1Physics E-1a Expt 1: Measuring from a Distance Fall 2006

Due Tuesday, Oct 10, by 6 pm in your lab TFs mailbox

Introduction

There are many objects in the universe that simply aren't easy to measure. You can not goout with a ruler and easily determine the radius of the Earth (although, it can be done!).Similarly, the size of the Moon is beyond the reach of your meter stick, and even theheights of the trees in the Yard are hard to sample with even the longest of tape measures.

Since direct measurement isn't always possible, people have found indirect techniques forgetting at various measurements. In this experiment, we will explore how an object'sangular size can be used to determine its physical size, or its distance.

ObjectivesThe measurements for this experiment will not be performed in the lab. You will conductthem on your own (or with a partner), on your own time. We call this a home experi-ment and hope to accomplish the following: If you have been away from math for a while, this experiment will help you review

the geometry and trigonometry. It will start you off on the right track concerning error analysis learning to identify

and deal with uncertainties in your measurements. You'll also get some practice in plotting data and extracting information from a graph. And finally, you will discover that its possible to measure seemingly difficult things

with simple tools (this is the fun part). Its actually pretty amazing what one can learnabout our physical world using only the simplest of apparatus, good observations, andmathematics.

MethodThe tool you will make is a quadrant. The quadrant uses the basic principles of surveying:that light travels in straight lines, and that some lengths and angles of a triangle (or othershape) can be measured so that unknown lengths, often not directly measurable, can becalculated. The simplest cases are (1) those in which one angle of a triangle is verysmall, or (2) one angle is 90. By small we mean that the angle (measured inradians) differs from its sine or tangent by so little that, for all practical purposes, thesethree quantities can be considered equal this small angle approximation will be used inthe moon measurement (and many times in this course throughout the year). Secondly, ifone of the angles is 90, we simply apply the rules governing right triangles.

Order of Tasks:

2First make your quadrant (page 10) and learn how to use it by measuring a window. Nextmeasure the height of the Science Center with this new tool. Then, using only your arm, apiece of paper, and a ruler, measure the size of the moon. Please dont leave the moonmeasurement until the last minute as the weather and/or phases of the moon may notcooperate with you. Finally, answer all the bold numbered questions on a separate paperwith a few concise sentences. You will hand in these answers along with a copy of yourgraph from the procedure section (there is a check-list for this report on page 8).

QuadrantSurveying began with instruments like the one youll make. The basic quadrant consistsof some kind of angle measuring device (like a protractor) that you can sight along anddetermine the angle between a reference point (or direction) and the object youresighting. It has been improved by adding telescopes with cross-hairs, graduated circleswith magnifiers and verniers, magnetic compasses, levels and plumb-bobs, and recentlylasers and corner reflectors. Ours shall remain simple; instructions on how to make it areon the last page of this write-up.To use the quadrant, sight along the edge of thecardboard the angular height of the object beingsighted is read off where the plumb bob stringintersects the protractor scale. A sample quadrantwill be shown in lecture. [Tip: Instead of puttingyour eye up close to the cardboard edge, try movingyour head back a bit so that the entire edge will bein focus as you try to sight along it. If you findsighting to be tricky, tape a drinking straw along theedge and sight through it.]

Preliminary Observations andAnalysis

As with any new tool, you should try it out first and measure the size of things that youalready know (or can easily verify) to make sure it works. This means learning how touse your new tool well; learning the techniques and tricks, and coming to understand itslimitations. Test your quadrant by trying to determine the height of a window from afar.Measure the distance between you and the window; call it D. To determine the angularheight of the window, start by sighting the bottom of the window with the quadrant level(youll probably have to adjust your viewing height by crouching down). Then, withoutchanging your position, sight the top of the window and record the angle; call it .Practical Hint: When your line-of-sight is on the point of interest, pinch the string againstthe quadrant (to hold it in place) and then look to see what angle it is lined up with. Thinkabout what you can do to improve your technique to minimize the uncertainty in yourangle measurements. If you're doing this with a partner, take turns practicing makingmeasurements and critique each other's style so that you can better understand how toavoid measurement errors. Repeat your angle measurements to see how reproducible theyare. Having measured , the angle subtended by the window, you can calculate the heightof the window using H = D tan. Now measure the height of the window with a tapemeasure and compare this measurement with the calculated value of H.

