Copy of Estimating Angles

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Feb 20, 2024

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Name: Natalie Zordani Date: February 5 2024 Class: ASTR100L Estimating Angles Instructions: Work with your group to answer the following questions. Estimate the angular size subtended by your pinky finger. Work with your group to estimate the angular sizes and distances of classroom objects. Part 1: Background Information 1. How many centimeters make up a meter? 100 How many millimeters in one centimeter? 10 How many millimeters in a meter? 1000 Figure 1: Portion of a meter stick 2. Fill in the table below. Radians Degrees Arc Minutes Arc Seconds 1 Circle is… 2pi 360 21600 1296000 1 Radian is… 1 57.3 3437.8 206265 Table 1: Angle Conversions 3. Label the following diagram. Figure 2: Angular Size 1

Name: Natalie Zordani Date: February 5 2024 Class: ASTR100L 4. What is the relationship between an object’s size, distance, and the angle it subtends in your field of vision? the relationship between an object's size, distance, and the angle it subtends in your field of vision is the formula Θ =D/d, Θ being the angle it subtends in your field of vision, D being the objects size, and d being the distance. VALUE +- ERROR AVERAGE VALUE +- ERROR ERROR-> %ERROR AND/OR %DIFFERENCE Part 2: Angular Size of Your Pinky Finger 5. Get a ruler. Measure the width of your pinky finger in millimeters. This would be the distance across the top of your pinky finger (not the length of your pinky fingernail). Width of Pinky: _________13______ mm 6. Using a meter stick, measure the distance between your pinky finger and your eyes. You need to measure this distance with your arm fully outstretched. Hold your arm straight and have your lab partner measure this distance for you. Distance between pinky and eye: ___________660.4__________ mm 7. What is the ratio of the width of your pinky to the distance between your eye and pinky? Ratio = width of pinky/distance between pinky and eye: _________.02___________ 8. What angle does your pinky subtend when your hand is outstretched? Give your answer in radians, degrees, and arc seconds. Angular size of pinky: ___________.02__________ rad Angular size of pinky: ___________1.146__________ degree Angular size of pinky: ____________4125.3___________ arc seconds 9. Compare answers with all of your group members. Record the largest and smallest angular pinky sizes for your group. Use degrees as your unit. Find your group’s average pinky angle and find the percent difference for your group. (Show your work when computing the percent difference) 2

Name: Natalie Zordani Date: February 5 2024 Class: ASTR100L Group Member Name Angular Pinky Size natalie 1.146 diana 1.2 david 1.587 Table 2: Group Comparison Largest pinky size: _____________1.587__________ degree Smallest pinky size: ___________1.146____________ degree Average of all group members: ___________1.311____________ degree 𝑃?????? 𝐷𝑖???????? = 𝐿𝑎????? ?𝑖𝑧? − 𝑆?𝑎????? ?𝑖𝑧? | | 𝐴𝑣??𝑎?? ?𝑖𝑧? × 100 Percent Difference: ____________33.64___________ % 10. Why did we use percent difference instead of percent error when comparing the angular sizes of all group members? you cannot have correct accuracy while measuring from the eye, so using precision would be a better measure of finding similar answers. 3

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Name: Natalie Zordani Date: February 5 2024 Class: ASTR100L 11. With your pinky outstretched, spin your body to trace a full circle around you. 12. What is the radius of the circle you have traced in millimeters? (Hint: use your answer from question 6) Radius: __________660.4___________ mm Figure 3: Point your pinky forward and spin! 13. What is the diameter of this circle? (use millimeters) Diameter: ___________1320.8____________ mm 14. What is this circle’s circumference? (use millimeters) Circumference = 2 x 𝝅 x radius = __________4149.42_____________ mm 15. Imagine cutting this circle into 360 equal parts. How big (in millimeters) would each of these parts be? Width 1/360th of the circumference of your circle: _________11.53_________ mm 16. How many of your pinky widths would fit within the circle you have traced out? Find this by dividing your answer above by the angular size of your pinky in degrees. Number of pinky fingers: _______________________ 17. How close is your outstretched pinky to one degree? 1.47 18. Look at Table 2 and your answer to question 17. Is the general approximation that an outstretched human pinky subtends an angle of 1 degree valid? Explain your answer. Look at Figure 4. These are other useful angle approximations commonly used by sky observers. I think that it is valid because usually I think on a larger scale the average of all the pinky degrees would probably be close to one degree, therefore using an average of one 4

