Calculating Astronomical Unit from Venus Transit

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Calculating Astronomical Unit from Venus Transit"

Transcription

1 Calculating Astronomical Unit from Venus Transit A) Background 1) Parallaxes of the Sun (the horizontal parallaxes) By definition the parallaxes of the Sun is the angle β shown below: By trigonometry, sin β = R / r but the angle β is small and sin β can be approximated by β measured in radians. R is the radius of Earth and r is the distance from the observer to the object. We can get r using the relationship r = R/ β 2) Observation from the Earth Let's consider two observers on the Earth situated in points A and B on the same longitude (meridian), but at very different latitudes. The alignment of AB should be approximately perpendicular to the line of sight to the transit so as to keep the errors as small as possible. Venus is seen as a small disk on the face of the Sun at two different points A' and B'. This is because the lines of sight of A and B towards Venus are not identical. 1

2 Putting the two observations together using the reference stars it is possible to measure this parallax displacement. 3) A Geometrical Problem Let's consider the plane defined by three points: the Earth's centre O, the Sun's centre C and the Venus centre V. The triangles APV and BPC have the same external angles at P, hence βv + β1 = βs + β2 βv - βs = β2 - β1 = Δβ Where angle Δβ measures the distance between the different positions of Venus's trace on the face of the Sun. Rearranging the last equation gives Δβ = βs (βv/ βs - 1) Now Venus's parallax is βv = AB/(r e - r v ) and the Sun's parallax βs = AB/r e, hence the quotient βv/ βs = r e /(r e - r v ). Substituting this into the equation above gives 2

3 Δβ = βs (r e /(r e - r v ) - 1) = βs r v /(r e - r v ) In particular, we can get the solar parallax, βs = Δβ (r e / r v - 1) Note that in order to measure Δβ it is necessary to superimpose the two Sun centres at C and then Δβ is the distance between the two traces of Venus observed at same time from A and B. 4) Kepler's Third Law Let's take r e as the Earth-Sun distance and r v the Venus-Sun distance. We can calculate the ratio (r v / r e ) 3 by using Kepler's Third Law as we know that the periods of revolution of Venus and of the Earth are days and days respectively. (r e / r v ) 3 = (365.25/224.7) 2 therefore r e / r v = ) Final formulae for the Earth-Sun distance. Using this result in the parallax formula from section 3, we get βs = Δβ((r e / r v ) - 1)= Δβ ( ) therefore βs = Δβ And finally using the parallax formula from section (1), the distance from the Earth to the Sun r e is r e = AB/βs 3

4 So we need to find AB from the location of the observers and to measure Δβ from observational data of the transit. B) Observational data needed 1) Distance between observers at points A and B The distance AB can be deduced from the latitude of the two points of observation. In the diagram, ϕ1 and ϕ2 are the latitudes of A and B, and R is the radius of the Earth. In the right angled triangle that divides the isosceles triangle RAB sin ((ϕ1 + ϕ2)/2) = AB/2)/R. Then the distance AB is AB = 2 R sin ((ϕ1 + ϕ2)/2) Be careful. If both cities are in the same hemisphere, the angle is (ϕ1 - ϕ2)/2 and also the geometrical situation changes if both cities are on different longitudes. 2) Distance Δβ between two observed paths of Venus In order to calculate Δβ we need the data obtained by two observers at the points A and B on the same longitude (meridian). In any case it is necessary to have a "photograph" of the paths of Venus visible from each location or the times that Venus crossed the Sun's disk. 4

5 Calculation of Δβ by direct measurement Measure the diameter of the Sun D and the distance between the two paths Δβ, that is to say A'B', on a photograph. The angular diameter of the Sun, seen from the Earth is 30' (minutes of arc or 30 / 60 ). By means of simple proportion, the distance between the observations of Venus is linked to the Sun's diameter by Δβ /30' = A'B'/D therefore Δβ = (30') (A'B'/D ) but the formula requires the Sun's angular diameter to be expressed in radians. Therefore Δβ = (30 π/10800) (A'B'/D) Δβ = (π/360) (A'B'/D) C) Observations that you can make in ) Single observations that are quick and easy to make Make plans to record an image of the transit when Venus is on the mid-line of the face of the Sun (point A'). You then need to share your results with another observer who is located on the same longitude (meridian) and who will observe at more or less the same time (point B'). You have to measure the distances DA and Δβ (from the centre of the Sun CA and CB) as shown in the diagram. Obviously the diagrams must be adjusted to be the same size to allow DA and Δβ to be compared. To obtain the highest accuracy you will need to take a photograph or make a still video image. 5

