Measuring Distance Using Parallax

Transcription

1 Mesuring Distnce Using Prllx Prllx is the pprent motion of sttionry object due to chnge in the observer s position. In stronomy prllx is the gold stndrd (best) wy to mesure distnces. Tht is, until it doesn t work ny more. The ccurcy of prllx mesurements depends very strongly on the distnce itself, s you will find out from your work. In this exercise you will use prllx to determine distnces to three different objects chosen to be t VERY different distnces. You will relte the ccurcy of your mesurements to the ccurcy of the distnce mesurement. Historicl Bckground: The Greeks knew tht if distnt object is viewed from two different points, the direction to it will chnge, s is shown in the figure. They (nd you) could figure out ll the sides of tringle whose corners re t the observing positions nd the nose of the niml given the length of the bseline nd two ngles. This procedure is used for finding distnces nd for surveying nd mesuring lnd. The Greeks tried to mesure the prllx of the Sun, the strs, the plnets etc. They did not get ccurte prllx mesurements for ny of these objects. Since the strs did not seem to move AT ALL over the yer they thought tht the Erth stnds still in the middle of sphere of strs. Why couldn t the Greeks mesure the prllx of the strs? The problem is tht s something gets further wy, the pprent motion due to prllx gets smller. Eventully it becomes too smll to mesure. The distnce which is becomes too fr to mesure depends on the equipment nd cre used. Mterils: Ruler or tpe mesure, protrctor, clcultor, pper, tpe or chlk to mrk the bseline. Procedure: You will mesure the distnce to three objects nd then estimte the ccurcy of these mesurements. The ide for mesuring the distnces is to mesure one side nd two ngles of tringle, then you will use this informtion (nd mth) to figure out the distnce to n object. The ngles nd distnces to mesure re shown in the picture. You my use EITHER type of protrctor. Becuse you re exploring the effect of distnce on mesurement ccurcy, choose 3 objects t different distnces, one t distnce of bout 5 times the bseline, one t bout 10 times, nd one Much further.. Guess the distnces to decide wht to mesure. Protrctor Bseline, Distnce between points where observtions re mde Mesure 3 times to nerest 0.1 deg Protrctor, different type Decide on bselines to use to mesure your objects. The bseline must be AT LEAST 200 cm. You MAY use the sme bseline for ll three. You must be ble to see the object from both ends of the bseline. Mrk the ends of the bseline with tpe or chlk nd be certin to hve something to show the direction from one end to the other ( fence or riling is good). Mesuring Distnce Using Prllx 1

2 Mesure the length of the bseline to the nerest millimeter (0.1 centimeter). So the number of centimeters in the length tht you mesure will not normlly end with zero. You must hve three different vlues for the length. (The length does not chnge, but if you mesure with enough precision, your mesurement errors will cuse the vlues to be different.) Look t the object from ech end of the bseline. Mesure the ngle between the bseline nd the direction to the object by putting the center of protrctor t the end of the bseline, lining up the zero degree line with the bseline, Looking cross the center of the protrctor to the distnt object nd mrking the spot on the protrctor where the line of sight crosses the protrctor. Decide whether the ngle is cute(<90deg) or obtuse(>90deg) before reding the protrctor so tht the mny scles on protrctor do not confuse you. SKETCH your set ups. Show your bseline, some lndmrks in the bckground (like trees or buildings) nd identify the objects which you re mesuring. Do NOT try to mesure of f the ngles nd sides but DO be sure to show which ngles re cute (less thn 90 o ) nd which re obtuse. Record your mesurements in the following tble. Be sure tht the bseline is mesured in centimeters to the nerest millimeter nd tht the ngles re mesured nd computed to the nerest 1/10 degree. Object # 1 Object # 2 Object #3 Nme of Object Bseline Length Middle Vlue Lrgest Vlue Bseline Length Smllest Vlue Bseline Length Middle Vlue Lrgest Vlue Smllest Vlue Middle Vlue Lrgest Vlue Smllest Vlue Angle A Middle Vlue (180 o -Middle B-Middle C) Lrgest A Vlue (180 o -Smllest B- Smllest C) Smllest A Vlue (180 o -Lrgest B- Lrgest C) Getting the distnce when you know two ngles nd side Your mesurements for ech object mount to two ngles nd the included side (the bseline) for tringle. Your observtion points re t two of the vertices nd the object which you re trying to mesure is the third. This is enough informtion tht, there is only one tringle which corresponds to the mesurements. When the tringle is completed, the position of the distnt object will be determined. But drwing is wkwrd nd introduces its own errors. Mesuring Distnce Using Prllx 2

