming a constant tension, a wave traveling from a light rope to a heavier rope will decrease in velocity and decrease in wavelength.

•

Conversely, a wave traveling from a heavy to light rope will increase in both velocity and wavelength.

•

The properties of transmitted waves can be derived because we know the frequency is constant.

•

Consider the following diagrams (we assume all energy is transmitted):

Slow Medium

Fast Medium

v1Æ

v2Æ

λ1

λ2

v2 > v1

λ2 > λ1

v1 = fλ1 Solving for f :

f =

v1

λ1

and

v2 = fλ2

and

f =

Combining :

v1 v2 = λ1 λ2

or

v1 λ1 = v2 λ2

v2

λ2

Examples 1. A slinky is stretched and connected to a string. A wave travels at 3.5 m/s in the slinky. It is observed that the velocity in the string is 7.8 m/s. (a)

Find the transmitted wavelength, λ2, if the original wavelength was 0.24 m. What is the frequency of the original and transmitted wave?

Waves & Sound (b)

Mr. MacDonald

27

Suppose the transmitted wavelength was 0.92 m, what is the original λ1?

•

Waves that propagate and encounter a rigid barrier undergo fixed-end reflection. o The reflected wave/pulse will be inverted. A crest will become a trough and vice-versa. Amplitude, velocity, and frequency will be conserved in an ideal system (no loss of energy to outside forces, i.e. friction).

•

If there is no fixed boundary, the wave/pulse undergoes free-end reflection. o A crest will be reflected as a crest and a trough as a trough. All wave properties will be conserved (ideally).

•

Usually when waves propagate into another medium there is a transmitted and reflected wave. This is called partial reflection. o Some of the energy is transmitted, and some is reflected. Recall the larger the amplitude, the more energy a wave has. o There are two cases:

1. Fast to a slow medium.

Fast

Slow

v1Æ

V2Æ

Å v1

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28

Ö The material in the slower medium has more inertia to resist the motion of the wave. This acts like a rigid obstacle (fixed-end). Ö The reflected wave is inverted, but the transmitted wave (if any) is not inverted.

2. Slow-to-fast medium

Slow

Fast

v1Æ

Å v1

V2Æ

Ö The fast medium acts like a free-end reflection. There is no inversion in either the transmitted or reflected wave/pulse. Ö Recall there is a change in λ and v in both cases.

Transmission of Waves Ö Wave interference is when two or more waves act simultaneously on the same particles of a medium. Ö Principle of Superposition: The resultant displacement of a given particle is equal to the sum of the displacements that would have been produced by each wave acting independently.

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Mr. MacDonald

¾ Constructive interference results when two of more waves interfere to produce a resultant displacement greater than the displacement caused by either wave itself.

www.physicsclassroom.com

¾ Destructive Interference is when the resultant displacement is smaller than the displacement that would be caused by one wave by itself.


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Mr. MacDonald

Standing Waves: Interference in One Dimension Ö Demo and discuss with use of slinky and/or string. Ö Computer simulations of the standing wave phenomenon. Discuss how many nodes there may be and how to get the wavelength from that information. Standing Waves: Interference in One Dimension Ö A standing wave interference pattern occurs if interfering waves have the same amplitude, wavelength, frequency, and are traveling in opposite directions.

Called a standing wave for short.

electron4.phys.utk.edu/ 141/dec1/December%201.htm

Ö The node, or nodal point, is where crests and troughs of equal amplitude interfere destructively. For one-dimensional waves the fixed ends are nodal points. Ö The antinodes, or loops, are areas of constructive interference. Ö The number of nodal points for a given medium depends on the physical structure of that medium, thus only certain frequencies will produce a standing wave pattern. Such frequencies are resonance frequencies for that medium.

Waves & Sound

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Mr. MacDonald

Ö If one antinode were created with a certain frequency, say f1, then to create two or three antinodes (etc.) the frequency would have to be 2f1, or 3f1 respectively. Note the decrease in amplitude as more antinodes are created.

sol.sci.uop.edu/.../ soundinterference.html

Ö The distance between two successive nodes in a vibrating string is ½λ. Ö The point of maximum displacement from a node is ¼λ.

Waves & Sound

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Mr. MacDonald

Examples 1. A standing wave interference pattern is produced in a rope by a vibrator with a frequency of 45 Hz. If the wavelength is 55 cm, what is the distance between successive nodes? What is the velocity of the wave?

2. The distance between the first and sixth nodes in a standing wave is 75 cm. What is the wavelength of the waves? What is the velocity if the source has a frequency of 12 Hz?