6 You can try to make a pencil sketch of a projected image but the problem will be that the image will drift across your screen as the Earth turns. This will make it difficult but not impossible to be precise. We suggest you try a combination of methods in case one of them lets you down! To calculate Δβ you need to measure the diameter of the image D, and D A and D B to the same scale. Then taking the angular diameter of the Sun, seen from the Earth as 30' (minutes of arc) then and the distance AB is Δβ = (π/360)((d B - D A )/D) =... radians βs = Δβ =... radians AB = sin ((ϕ1 - ϕ2) / 2) =... km where ϕ1 and ϕ2 are the latitudes of the observers and the radius of the Earth, R = 6378 km. If the observers are in opposite hemispheres, the angle is (ϕ1 + ϕ2) / 2. and finally the Earth-Sun distance r e = AB/βσ =... km If you want, you can find your own value for the angular diameter of the Sun but you will need to know the focal lengths of your telescope's eyepiece and objective lens and you will have to refer to an optics text book for the necessary formula. 2) Longer observations that are quite easy to make 6

7 If you are able to make observations throughout the transit, you will be able to plot the path of Venus and note the times of first and second contacts. If you can observe for at least half an hour you can reconstruct the whole transit as follows. Record the exact starting and finishing times of your observations. On a scale drawing, mark a line to represent the path of Venus during your observations. Now extend this straight line until it touches the limbs (edges) of the Sun's image. This is shown in the diagram. By simple ratios you can find the time for the total transit (for instance t A ). You then need to share your results with an observer who is located at a different latitude. You will then have their transit time t B and so will be able to calculate the Earth-Sun distance, also known as the astronomical unit (1 AU). The great advantage of this method is that you do not need to be on the same longitude (meridian) as the other observer. [Source: 7

The Classroom Astronomer Magazine s Transit of Venus Lab Exercise

The Classroom Astronomer Magazine s Transit of Venus Lab Exercise Transit of Venus starting photo taken in Marietta, GA June 5, 2012 Transit of Venus ending photo taken in Marietta, GA June 5, 2012 Transit of Venus starting photo taken in Cairns, Australia June 6, 2012

More information

4 The Rhumb Line and the Great Circle in Navigation

4 The Rhumb Line and the Great Circle in Navigation 4 The Rhumb Line and the Great Circle in Navigation 4.1 Details on Great Circles In fig. GN 4.1 two Great Circle/Rhumb Line cases are shown, one in each hemisphere. In each case the shorter distance between

More information

Coordinate Systems. Orbits and Rotation

Coordinate Systems. Orbits and Rotation Coordinate Systems Orbits and Rotation Earth orbit. The earth s orbit around the sun is nearly circular but not quite. It s actually an ellipse whose average distance from the sun is one AU (150 million

More information

Transit of Mercury. A Lab for College Intro Astronomy Students Or Secondary Physics Students. Educator Instructions

Transit of Mercury. A Lab for College Intro Astronomy Students Or Secondary Physics Students. Educator Instructions Transit of Mercury A Lab for College Intro Astronomy Students Or Secondary Physics Students Educator Instructions What follows is a lab experience developed for intro astronomy students by Katherine Benson

More information

Which month has larger and smaller day time?

Which month has larger and smaller day time? ACTIVITY-1 Which month has larger and smaller day time? Problem: Which month has larger and smaller day time? Aim: Finding out which month has larger and smaller duration of day in the Year 2006. Format

More information

SIERRA COLLEGE OBSERVATIONAL ASTRONOMY LABORATORY EXERCISE NUMBER III.F.a. TITLE: ASTEROID ASTROMETRY: BLINK IDENTIFICATION