Name: Natalie Zordani Date: February 5 2024 Class: ASTR100L degree to measure would be okay, although the accuracy of using this method would vary individually. Figure 4: Angles subtended by a hand. Part 3: Predicting the Distances to and Sizes of Objects Using Their Angular Size 19. Get an index card and ruler from the front of the room. Measure the long side of the index card in millimeters. Length of card: ___________126____________ mm 20.How many of your pinky finger widths would fit across this card? Divide the length of your card by the width of your pinky in mm (see question 5). Number of pinky finger widths across card: __________9.7____________ 21. Make an angular ruler using the information from questions 17 and 20. Mark off how many of your pinky fingers would fit along the length of the card. Label each division with the angular size of your pinky times how many divisions from the bottom of your card. 22. Look around the lab room. Find an object that you believe you can estimate the size of (but do not measure this object’s actual size). 23. Write down your size estimates in the table below. Using your angular ruler at arm's length, measure the angular size of each object. Record it below. 24. Using the equation from question 4, predict the distance to each object. Record these predictions below. Name of (or description of) object chosen Estimate of size (height or length) in centimeters. Call this size, ? The angle subtended by object (from your location, use your angular ruler θ (???????) Predicted distance to object (in centimeters) 𝐷 = ? θ ??????? 57.3 (???????) Object paper towel dispenser 25 cm 2.865 500 Table 3: Distance Predictions 5

Name: Natalie Zordani Date: February 5 2024 Class: ASTR100L 25. Now, measure the size and distance of the object in Table 3. How close were your predictions to the real size distance? Actual Size (cm) Percent Error Actual Distance (cm) Percent Error Object 25 cm 0% 673 34.6% Table 4: Estimating Errors 26. What do you think contributed to your errors in your predicted distances? Explain. for the predicted distance I think that the contributing factors that led to error was the error of not being able to accurately measure size or be able to accurately measure distance based off of my naked eye. there is also probably an error in my ruler, just by human mistake. 27. Due to a happy coincidence, the Sun and Moon have the same angular size in our sky (about 0.5 degrees). This fact produces the most spectacular total solar eclipses in the solar system. Look up the size (diameter) of the moon, the average distance to the moon, and the diameter of the Sun (all in km). Use this information to confirm that, indeed the Moon does subtend an angle of 0.5 degrees on the sky and then predict the distance to the Sun. Distance to the Moon from Earth: ____238,900___________ km Diameter of the Moon: __________2159.1___________ km Diameter of the Sun: ______865370___________ km 28. Calculate the angular size of the Moon in degrees. (Show work) Angular diameter= 206265*d/D 206265*(2159.1/238900) 1864.15 afc seconds -> degrees = .52 Figure 5: Solar Eclipse 6

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Name: Natalie Zordani Date: February 5 2024 Class: ASTR100L The angular diameter of the Moon: _________.52______ degrees 29. Calculated distance to the Sun assuming that the angular size of the Sun is exactly the same as the Moon. (show work) D=d(57.3/ Θ ) D=865370(57.3/.52) D=95357117.31 Distance to the Sun: _______95357117.31_________ km (Show your work) 30. Look up the actual distance to the Sun (in km) and report it here. 150 million km 31. How close was your calculated distance to the Sun’s real distance? Use percent error to estimate the difference. Why do you think we should use percent error and not percent difference? my calculated distance of the Sun was about 36.43% off, the reason we should use percent error is because we are measuring the accuracy of the distance compared i found to what the the actual distance is, and precision wouldn’t really make sense in this context. 7

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