3. A standing wave pattern is produced. It is observed to have 10 loops with a node at each end. The distance between the first and last node is 75.0 cm and the waves have a velocity of 6.25 m/s. What frequency is needed to observe four loops?

Waves & Sound

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33

Reflection of Waves Ö When waves strike a barrier they undergo reflection. Ö The direction of the reflected wave obeys the law of reflection: The angle of reflection is equal to the angle of incidence.

Normal

Incident wavefront

θi

θr

Reflected wavefront

Barrier θi = θr Ö All parts of the waves are reflected Æ The reflected wave will have the same characteristics as the initial wave.


•

Consider a parabolic barrier: All the waves reflect and intersect at a common point, called the focal point.

Waves & Sound

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Mr. MacDonald


•

This has many applications. One of which is a sound-collecting dish used by the military and media at sport games.

Refraction of Waves Ö Refraction is the change of direction of waves at the boundary between two different media. Ö Take, for example, deep waves traveling into shallow water:


Ö It is like a pulse traveling from a fast to slow medium, but here the boundary is at an angle to the wave front. Note that there will be reflected waves as well.

Waves & Sound

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Mr. MacDonald

Diffraction and Interference of Waves •

When waves encounter a small hole in a barrier, they will not pass right through.

•

They bend around the edges of the barrier and form circular waves. This is called diffraction.

•

Diffraction can also occur around the edges of an obstacle in a waves path. Ö The smaller the wavelength in comparison to the size of the obstacle, the less the diffraction. Ö Consider a doorway. Small sound wave-lengths will be less diffracted than longer sound wavelengths.

•

Wave interference occurs when waves pass through two or more holes in a barrier. Nodes and antinodes form at regular intervals (see image on page 301 of text).

cougar.slvhs.slv.k12.ca.us

Waves & Sound

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Mr. MacDonald

Sound ¾ The speed of sound varies in different types of media. Generally, sound travels fastest in solids, slower in liquids, and slowest in gases. ¾ Recall that sound is a longitudinal pressure wave, hence it requires a medium to travel though. ¾ The speed of sound in air is approximated by:

v air = 331 + 0.610Tair ¾ where Tair is the temperature of the air in degrees Celsius (not the period) and the units for the velocity are m/s. (Note: The numbers 331 and 0.610 are experimentally derived and have units such that the velocity comes out in m/s. This formula can be used only for a small range in temperature.) ¾ Using the Kelvin temperature scale the formula becomes:

v air = 20.0 Tair ¾ where once again there are units associated with the value of 20.0 that give then velocity in m/s. This equation is useful for a wider temperature range than using the Celsius scale. ¾ Objects traveling faster than the local speed of sound are said to be traveling at supersonic speeds. Supersonic objects generate a discontinuity of pressure and temperature in the immediate area of the moving object. The pressure drops and the temperature increases and this is generally called a shock wave. ¾ When objects travel faster than sound their generated sound waves bunch up into a small area. When such an object passes a listening device all of the sound is heard at once, called a sonic boom. Example 1. What is the speed of sound in air that is 25 ºC? 298 ºK?

2. A fighter pilot wants to travel three times the speed of sound. How fast must she travel if the air temperature is 15 ºC?

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Mr. MacDonald

3. What must the temperature be, in Celsius, for the speed of sound to equal the speed of light (c = 3.00 × 108 m/s)?

4. Before a going supernova, a star’s core temperature can reach upwards to 5.0 × 109 ºC. Calculate the speed of sound within the core.

¾ Example four shows one reason why the formula fails at high temperatures. The answer comes out to be greater than the speed of light, but our current understanding of physics dictates that nothing can travel faster than light. ¾ Using the Kelvin-scale formula will yield a physics friendly solution, but it too is incorrect for that extreme temperature. The formulas would be more accurate if the gas density and pressure were also taken into account (as these values are very large for the core of a star). ¾ Both equations are adequate for the confines of this course, but both will fail for extreme cold (near absolute zero) and extreme high temperatures. Doppler Shift ¾ A source generating waves moves relative to an observer, or vice – versa, there is an apparent shift in the source’s frequency. ¾ If the separation between source and observer is increasing, then the frequency apparently decreases. ¾ If the separation between source and observer is decreasing, then the frequency apparently increases. ¾ This can be seen by visualizing what happens to sound waves of a moving object.

http://en.wikipedia.org/wiki/Doppler_shift

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Mr. MacDonald

¾ The waves compress in the direction of travel and expand in the other direction. ¾ This is a phenomenon familiar to many people if you’ve ever stood on a road with traffic going by.

http://static.howstuffworks.com/gif/doppler.gif

¾ The relationship between the frequency of a moving source and an observer (in one dimension) is represented by the Doppler shift formula as two cases: The observer and source are approaching or receding.