SIERRA COLLEGE OBSERVATIONAL ASTRONOMY LABORATORY EXERCISE NUMBER III.F.a. TITLE: ASTEROID ASTROMETRY: BLINK IDENTIFICATION SIERRA COLLEGE OBSERVATIONAL ASTRONOMY LABORATORY EXERCISE NUMBER III.F.a. TITLE: ASTEROID ASTROMETRY: BLINK IDENTIFICATION DATE- PRINT NAME/S AND INITIAL BELOW: GROUP DAY- LOCATION OBJECTIVE: Use CCD

More information

O5: Lenses and the refractor telescope

O5: Lenses and the refractor telescope O5. 1 O5: Lenses and the refractor telescope Introduction In this experiment, you will study converging lenses and the lens equation. You will make several measurements of the focal length of lenses and

More information

Vik Dhillon, Sheffield University - UK

Vik Dhillon, Sheffield University - UK Spherical Trigonometry Vik Dhillon, Sheffield University - UK http://www.shef.ac.uk/uni/academic/n-q/phys/people/vdhillon/teaching/phy105/phy105_sphergeom.html INTRODUCTION Before we can understand the

More information

HELIOSTAT II - MEASURING THE SOLAR ROTATION

HELIOSTAT II - MEASURING THE SOLAR ROTATION HELIOSTAT II - MEASURING THE SOLAR ROTATION SYNOPSIS: In this lab you will map sunspots, and from the movement of the spots over several days, you will determine the rotation rate of the Sun. EQUIPMENT:

More information

y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)

y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement) 5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If

More information

Binary Stars. Kepler s Laws of Orbital Motion

Binary Stars. Kepler s Laws of Orbital Motion Binary Stars Kepler s Laws of Orbital Motion Kepler s Three Laws of orbital motion result from the solution to the equation of motion for bodies moving under the influence of a central 1/r 2 force gravity.

More information

Latitude and Longitude Vocabulary. Equator imaginary line separating the northern and southern hemispheres.

Latitude and Longitude Vocabulary. Equator imaginary line separating the northern and southern hemispheres. Latitude and Longitude Vocabulary Degrees the unit for measuring distance on a map Equator imaginary line separating the northern and southern hemispheres. Globe a miniature model of the Earth Hemisphere

More information

Celestial Coordinate Systems

Celestial Coordinate Systems Celestial Coordinate Systems Craig Lage Department of Physics, New York University, csl336@nyu.edu January 6, 2014 1 Introduction This document reviews briefly some of the key ideas that you will need

More information

ASTR 693A Coordinate systems

ASTR 693A Coordinate systems ASTR 693A Coordinate systems The following notes contain the essential information you ll need to understand where astronomical sources are in the sky. Further details can be found in Textbook on Spherical

More information

Paths Between Points on Earth: Great Circles, Geodesics, and Useful Projections

Paths Between Points on Earth: Great Circles, Geodesics, and Useful Projections Paths Between Points on Earth: Great Circles, Geodesics, and Useful Projections I. Historical and Common Navigation Methods There are an infinite number of paths between two points on the earth. For navigation

More information

Lesson 19: Latitude and Longitude

Lesson 19: Latitude and Longitude 277 Lesson 19: Latitude and Longitude Every point on a globe (with the exception of the north and south poles) has latitude and longitude coordinates. (The north and south poles have latitude coordinates,

More information

Solutions to Exercises, Section 5.1

Solutions to Exercises, Section 5.1 Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

Aphelion The point in the orbit of a planet or other celestial body where it is furthest from the Sun.

Aphelion The point in the orbit of a planet or other celestial body where it is furthest from the Sun. SKYTRACK Glossary of Terms Angular distance The angular separation between two objects in the sky as perceived by an observer, measured in angles. The angular separation between two celestial objects in

More information

A Fortran program to calculate sunrise and sunset

A Fortran program to calculate sunrise and sunset A Fortran program to calculate sunrise and sunset Nicholas Moe 23 April 2007 1 Introduction This is a description of a program written in Fortran 90/95 to calculate the rise and set ties of the sun, accurate

More information

Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

More information

A model of the Earth and Moon

A model of the Earth and Moon A model of the Earth and Moon Background Information This activity demonstrates the relative sizes of the Earth and Moon and the distance between them. The Moon is our nearest neighbour. It orbits the

More information

Ray Tracing: the Law of Reflection, and Snell s Law

Ray Tracing: the Law of Reflection, and Snell s Law Ray Tracing: the Law of Reflection, and Snell s Law Each of the experiments is designed to test or investigate the basic ideas of reflection and the ray-like behavior of light. The instructor will explain