31 What is the percentage error in your calculated value of H?2 Is this error in H consistent with the uncertainty you found in your anglemeasurements? If not, why not?

The Height of the Science CenterNow for something a little more challenging... measuring the height of the ScienceCenter. First make a few order of magnitude estimates of its height. Though you may bemore comfortable thinking in feet, most scientific measurement is done with the metricsystem, so try to make your estimates in meters as well. Remember, 1 m 3 feet.Here are some possibile ways of estimating the height of the Science center you neednot do them all and you may have better ways of estimating: Guessing gives the height of the Science Center as more than _____ft (or m) and less

than ____ft (or m). It is about ____ft (or m) as judged by the number of stories. It is about ____ft (or m) from a knowledge of the height of a stair and the number of

stairs I need to climb to the top.

3 What is your estimate of the height (in meters) of the Science Center and howdid you arrive at this number?

4GeometryThe measurement scheme is similar to your preliminary observations, except that in thiscase you don't make a direct measurement of the distance from the origin (the base ofScience Center) to your observation points. To measure the length of the baseline, useyour normal stride as the unit of length. [You can figure out now, or later, what theconversion is between your stride length and the number of meters by measuring off, sayten paces.] As you can see from the geometry in the figures, it would be easy todetermine D if we knew L. But we dont know L, and theres no easy way to measure it,because you dont know where the origin is (the point directly below the dome).

Now, if you measure several angles at various distances, you could turn this into ageometry problem and deduce D by using appropriate triangles and trigonometricidentities. However, a better solution to this little conundrum is to use all the angle anddistance measurements and plot them on a graph. The reason this is better is because,by incorporating many measurements in your analysis, you will be minimizing theuncertainty in the final result. By drawing a suitable graph, you can make use of all yourdata to obtain a single best value for the height of the Science Center.

The parameters needed for the analysis are shown in the following diagram.

5x

D

L

starting point origin 123

3

4

According to the diagram above, if your point of measurement is a distance (L+x) fromthe origin, then

tan = DL+x and thus cot = L+xD .

Here, x is the distance from the starting point and is the angular height of the ScienceCenter dome at that value of x.1 L is the unknown distance between the starting point andthe origin. This expression can be rewritten in the form of a straight line, (y = ax + b):

cot = 1D x + LD .

When we plot the cotangent of vs the distance from the starting point, we obtain astraight line whose slope is

1D and whose y-intercept is

LD . The height D can be

determined from the graph of such a line. Note that the quantities x/D and L/D are bothpure numbers, i.e. they have no units. If x, D, and L are all given in meters, then metersover meters cancel. This means, of course, that cotangent (like all trigonometricfunctions) is a pure number. However, the slope, 1/D, does have units. If x is in meters,D and L should also be in meters, and the slope has units of reciprocal meters (1/m=m-1).Further note that it is not necessary to know L, although it can also be determined fromyour data. 4 How can you determine L from the graph?

1 A reminder: the cotangent is 1/tan, not tan-1. Don't make that mistake on your calculator!

6ProcedureWalk far enough outside the Science Center so that you can see the telescope dome onthe roof. Let this spot be your starting point (see figure), a distance L away from theorigin (the foot of the perpendicular from the telescope dome to your baseline). Measure(and record) the angle between the sight-line to the top of the telescope dome and thevertical. Repeat the angle measurement two or three times and find the average anglefrom this point of observation. Now move to another observation point farther awayfrom the building (say 10 or 15 paces - record how many) and make a new set of anglemeasurements. You need to move far enough to produce a significant change in theangle. Repeat this procedure until you have measured angles from at least 4 differentpoints.Graph your data in the following way: Take, as horizontal axis (abscissa), the distance xfrom your starting point to each point of observation along the base-line. You can use, asyour unit of x, one stride; you do not need to convert distances into meters until later.Take, as a vertical axis (ordinate), the cotangent (1/tan) of your measured angles.Choose scales such that your graph occupies most of the width and height of a page ofgraph paper. (But use sensible scales based on multiples of 5 or 10. Avoid, for example,using 3 or 4 squares to represent a distance of 10 units.) For each point of observation,plot cot against x. Draw a small circle around each point.5 What are the sources of uncertainty and explain by howmuch they are likely to affect your measurement? To show theeffect of uncertainties in the angle measurements, plot, for eachobservation point, not only the cotangent of the average angle butalso the max and min values cotangent of from your data. Do notcircle these additional points, but connect them with a short verticalline this is your so-called error bar. There will also be anuncertainty in converting from number of paces to meters. Addhorizontal error bars to each data point to indicate that uncertainty.Each point will then look like a little cross as shown in the figure tothe right.Fit, by eye, the best straight line through the points. The magnitudeof the slope of the graph is the reciprocal of the height of the Science Center, measured inwhatever units you used for x.6 What do you determine the height of the Science Center to be?7 From the slopes of the steepest and least-steep lines that you can draw throughyour data points (while staying within your error bars), estimate the percentageerror in your determination of D.