⎛ 1 + vo v ⎞ ⎜ ⎟⎟ f f = Approaching: o s⎜ v v − 1 s ⎝ ⎠

Receding: ¾ ¾ ¾ ¾ ¾ ¾

⎛ 1 − vo v ⎞ ⎟⎟ f o = f s ⎜⎜ 1 + v v s ⎝ ⎠

fo = observed (heard) frequency fs = source frequency vo = observer’s velocity vs = velocity of the source v = speed of sound in medium. We do not need to associate a sign notation with the moving objects, the formula takes that into account.

¾ The above are the general formulas for moving observers and sound sources. The formulas become much simpler of one object is moving and the other is not. ¾ What happens to fo if if vs ≥ vo, vo ≥ vs, vs ≥ v or vo ≥ v?

Waves & Sound

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Mr. MacDonald

Examples 1. What is the observed frequency of a 525 Hz source moving towards a stationary observer at 75 m/s? Take the speed of sound to be 375 m/s.

2. How fast must someone be driving so that a stationary observer hears a frequency twice that of the source frequency? Take the speed of sound in air to be 345 m/s. For a reference convert your answer into km/h.

3. A fighter plane emits an audible frequency of 2.5 ×103 Hz. What frequency is heard by a stationary observer if a fighter plane is approaching him at Mach I? Mach II? (Mach I is one times the speed of sound, Mach II is twice the speed of sound.)

4. A police siren has a frequency of 1.8 × 104 Hz. A crook in his getaway car drives away from the police at 105 m/s. What frequency is heard by the crook is the police car is driving at 85 m/s? The temperature today is 25 ºC.

Physics 20 – Light

40

Sources of Light


41

Light Ö

Light is the range of frequencies (wavelengths) that can be seen with the human eye. Light travels as an electromagnetic wave and can be thought of as a packet of energy. o For our purposes, we will use the properties of transverse waves to describe light. o The main difference is that energy light carries depends on it wavelength (or frequency), not its amplitude. o The science of astronomy/astrophysics is all based on observing the EM spectrum that reaches the Earth.

Ö

We see only a small part of the electromagnetic spectrum. The EM spectrum consists of radio, micro, infrared, visible (colour), ultraviolet, X, and gamma rays.

Ö

Visual Spectrum (approximate) o Red: 600 – 700 nm o Yellow: 575 – 600 nm o Green: 500 – 575 nm o Blue: 425 – 500 nm o Violet: 375 – 425 nm

Matt Oltersdorf et al.: www.twcac.org/ Tutorials/notes(3).htm


42

universe.gsfc.nasa.gov/ lifecycles/technology.html

Mr. MacDonald

Physics 20 – Light Ö

43

We see objects because they reflect light (non-luminous bodies) or they emit light (luminous bodies) o We see the visual spectrum of light. Certain colours appear because an object absorbs all light except that colour. o Black objects absorb all light and reflect none. o White objects reflect all light and absorb none.

Ö

Light travels in a straight-line path (except in extreme gravitational fields); this is called rectilinear propagation. o We use the ray model. We use rays to represent the path followed by light. A collection of rays is called a beam of light. Ray

Beam

Ö

The speed of light, c, is 299 792 458 m/s in a perfect vacuum Æ This is the fastest anything can travel according to Einstein’s Theory of Relativity. We round to c = 3.00 × 108 m/s. o The speed of light changes as it enters different media, but all types of EM radiation travels at the same speed in the same medium. o c = fλ, like we studied for waves. o Since c is constant, if the frequency increases, wavelength decreases and vise – versa. f and λ are inversely proportional.

Ö

To describe the properties of the EM spectrum we use the unit of nanometers, nm (1 nm = 10-9 m)


44

Sources of Light Ö

Consider a light source: o The rate at which light is emitted is called the luminous flux, P (flux means a continuous flow of something per unit area). The unit of luminous flux is the lumen, lm. o In the picture below, if you add up all of the light striking the dotted area per unit time, that is the luminous flux.

International Light: www.intl-light.com/handbook/flux.html

Ö Ö Ö

A light bulb has approximately 22 lumens/watt (Watt is energy/sec). A lumen is a measure of visible light from a source; a watt is the total light energy from a source. Often we are only interested in the illumination of a specific area (like a book, desk, roads, etc.) Light diverges from its source in a spherical shape. o So to find the illumination on a surface, the illuminance E, we divide the luminous flux by the area of a sphere a distance d from the light source. The unit is lm/m2, or lx.