More information

AST1100 Lecture Notes

AST1100 Lecture Notes AST1100 Lecture Notes 3 Extrasolar planets 1 Detecting extrasolar planets Most models of star formation tell us that the formation of planets is a common process. We expect most stars to have planets orbiting

More information

Determining Polar Axis Alignment Accuracy

Determining Polar Axis Alignment Accuracy Determining Polar Axis Alignment Accuracy by Frank Barrett 7/6/008 Abstract: In order to photograph dim celestial objects, long exposures on the order of minutes or hours are required. To perform this

More information

Basic numerical skills: TRIANGLES AND TRIGONOMETRIC FUNCTIONS

Basic numerical skills: TRIANGLES AND TRIGONOMETRIC FUNCTIONS Basic numerical skills: TRIANGLES AND TRIGONOMETRIC FUNCTIONS 1. Introduction and the three basic trigonometric functions (simple) Triangles are basic geometric shapes comprising three straight lines,

More information

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

Trigonometry LESSON ONE - Degrees and Radians Lesson Notes 210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:

More information

Exercise: Estimating the Mass of Jupiter Difficulty: Medium

Exercise: Estimating the Mass of Jupiter Difficulty: Medium Exercise: Estimating the Mass of Jupiter Difficulty: Medium OBJECTIVE The July / August observing notes for 010 state that Jupiter rises at dusk. The great planet is now starting its grand showing for

More information

PREVIOUS 8 YEARS QUESTIONS (1 mark & 2 marks) 1 mark questions

PREVIOUS 8 YEARS QUESTIONS (1 mark & 2 marks) 1 mark questions 230 PREVIOUS 8 YEARS QUESTIONS (1 mark & 2 marks) 1 mark questions 1. An object is held at the principal focus of a concave lens of focal length f. Where is the image formed? (AISSCE 2008) Ans: That is

More information

6.1 Basic Right Triangle Trigonometry

6.1 Basic Right Triangle Trigonometry 6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at

More information

This paper is also taken for the relevant Examination for the Associateship. For Second Year Physics Students Wednesday, 4th June 2008: 14:00 to 16:00

This paper is also taken for the relevant Examination for the Associateship. For Second Year Physics Students Wednesday, 4th June 2008: 14:00 to 16:00 Imperial College London BSc/MSci EXAMINATION June 2008 This paper is also taken for the relevant Examination for the Associateship SUN, STARS, PLANETS For Second Year Physics Students Wednesday, 4th June

More information

Predicting Sunrise and Sunset Times

Predicting Sunrise and Sunset Times Predicting Sunrise and Sunset Times Donald A. Teets donald.teets@sdsmt.edu, South Dakota School of Mines and Technology, Rapid City, SD 57701 For a given location and day of the year, can you predict the

More information

Lecture 5 Chapter 1. Topics for today. Phases of the Moon. Angular Measure and Parallax. Solar and Lunar Eclipses

Lecture 5 Chapter 1. Topics for today. Phases of the Moon. Angular Measure and Parallax. Solar and Lunar Eclipses Lecture 5 Chapter 1 Topics for today Phases of the Moon Angular Measure and Parallax Solar and Lunar Eclipses ! Day, year, seasons Our Calendar! How about months? Phases of the Moon Why do we see phases?!

More information

CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session. Geometry, 17 March 2012 CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

More information

Astron 100 Sample Exam 1 1. Solar eclipses occur only at (A) New moon (B) 1 st quarter moon (C) Full moon (D) 3 rd quarter moon (E) The equinoxes 2.