slope = 1/D

x

cot

7Measuring the Size of the MoonIf you know the distance to the moon2, its relatively easy to determine its size(diameter). This calculation makes use of the small angle approximation (seeIntroduction: Method) diagrammed below. Remember that the angle must be expressed inradians (1 radian = 57.3o).

The angle subtended by the moon is pretty small. Incidentally, if you saw the movieApollo 13, you may remember Tom Hanks (Lovell) lying in a lawn chair after a party oneevening comparing the size of the moon with his outstretched thumb. Try it yourself.Your outstretched thumb subtends an angle a little less than 2 (measure it and apply thesmall angle approximation to check this). The moons diameter is between 1/4 and 1/3the thickness of your thumb. Now you can make a reasonable guess as to the anglesubtended by the moon (order of magnitude approximation). Going back to the movieApollo 13, Lovell compared the size of the earth with his thumb when he came aroundthe moon. 8 How did the size of the earth compare with the size of his thumb? If you didntsee the movie, how should the sizes compare?To actually measure the angle subtended by the moon (), cut yourself a rectangular stripof paper. Make the narrow dimension (D in the drawing) small say, 1/2 cm or 1/4.The longer side can be any length. Hold the piece of paper up at arms length and line upthe narrow side (D) across the moons diameter. Move the paper closer to your eye untilthe width of the paper just matches the width of the moon. Measure the distance betweenyour eye and the piece of paper this distance is R. Now you can determine the angle ,which is the angle subtended by the moon.

2Giancoli gives a value of 384x103 km (inside front cover). Actually, the distance to the moon varies by about 14%from its apogee to perigee. Since you (presumably) don't have the appropriate astronomical data at hand that tell inwhich part of the orbit the moon is at this time, you might as well round off this number to 400,000 km, or 4x108m.For your information, the moon's diameter is 3.48x106 m.

the angle subtended

D R, measured in radians

distance to eye

width of paper

R

D

Ldistance to the Moon

Moon

8However, before measuring the moon, test your technique by measuring the thickness ofa tree. You can confirm the accuracy of the method with a direct measurement of thetrees thickness.9 What is the thickness of the tree using this method?10 How does this value compare to the direct measurement?Knowing the distance to the moon L, you can now calculate its diameter.11 What is the size of the moon?12 How does your number compare to the known size? Again, quote uncertainties.Be careful here and dont be too optimistic. Remember that no matter how good yourmeasurements are, the final result is no better than the least accurate number in yourcalculation.

Check List for the Report Answer the 12 bold face questions on a separate paper. Be sure to answer all

parts of each question. Include tables of all measurements as well as calculations. Include the graph used to determine the Science Center height. Prepare the graph

following the directions on pp. 11-13 of the Lab Companion.

9Lunar Myths and Measurements

You may have noticed that the moon looks much bigger when it rises over the horizon its particularly dramatic with a full moon. If you have a chance, measure it anddetermine how much bigger it appears to be. Youll be surprised.

10

Quadrant PatternTo construct the quadrant, cut out the pattern below and paste it onto a piece of card stock orcardboard. Attach the top side of the quadrant (labeled top), to a pencil or small rod. Tie a key toone end of a 1.5 ft piece of string, and attach the other end of the string to the pencil on the rightside where the circle with the cross is located on the pattern. Tape the string to the pencil so thatit can't slide. You are done! Your finished quadrant should look like the quadrant in the figure onpage 2.