Luminous Flux Surface Area P E= 4πd 2

Illuminance =

where d is the distance from the light source to the surface.

o Illumination of a surface is an example of the inverse square law.


45

cougar.slvhs.slv.k12.ca.us/. ../properties.html

Examples 1. The luminous flux from a light source is 445 lm. a. What is the illumination on a surface 1.5 m away?

b. What is the illumination on a surface 4.5 m away?

2. The illumination on a desk is 125 lm/m2. What is the luminous flux of the desk lamp if it is 75 cm above the desk?


46

3. Suppose a streetlight has a luminous flux of 2275 lm. How high is it from the street if the street has an illumination of 2.83 lm/m2?

Light Intensity Ö

This is measured in candela, cd (candle power). o It is the luminous flux that falls on one square meter one meter in radius. Thus the equation becomes:

o

I=

P , note that the intensity is the same as the illuminance at a 4π

distance of 1 m. o If we are given an intensity, the illuminance is: o

E=

I d2

Examples 1. What is the intensity of a 1695 lm bulb?

2. The illumination on a person from a light source 3.2 m away is 345 lx, what is the intensity of the light?


47

3. Two light sources are equally intense on a wall. One bulb has a luminous flux of 3.5. × 103 lm and is 3.20 m from the wall. What distance is the other bulb from the wall if its luminous flux is 9.8 × 103 lm.

Shadows: Umbra & Penumbra Ö

A shadow is a region of space within which rays from a source of light are obstructed by an opaque body. o Shadows are only visible if there is an object to intersect it. o Light from a point source cannot be seen anywhere in the shadow.

Point Source

Body

Shadow

o If the light is coming from source with a finite angular size then two shadow regions will form. o Umbra Æ Where complete darkness prevails, no part of the source can be seen. o Penumbra Æ A part, but never all, of the light source is visible. o Out side of those two regions, the entire source is visible.


48


Ö

There are two types of solar eclipses, total and partial. o Total occurs when the Moon is in the umbra. o Partial occurs when the Moon only enters the penumbra. The closer the Moon is to the umbra, the more of the Sun is blocked out. o The length of the umbra has a finite size, should the Earth be directly behind the Moon, but behind the umbra, a ring eclipse forms.


Ö Ö

Here the Moon travels through a much larger Earth’s shadow Æ which is why lunar eclipses last many hours. The colour change of the Moon is due to refraction of light by the Earth’s atmosphere (same reason we see red/orange sunsets and sunrises).


49

Example Sketch the shadow cast by an opaque object in front of a finite light source. Then mark the points in the shadow were the following eclipses would be seen by an observer.

A

C

B

D


50

The Pinhole Camera Ö

Simply consists of a light proof box with a pinhole at one end and a screen (tracing paper or film) at the other.

Ö

The image is formed on the screen by light following a straight-line path from the object.


Ö

We will examine the geometry of the pinhole camera.

Pinhole Camera

hi

di

do

ho Object

Ö

Similar triangles reveal: height of image distance of image from pinhole =− height of object distance of object from pinhole or hi d =− i ho do

Physics 20 – Light Ö Ö

51

The negative sign is there to represent the lateral inversion of the image. The above equation is called the magnification equation. Either ratio gives you how many times larger or smaller than the object the image will be. M =

hi , ho

M =−

di do

M = magnificat ion

Example 1. What is the size of the image of a tree that is 8.0 m high and 80.0 m from a pinhole camera that is 20 cm long? What is the magnification?

2. The size of a flagpole’s image is 15 cm on the screen of a pinhole camera that is 25 cm long. What is the original height of the flagpole if it 12 m from the camera?


52 Solar System Data Earth – Sun Distance: 1.496 × 1011 m Earth – Moon Distance: 3.84 × 108 m Radius of the Earth: 6.371 × 106 m Radius of the Moon: 1.74 × 106 m

Planet Distances Distance from Sun (AU)

Planet Mercury

0.4

Venus

0.7

Earth

1.0

Mars

1.5

Asteroid Belt

2–5

Jupiter

5.2

Saturn

9.5

Uranus

19.2

Neptune

30.1

Pluto

39.4

Physics 20 – Mirrors

53

Mr. MacDonald

Mirrors and the Reflection of Light


Mr. MacDonald

54

Mirrors Normal

Incident ray

θi

θr

Reflected ray

Barrier/Mirror θi = θr •

Incident ray: a ray approaching a surface.

•

Point of incidence: where incident ray strikes a surface.