Astron 100 Sample Exam 1 1. Solar eclipses occur only at (A) New moon (B) 1 st quarter moon (C) Full moon (D) 3 rd quarter moon (E) The equinoxes 2. Astron 100 Sample Exam 1 1. Solar eclipses occur only at (A) New moon (B) 1 st quarter moon (C) Full moon (D) 3 rd quarter moon (E) The equinoxes 2. If the Moon is at first quarter tonight in Amherst,

More information

MIRRORS AND REFLECTION

MIRRORS AND REFLECTION MIRRORS AND REFLECTION PART 1 ANGLE OF INCIDENCE, ANGLE OF REFLECTION In this exploration we will compare θ i (angle of incidence) and θ r (angle of reflection). We will also investigate if rays are reversed

More information

CHAPTER 8 PLANETARY MOTIONS

CHAPTER 8 PLANETARY MOTIONS 1 CHAPTER 8 PLANETARY MOTIONS 8.1 Introduction The word planet means wanderer (πλάνητες αστέρες wandering stars); in contrast to the fixed stars, the planets wander around on the celestial sphere, sometimes

More information

Mathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings

Mathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings Mathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings Trigonometry in Right Triangles A In right ABC, AC is called the hypotenuse. The vertices are labelled using capital letters.

More information

Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)

Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes) Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.

More information

Trigonometry I. MathsStart. Topic 5

Trigonometry I. MathsStart. Topic 5 MathsStart (NOTE Feb 0: This is the old version of MathsStart. New books will be created during 0 and 04) Topic 5 Trigonometry I h 0 45 50 m x MATHS LEARNING CENTRE Level, Hub Central, North Terrace Campus

More information

10-4 Inscribed Angles. Find each measure. 1.

10-4 Inscribed Angles. Find each measure. 1. Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

More information

Solution Derivations for Capa #13

Solution Derivations for Capa #13 Solution Derivations for Capa #13 1) A super nova releases 1.3 10 45 J of energy. It is 1540 ly from earth. If you were facing the star in question, and your face was a circle 7 cm in radius, how much

More information

Lecture PowerPoints. Chapter 23 Physics: Principles with Applications, 7th edition Giancoli

Lecture PowerPoints. Chapter 23 Physics: Principles with Applications, 7th edition Giancoli Lecture PowerPoints Chapter 23 Physics: Principles with Applications, 7th edition Giancoli This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching

More information

Solution Guide for Chapter 6: The Geometry of Right Triangles

Solution Guide for Chapter 6: The Geometry of Right Triangles Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E-. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab

More information

PHYS 3324 Experiment 2: Atomic Spectra

PHYS 3324 Experiment 2: Atomic Spectra PHYS 3324 Experiment 2: Atomic Spectra Background Reading: Krane, pp. 185-189 Apparatus: Spectrometer, sodium lamp, hydrogen lamp, mercury lamp, diffraction grating, watchmaker eyeglass, small flashlight.

More information

Reflection and Refraction

Reflection and Refraction Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,

More information

APPENDIX D: SOLAR RADIATION

APPENDIX D: SOLAR RADIATION APPENDIX D: SOLAR RADIATION The sun is the source of most energy on the earth and is a primary factor in determining the thermal environment of a locality. It is important for engineers to have a working

More information

REFLECTION & REFRACTION

REFLECTION & REFRACTION REFLECTION & REFRACTION OBJECTIVE: To study and verify the laws of reflection and refraction using a plane mirror and a glass block. To see the virtual images that can be formed by the reflection and refraction

More information

Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)

Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.) 1- The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview

More information

CHAPTER 38 INTRODUCTION TO TRIGONOMETRY

CHAPTER 38 INTRODUCTION TO TRIGONOMETRY CHAPTER 38 INTRODUCTION TO TRIGONOMETRY EXERCISE 58 Page 47. Find the length of side x. By Pythagoras s theorem, 4 = x + 40 from which, x = 4 40 and x = 4 40 = 9 cm. Find the length of side x. By Pythagoras

More information

Earth-Sun Relationships. The Reasons for the Seasons

Earth-Sun Relationships. The Reasons for the Seasons Earth-Sun Relationships The Reasons for the Seasons Solar Radiation The earth intercepts less than one two-billionth of the energy given off by the sun. However, the radiation is sufficient to provide

More information

INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE

INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE ABSTRACT:- Vignesh Palani University of Minnesota - Twin cities e-mail address - palan019@umn.edu In this brief work, the existing formulae

More information

EDMONDS COMMUNITY COLLEGE ASTRONOMY 100 Winter Quarter 2007 Sample Test # 1

EDMONDS COMMUNITY COLLEGE ASTRONOMY 100 Winter Quarter 2007 Sample Test # 1 Instructor: L. M. Khandro EDMONDS COMMUNITY COLLEGE ASTRONOMY 100 Winter Quarter 2007 Sample Test # 1 1. An arc second is a measure of a. time interval between oscillations of a standard clock b. time