•

Normal: It is a line drawn perpendicular to the surface at the point of incidence.

•

Reflected Ray: The portion of the incident ray that leaves the surface at the point of incidence.

•

Angle of Incidence: The angle between the incident ray and the normal.

•

Angle of Reflection: The angle between the reflected ray and the normal.

•

Laws of Reflection: o The angle of incidence is equal to the angle of reflection. o The incident ray, the normal, and the reflected ray are coplanar.


55

Mr. MacDonald

www.fas.harvard.edu/.../ OpticsDisk04.jpg

• • •

The above image is an example of specular reflection. The surface is very smooth so incident parallel rays reflect as parallel rays. If parallel rays strike a rough surface (on the microscopic level) they will not be reflected in parallel. This is called diffuse reflection. Both specular and diffuse reflection obeys the laws of reflection.

library.thinkquest.org/ 27356/p_reflection.htm


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Mr. MacDonald

Images in a Plane Mirror • •

When you look into a plane mirror, your image appears to be behind the surface of the mirror. To understand why, and find the position of objects in a plane mirror, we need to consider how light is seen in the eye. Light emanates from an object in all directions, but the eye only sees a diverging cone of rays.

physics.usc.edu/.../ LightOptics.html

• •

When you see an object in the mirror, the mirror reflects the cone of rays. You cannot see that light was reflected, and assumes the cone of rays originated behind the mirror.

physics.usc.edu/.../ LightOptics.html

•

The resulting image is called a virtual image. That is because the image is produced at the point where the reflected rays, extended behind the mirror, intersect.

Physics 20 – Mirrors • •

57

Mr. MacDonald

A real image (like the one formed with the pinhole camera), is an image that can be formed on a screen (where the actual rays intersect). Images in plane mirrors are laterally inverted Æ What was on the right now appears to be on the left from the mirrors perspective.

Sample Problem Ö

Given an object located in front of a plane mirror, locate the image and show how the eye “sees” it.

1. When drawing rays that don’t actually exist, draw them as dotted lines. Rays that do exist are solid lines 2. Draw a perpendicular from the object to the mirror. 3. Extend this line an equal distance behind the mirror to locate the image. 4. The eye considers that the light originates from a point source image behind the mirror. The light rays from a point source travel out in all directions, and they include a cone of rays traveling towards the eye. 5. Rays originate from the object. Note the laws of reflection.


58

Mr. MacDonald

Curved Mirrors • • •

Converging mirror: Are concave in shape and focus parallel rays at a single point. Diverging mirror: Is convex in shape and makes parallel rays diverge. Both types can be thought of as a section of a spherical mirror.

Geometry and Terminology of Spherical Mirrors • • •

The centre of a curved reflecting surface is called the centre of curvature, C, and the radius of curvature is the shortest distance from the centre to the curved surface. The geometric centre of the curved mirror is called the vertex, V. The straight line passing through V and C is called the principal axis.


59

Mr. MacDonald

physics.bu.edu/~duffy/ PY106/Reflection.html

Reflection in a Converging Mirror • • •

Rays that are parallel to the principal axis will reflect and converge at the same point called the principal focus, F. The principal focus will be halfway between V and C. Rays that are not parallel to the P.A. but parallel to each other will meet at a focus (not the principal focus). The line connecting all the focal points is called the focal plane.

Images in a Converging Mirror •

Image beyond principal focus: o Inverted & real. • Image between principal focus and vertex: o Erect & virtual. • The further the object is from the vertex, the smaller the image. Ö To determine the position of an image in a converging mirror, it is necessary to use only two rays that intersect. Ö We use two rays that emanate from the tip of the object. One that is parallel to the principal axis, and one that passes through F. Another ray is the ray along the radius of curvature. Rules for Rays in a Converging Mirror 1. A ray that is parallel to the principal axis is reflected through the principal focus. 2. A ray that passes through the principal focus is reflected parallel to the principal axis. 3. A ray the passes through the centre of curvature is reflected back along the same path.


Mr. MacDonald

60

www.batesville.k12.in.us/ physics/PhyNet/Optic..

1. Object between F and C.

C

Characteristics Æ

O

F

V


Mr. MacDonald

61

2. Object at C.

O

C

F

V

C

F

V

Characteristics Æ

3. Object beyond C.

O

Characteristics Æ


Mr. MacDonald

62

4. Object at F.

O

C

F

Characteristics Æ

5. Object between F and V.

C

Characteristics Æ

F

O

V

V


Mr. MacDonald

63

Images Formed by Diverging Mirrors •

The principal focus and centre of curvature are virtual.