More information

CHAPTER 28 THE CIRCLE AND ITS PROPERTIES

CHAPTER 28 THE CIRCLE AND ITS PROPERTIES CHAPTER 8 THE CIRCLE AND ITS PROPERTIES EXERCISE 118 Page 77 1. Calculate the length of the circumference of a circle of radius 7. cm. Circumference, c = r = (7.) = 45.4 cm. If the diameter of a circle

More information

Mathematical Procedures

Mathematical Procedures CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

More information

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

More information

8-3 Perimeter and Circumference

8-3 Perimeter and Circumference Learn to find the perimeter of a polygon and the circumference of a circle. 8-3 Perimeter Insert Lesson and Title Circumference Here perimeter circumference Vocabulary The distance around a geometric figure

More information

Catholic Schools Trial Examination 2004 Mathematics

Catholic Schools Trial Examination 2004 Mathematics 0 Catholic Trial HSC Examination Mathematics Page Catholic Schools Trial Examination 0 Mathematics a If x 5 = 5000, find x correct to significant figures. b Express 0. + 0.. in the form b a, where a and

More information

Set 1: Circumference, Angles, Arcs, Chords, and Inscribed Angles

Set 1: Circumference, Angles, Arcs, Chords, and Inscribed Angles Goal: To provide opportunities for students to develop concepts and skills related to circumference, arc length, central angles, chords, and inscribed angles Common Core Standards Congruence Experiment

More information

UTM and UPS James R. Clynch 2003

UTM and UPS James R. Clynch 2003 UTM and UPS James R. Clynch 2003 I. Introduction The Universal Transverse Mercator (UTM) projection coordinates occur on most topographic maps. This is the Northing and Easting coordinates discussed below.

More information

Lab Module 1: The Observing Project

Lab Module 1: The Observing Project Lab Module 1 The Observing Project The Location and Time of Sunset OR Fremont Peak Observatory (Worth 15 points) Due Date: The last lab class before finals I. The Location and Time of Sunset Background

More information

Geometric Optics Physics 118/198/212. Geometric Optics

Geometric Optics Physics 118/198/212. Geometric Optics Background Geometric Optics This experiment deals with image formation with lenses. We will use what are referred to as thin lenses. Thin lenses are ordinary lenses like eyeglasses and magnifiers, but

More information

INTRODUCTION TO THE TELESCOPE

INTRODUCTION TO THE TELESCOPE AST 113/114 Fall 2014 / Spring 2016 NAME: INTRODUCTION TO THE TELESCOPE What will you learn in this Lab? For a few of the labs this semester, you will be using an 8-inch Celestron telescope to take observations.

More information

Chapter 6 Trigonometric Functions of Angles

Chapter 6 Trigonometric Functions of Angles 6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.

More information

ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE

ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE Keywords: Central angle An angle whose vertex is the center of a circle and whose sides contain the radii of the circle Arc Two points on a circle

More information

1.2 Chord Tables of Hipparchus and Ptolemy (Copyright: Bryan Dorner all rights reserved)

1.2 Chord Tables of Hipparchus and Ptolemy (Copyright: Bryan Dorner all rights reserved) 1.2 Chord Tables of Hipparchus and Ptolemy (Copyright: Bryan Dorner all rights reserved) Hipparchus: The birth of trigonometry occurred in the chord tables of Hipparchus (c 190-120 BCE) who was born shortly

More information

Today. Appearance of the Sky. Orientation. Motion of sky. Seasons. Precession. Phases of the Moon

Today. Appearance of the Sky. Orientation. Motion of sky. Seasons. Precession. Phases of the Moon Today Appearance of the Sky Orientation Motion of sky Seasons Precession Phases of the Moon The Appearance of the Sky The Local Sky An object s altitude (above horizon) and direction (along horizon) specify

More information

Refractive Index and Dispersion: Prism Spectrometer

Refractive Index and Dispersion: Prism Spectrometer Refractive Index and Dispersion: Prism Spectrometer OBJECTIVES: The purpose of this experiment is to study the phenomenon of dispersion i.e. to determine the variation of refractive index of the glass

More information

Lesson 5 Rotational and Projectile Motion

Lesson 5 Rotational and Projectile Motion Lesson 5 Rotational and Projectile Motion Introduction: Connecting Your Learning The previous lesson discussed momentum and energy. This lesson explores rotational and circular motion as well as the particular

More information

Stereographic projections

Stereographic projections Stereographic projections 1. Introduction The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. The projection is defined as shown in Fig. 1. If

More information

THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE

THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE / 24 THE GOLDEN RATIO AND THE FIBONACCI SEQUENCE Todd Cochrane Everything is Golden 2 / 24 Golden Ratio Golden Proportion Golden Relation Golden Rectangle Golden Spiral Golden Angle Geometric Growth, (Exponential

More information

not to be republished NCERT Latitude, Longitude and Time Chapter 3

not to be republished NCERT Latitude, Longitude and Time Chapter 3 26 Chapter 3 Latitude, Longitude and Time THE EARTH is nearly a sphere. It is because of the fact that the equatorial radius and the polar radius of the earth is not the same. The rotation of the earth

More information

GAP CLOSING. 2D Measurement. Intermediate / Senior Student Book

GAP CLOSING. 2D Measurement. Intermediate / Senior Student Book GAP CLOSING 2D Measurement Intermediate / Senior Student Book 2-D Measurement Diagnostic...3 Areas of Parallelograms, Triangles, and Trapezoids...6 Areas of Composite Shapes...14 Circumferences and Areas

More information

Lecture PowerPoints. Chapter 23 Physics: Principles with Applications, 7 th edition Giancoli

Lecture PowerPoints. Chapter 23 Physics: Principles with Applications, 7 th edition Giancoli Lecture PowerPoints Chapter 23 Physics: Principles with Applications, 7 th edition Giancoli This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching

More information

Spectroscopy, the Doppler Shift and Masses of Binary Stars.

Spectroscopy, the Doppler Shift and Masses of Binary Stars. Spectroscopy, the Doppler Shift and Masses of Binary Stars http://apod.nasa.gov/apod/astropix.html Doppler Shift At each point the emitter is at the center of a circular wavefront extending out from its

More information

DETERMINING SOLAR ALTITUDE USING THE GNOMON. How does the altitude change during the day or from day to day?

DETERMINING SOLAR ALTITUDE USING THE GNOMON. How does the altitude change during the day or from day to day? Name Partner(s) Section Date DETERMINING SOLAR ALTITUDE USING THE GNOMON Does the Sun ever occur directly overhead in Maryland? If it does, how would you determine or know it was directly overhead? How

More information

3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?

3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled? 1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, co-ordinate geometry (which connects algebra

More information

Solar Angles and Latitude

Solar Angles and Latitude Solar Angles and Latitude Objectives The student will understand that the sun is not directly overhead at noon in most latitudes. The student will research and discover the latitude ir classroom and calculate

More information

The Disk Rotation of the Milky Way Galaxy. Kinematics of Galactic Rotation

The Disk Rotation of the Milky Way Galaxy. Kinematics of Galactic Rotation THE DISK ROTATION OF THE MILKY WAY GALAXY 103 The Disk Rotation of the Milky Way Galaxy Vincent Kong George Rainey Physics Physics The rotation of the disk of the Milky Way Galaxy is analyzed. It rotates

More information

Universal Law of Gravitation

Universal Law of Gravitation Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies

More information

Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB.

Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. A B C **Using a ruler measure the two lengths to make sure they have

More information

MPE Review Section II: Trigonometry

MPE Review Section II: Trigonometry MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the

More information

Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013 Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

More information

Applications of Trigonometry

Applications of Trigonometry chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

More information

Objectives After completing this section, you should be able to:

Objectives After completing this section, you should be able to: Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding

More information

INDEX. Arc Addition Postulate,

INDEX. Arc Addition Postulate, # 30-60 right triangle, 441-442, 684 A Absolute value, 59 Acute angle, 77, 669 Acute triangle, 178 Addition Property of Equality, 86 Addition Property of Inequality, 258 Adjacent angle, 109, 669 Adjacent

More information

9/16 Optics 1 /11 GEOMETRIC OPTICS

9/16 Optics 1 /11 GEOMETRIC OPTICS 9/6 Optics / GEOMETRIC OPTICS PURPOSE: To review the basics of geometric optics and to observe the function of some simple and compound optical devices. APPARATUS: Optical bench, lenses, mirror, target

More information

Newton s Law of Gravity

Newton s Law of Gravity Gravitational Potential Energy On Earth, depends on: object s mass (m) strength of gravity (g) distance object could potentially fall Gravitational Potential Energy In space, an object or gas cloud has

More information

Axis of a coordination grid either of the two numgber lines used to form a coordinate grid. Plural is axes.