•

Rays directed towards the virtual principal focus are reflected parallel to the P.A.

•

Rays directed parallel to the P.A. are reflected in such a way that, when extended into the mirror, they go through the principal focus.

•

Rays directed towards the centre of curvature are reflected back along the same path.

Finding the Image in a Diverging Mirror

O

•

V

F

C

Images are always virtual, erect, smaller than the object, and located between the vertex and the principal focus.


Mr. MacDonald

64

Equations for Curved Mirrors Consider the following diagram:

C

ho

F

hi

V

f do

di f = the focal length of the mirror. do = the distance from the vertex to the object. di = the distance from the vertex to the image. ho = the height of the object. hi = the height of the image. Ö

Using the geometric properties of similar triangles we can arrive at the following:

1 1 1 + = do di f Ö

This is called the mirror equation.

Sign Convention 1. 2. 3. 4.

All distances are measured from the vertex of the curved mirror. Distances of real objects and images are positive. Distances of virtual images are negative. Object heights and image heights are positive when measured upward and negative when measured downward from the principal axis.

Physics 20 – Mirrors Ö

65

To take into account the sign convention, the magnification equation becomes:

M= Ö

Mr. MacDonald

hi − d i = ho do

A negative magnification means that the image is inverted from the original orientation.

Example 1 An object is located 30.0 cm from a converging mirror with a radius of curvature of 10.0 cm. (a) At what distance from the mirror will the image be formed?

(b) If the object is 4.0 cm tall, how tall is the image?

Example 2 A diverging mirror with a focal length of f = –5.0 cm produces an image of an object. If the object is located 15.0 cm from the mirror then, (a) What is the distance of the image from the mirror?

(b) What is the magnification?

Physics 20 − Refraction

66

Mr. MacDonald

Refraction of Light


67

Mr. MacDonald

Refraction •

Refraction is the change of direction of a ray of light as it travels into different media. (Different media means different densities).

www.physicsclassroom.com/Class/waves/u10l3b.html

o Waves change direction as they enter shallow water. o The same is true for light. Light changes direction as enters different media. o How the light will bend depends on how the two media compare in density & physical structure. o The ray diagrams below illustrate what happens to light for the two cases. Refraction from a less dense to denser medium:

Normal Incident ray

Reflected ray θi

Less Dense (air)

θr

Boundary

More Dense (water)

Refracted ray θR


68

Mr. MacDonald

Ö

The angle of refraction, θR, is less than the incident angle, θi.

Ö

The refracted ray bends towards the normal Æ light slows down.

Ö

This will always be the case if light is coming from air and into anther medium.

Refraction from a more dense to less dense medium:

Normal Incident ray

Reflected ray θi

More Dense (Oil)

θr

Boundary

Less Dense (air)

θR Refracted ray

Ö

The angle of refraction, θR, is more than the incident angle, θi.

Ö

The refracted ray bends away from the normal Æ light speeds up.

Ö

If the angle of incidence is zero, there is no change of direction, but there is a change of speed.

Ö

Principle of Reversibility: If a light ray is reversed, it travels back along its original path.


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Index of Refraction • • •

To understand the behavior of light in different properties, we refer to the index of refraction n. It is the ratio of the speed of light in a vacuum, c, to the speed of light in a given material, v. Mathematically,

n= •

c v

The higher the index of refraction, the more light is slowed down when it travels from a vacuum in to a substance.

Examples 1. The speed of light in a liquid is 2.25 × 108 m/s. What is the refractive index of the liquid?

2. Calculate the speed of light in Lucite (Plexiglas), if nlucite = 1.51

Indices of Refraction Substance Vacuum Air (0°C, 101 kPa) Water Ethyl alcohol Quartz (fused) Glycerin Lucite or Plexiglas Glass (crown) Sodium chloride Glass (crystal) Ruby Glass (flint) Zircon Diamond Note: For yellow light, wavelength = 589 nm

Index of Refraction (n) 1.0000 1.0003 1.33 1.36 1.46 1.47 1.51 1.52 1.53 1.54 1.54 1.65 1.92 2.42

Table from Fundamentals of Physics: An Introductory Course, Pg. 422


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Laws of Refraction •

Ö

Willebrod Snell (1591 – 1626) was able to determine the exact relationship between the angle of incidence and the angle of refraction. o This enables us to predict the direction a ray of light would take in various media. This is called Snell’s Law:

sin i = constant sin R Ö Ö

Where i = angle of incidence and R = angle of refraction. If light is traveling from a vacuum, the constant is the index of refraction of the material.