Axis of a coordination grid either of the two numgber lines used to form a coordinate grid. Plural is axes. Axis of a coordination grid either of the two numgber lines used to form a coordinate grid. Plural is axes. Base Line a line or number used as a base for measurement or comparison An average man, 6 feet

More information

AS101: Planetary Motions Page 1 PLANETARY MOTIONS

AS101: Planetary Motions Page 1 PLANETARY MOTIONS AS101: Planetary Motions Page 1 PLANETARY MOTIONS Goals: To develop a 3-D sense of the locations and motions of the planets in the solar system To recognize how solar illumination of planetary bodies and

More information

Measuring the Earth s Diameter from a Sunset Photo

Measuring the Earth s Diameter from a Sunset Photo Measuring the Earth s Diameter from a Sunset Photo Robert J. Vanderbei Operations Research and Financial Engineering, Princeton University rvdb@princeton.edu ABSTRACT The Earth is not flat. We all know

More information

Measuring Your Latitude from the Angle of the Sun at Noon

Measuring Your Latitude from the Angle of the Sun at Noon Measuring Your Latitude from the Angle of the Sun at Noon Background: You can measure your latitude in earth's northern hemisphere by finding out the altitude of the celestial equator from the southern

More information

UK Junior Mathematical Olympiad 2016

UK Junior Mathematical Olympiad 2016 UK Junior Mathematical Olympiad 206 Organised by The United Kingdom Mathematics Trust Tuesday 4th June 206 RULES AND GUIDELINES : READ THESE INSTRUCTIONS CAREFULLY BEFORE STARTING. Time allowed: 2 hours.

More information

Cambridge International Examinations Cambridge International General Certificate of Secondary Education

Cambridge International Examinations Cambridge International General Certificate of Secondary Education Cambridge International Examinations Cambridge International General Certificate of Secondary Education MATHEMATICS 0580/02 Paper 2 (Extended) For Examination from 2015 SPECIMEN PAPER Candidates answer

More information

Linear Motion vs. Rotational Motion

Linear Motion vs. Rotational Motion Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a

More information

Chapter 3 The Earth's dipole field

Chapter 3 The Earth's dipole field Chapter 3 The Earth's dipole field 1 Previously The Earth is associated with the geomagnetic field that has an S-pole of a magnet near the geographic north pole and an N-pole of a magnet near the geographic

More information

Eratosthenes: Estimating the Circumference of the Earth. Subject: Mathematics Topic: Geometry Grade Level: 8-12 Time: min Pre Show Math Activity

Eratosthenes: Estimating the Circumference of the Earth. Subject: Mathematics Topic: Geometry Grade Level: 8-12 Time: min Pre Show Math Activity Eratosthenes: Estimating the Circumference of the Earth. Subject: Mathematics Topic: Geometry Grade Level: 8-12 Time: 40-60 min Pre Show Math Activity Introduction: In Show Math, students will learn how

More information

Applications of Integration Day 1

Applications of Integration Day 1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

More information

CONTROL POINT SURVEYING AND TOPOGRAPHIC MAPPING

CONTROL POINT SURVEYING AND TOPOGRAPHIC MAPPING CONTROL POINT SURVEYING AND TOPOGRAPHIC MAPPING Shoichi Oki Land Bureau, National Land Agency, Japan Keywords: control point, survey, topography, mapping, geodesy Contents 1. Introduction 2. Geometric

More information

Waves and Modern Physics PHY Spring 2012

Waves and Modern Physics PHY Spring 2012 Waves and Modern Physics PHY 123 - Spring 2012 1st Midterm Exam Wednesday, February 22 Chapter 32 Light: Reflec2on and Refrac2on Units of Chapter 32 Today we will cover: The Ray Model of Light Reflection;

More information