The Laws of Refraction are: 1. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant (Snell’s Law). 2. The incident ray and the refracted ray are on opposite sides of the normal at the point of incidence, and all three lie in the same plane. Snell’s Law – A General Equation

Normal

θ1 Medium 1

n1

Medium 2

n2

θ2


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Mathematically we write:

sin θ1 n2 = sin θ 2 n1 or n1 sin θ1 = n2 sin θ 2 Examples 1. Light travels from crown glass into air. The angle of refraction in air is 60°. What is the angle of incidence in glass?

2. Light travels from crown glass into water. The angle of incidence in crown glass is 40°. What is the angle of refraction in water?

Total Internal Reflection and the Critical Angle Ö

When light travels into a medium where its speed changes, some light is reflected, and some is refracted.

Ö

As the angle of incidence increases, the intensity of the reflected ray increases. The intensity of the refracted ray decreases.

Ö

Total internal reflection occurs when light travels into a faster medium and the angle of refraction is 90° or greater.

Ö

When the angle of refraction is 90°, the angle of incidence is called the critical angle, angle C.

Ö

Angles of incidence greater than the critical angle result in total internal reflection.


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Virginia University website: galileo.phys.virginia.edu/. ../introduction.htm

Example 1. The critical angle for light traveling from crown glass into air can be found with Snell’s Law.

2. What is the critical angle of light traveling from a diamond to a ruby?

Lateral Displacement Ö

When light travels from air into glass and then back into air, it is refracted twice. o If the two refracting surfaces are parallel, the emergent ray is parallel to the incident ray, but not following the same path. o This is called lateral displacement. o The thicker the material, the greater the lateral displacement.


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Air

Glass

Mr. MacDonald

Air

Lateral Displacement

Angle of Deviation Ö

If the surfaces of the refracting material are not parallel, like with a prism, an emergent ray will take a completely different path. o The angle between the incident ray and the emergent ray is called the angle of deviation.

Angle of Deviation


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Finding Lateral Displacement 1. Draw the incident ray straight through as if it didn’t refract. 2. Construct a normal at the incident side and measure angle i, 3. Use Snell’s Law to determine the angle of refraction. 4. Measure and construct the refracted ray. 5. Construct normal #2 where the refracted ray meets the 2nd side. 6. The original incident angle will be the same as the angle made by the emerging ray back into the air. 7. Construct the emerging angle, measured from normal #2. 8. Measure the perpendicular distance between the emerging ray and the line where the incident ray would have traveled had it not refracted (step 1). Finding Angle of Deviation 1. Draw the ray straight through as if it had not refracted. 2. Construct a normal at the incident side and measure angle i. 3. Use Snell’s Law to determine the angle of refraction. 4. Measure angle R and construct the refracted ray. 5. Construct normal #2 where the refracted ray hits the opposite side of the triangle. 6. Measure the angle between normal #2 and the refracted ray in the triangle Æ this becomes the incident angle for the ray as it goes from the triangle into air. 7. Use Snell’s Law to determine the angle of refraction when the ray passes into air. 8. Measure this angle and construct the emerging ray. 9. Project the emerging ray backwards until it meets the original ray had it not refracted (step 1). 10. Measure the angle of deviation.


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Applications of Refraction 1. Apparent bending of a straight stick in water.

•

Refraction makes it look as if the light originates closer to the surface than it actually is.

2. Atmospheric Refraction •

As the Sun sets, its light is refracted as it passes from the vacuum of space into the Earth’s atmosphere.

•

The density of the air increases as the light gets closer to the surface of the Earth, so refraction increases, resulting in a curved path.

•

An observer “sees” the Sun coming from a point higher in the sky. When we see a sunset, the sunlight is coming from below the horizon!

•

This also explains the distortion of the Sun during a sunset.

cougar.slvhs.slv.k12.ca.us/.../ ch14.html

www.weather-photography.com/Photos/gallery.ph.


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3. Puddle Mirage on the Road •

Sometimes while driving you may see what looks like a puddle of water up ahead. Upon arriving there, the puddle disintegrates and it is in fact dry pavement.

•

Light has a slightly different index of refraction in warm air than in cool air.

•

As light propagates from the cool upper atmosphere towards the warmer air near the ground, it refracts and its path may take it upwards again.

•

When we drive through the refracted light we see atmosphere on the road in front of us. What we see is not water but the blue of the atmosphere.

Blue Sky Cool Air

Blue Water

Hot Air

Hot Surface/Pavement 4. Red/Orange Lunar Eclipse •

During a lunar eclipse the Moon takes on a red and orange colour. This happens because of light being refracted by the Earth’s atmosphere. Similar to when the Sun sets or rises when viewed from Earth.

http://www.mreclipse.com/MrEclipse.html

Physics 20 − Lenses

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Lenses

Mr. MacDonald


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Lenses Ö

Lenses are divided into two types: o Converging Lenses − Focuses light rays to a point. o Diverging Lenses − Spreads light rays apart.

Ö

Refraction in Lenses o Whether a lens focuses or spreads light rays depends on the shape of the lens Æ In other words the angle of incidence.

Tom Henderson: www.glenbrook.k12.il.us/. ../refrn/u14l5b.html

Tom Henderson: www.glenbrook.k12.il.us/. ../refrn/u14l5b.html


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Ö

A lens is usually a circular piece of glass with uniformly curved surfaces that refract light passing through.

Ö

A series of rays will be refracted by different amounts depending on where they hit the surface.

Ö

An image is formed where the light rays meet.

Rules for rays in curved lenses of both types: 1. A ray that is parallel to the principle axis is refracted so that it passes through (or appears to pass through) the principle focus (F). 2. A ray that passes through (or appears to pass through) the secondary principle focus (F΄) is refracted parallel to the principle focus. 3. A ray that passes through the optical centre goes straight through, without bending.

1. Object beyond 2F΄.

F O

Ö

2F΄

Characteristics o

F΄

2F I


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2. Object at 2F΄

O

F F΄

2F΄

Ö

2F I

Characteristics o

3. Object between F΄ and 2F΄.

O

2F΄

Ö

Characteristics o

F F΄

2F I


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4. Object at F΄,

O 2F΄

Ö

F

2F

F΄

Characteristics o

5. Object between lens and F΄.

I O 2F΄

Ö

Image is: o

F΄

F

2F

Physics 20 − Lenses Ö

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Mr. MacDonald

In a diverging lens, parallel rays are refracted so that they radiate out from a virtual focus (like with diverging mirrors).

F O

2F

F

2F

I

Ö For all positions of the object, the image is virtual, erect, smaller, and is always located between the principle focus and the optical centre. The Thin Lens Equation Ö

We must assume that all of the lenses we discuss are thin lenses; otherwise we will have to take into account lateral displacement of the light rays through the lens.

Ö

The equation is the same as the mirror equation.

Using the geometric properties of similar triangles we can arrive at the following:

1 1 1 + = do di f Ö

This is called the lens equation.

Sign Convention 1. All distances are measured from the optical centre of the lens. 2. Distances of real objects and images are positive. 3. Distances of virtual images are negative. 4. Object heights and image heights are positive when measured upward and negative when measured downward from the principle axis.

Physics 20 − Lenses Ö

83

To take into account the sign convention, the magnification equation becomes:

M= Ö

Mr. MacDonald

hi d =− i ho do

A negative magnification means that the image is inverted from the original orientation.

Sample Problems 1. An object 8.0 cm high is 18 cm from a converging lens having a focal length of 10.0 cm. a. How far is the image from the optical centre of the lens?

b. How tall is the image.

2. A diverging lens has a focal length of −4.0 cm. If an object is placed 8.0 cm from the lens, how from the optical centre is the image?


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Lens Makers Formula ¾ Lenses always consist of two sides, but the lens does not have to be symmetric. One side could be diverging and the other converging etc. ¾ Knowing the curvature for the lenses allows for a formula relating the focal length and index of refraction:

no

n

R2

R1

⎞⎛ 1 1 ⎛ nlens 1 ⎞ ⎟⎟ = ⎜⎜ − 1⎟⎟⎜⎜ + f ⎝ n0 ⎠⎝ R1 R2 ⎠ ¾ no is the index of refraction of the surrounding medium. ¾ Depending on the shape of the lens, the radius is measured as either positive or negative. o A positive radius measurement is used if the side is convex. o A negative radius measurement is used if the side is concave. o When solving for focal length: If f > 0, then the lens is converging. If f < 0 then the lens is diverging. ¾ Note that the focal length of a lens is a combination of the curvature for the two surfaces, so the radius of curvature of each side is not necessarily two times the focal length.


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http://laxmi.nuc.ucla.edu:8248/M248_03/HHtun_01/sld033.htm

Example 1. What is the focal length of a Plexiglas plano-convex lens that has a radius of 15.7 cm?

2. What is the measure of each radius of a double concave ruby lens that has a focal length of –25 cm?

3. In air a flint glass converging meniscus lens has a focal length of 14 cm. What will be the focal length of the lens if it is submerged in water?

Physics 20 Lecture Notes

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