3 You could drw the tringle to scle nd mesure the distnce on the scle picture. This is comprtively esy to think bout, but the process of drwing introduces dditionl errors nd the tringles re long nd skinny. The distnce to the object depends on whether the distnce is defined to be from one of the observtion points, from the bseline or from some other point. In stronomicl mesurements, the center of the erth is often used for nerby objects nd the center of the Erth s orbit is used for distnt objects. For your experiment, you will find side b nd cll tht the distnce. As the distnce gets lrge, it doesn t mtter much which side you choose for the distnce. Your picture my look like Figure 3. Notice the lbels on the ngles nd the sides. All of the ngles re known. Two hve been mesured t the observers. The third is found becuse: A+B+C= 180 degrees There is theorem (clled the Lw of Sines) which sys tht sine A = b sine B = c sine C. The lower cse letters represent the lengths of the sides of tringle; the uppercse letters represent the ngles. Side is cross from ngle A. Figure 3 shows the sitution. The nottion sine A mens to use function clled sine (bbrevited sin) nd to use the vlue for the ngle A. The sine of n ngle is the rtio (nswer) you get when the ngle is prt of right tringle nd you divide the length of the side opposite the ngle by the length of the hypotenuse, so the definition is s in ( n g l e ) le n g t h o f s i d e o p p o s it e le n g t h o f h y p o t e n u s e Once you know tht tringle is right tringle, nd you know one of the other ngles, you know the shpe of the tringle. This occurs becuse the third ngle must mke up the 180 degrees totl for the tringle. Since ll the ngles re known, the shpe is fixed, nd so re the reltive sizes of the sides. The trigonometric functions just tbulte the rtios of the sides of tringles. You could think of them s tbles which describe the shpes of right tringles. There is tble of the vlues of sine t the bck of this write up. Your scientific clcultor lmost certinly hs the sin function. Check out how it works nd use the tble to verify, Sine of 90 deg is 1. If you don t get tht from your clcultor, check to be certin tht the clcultor is in degrees mode. To find the distnce, we will use sine A = b sine B Bseline B Figure 3 C Here is the bseline, which you know. The ngle B is one of the mesured ngles, which you know. The ngle A is known becuse A = 180 degrees - B - C, nd you known B nd C both. The vlues of sine A nd sine B re found by looking t the tble in the bck of the write up. Thus you will know the vlues of ll the quntities except b, the distnce from one of the observers to the distnt object. This is exctly wht ws wnted. So now you cn solve the eqution nd find the distnce c b Angles B nd C re known Bseline length, is known A The distnt object is here Angle A = 180 degrees - B - C Mesuring Distnce Using Prllx 3

4 b sinb distnce sinb bseline Exmple Problem: The mesurements re s follows: Bseline = cm, cm, cm = 87.2 degrees, 88.1 degrees, 86.9 degrees = 85.4 degrees, 85.9 degrees, 85.1 degrees Let s just consider the middle vlues. First compute ngle A s 180 degrees= A + B+ C 180 degrees= A A = 180 degrees 87.2 degrees 85.4 degrees A= 7.4 degrees So now you re prepred to find sine A nd sine B. Your clcultor is the esiest wy (or you cn interpolte using the tble t the bck). sine A = sine 7.4 degrees = sine B = sine 87.2 degrees = The sines hve no units. The lengths of the two sides hd the sme units s one nother nd the units cncel. Using the formul distnce sinb bseline nd substituting the known vlues cm dis tnce dis tnce cm So the distnce to the object is found. The numericl vlue cm is not relly correct. We do not know the contributing quntities well enough to sy the distnce to the nerest thousndth of centimeter. You only mesured the bseline, cm to four digits, nd the vrition mong the mesurements tells us tht the uncertinty is bout 0.9cm (so more like 3 significnt figures). The B vlues in this exmple vry by round 0.6 degree(hlf the difference between lrgest nd smllest). This shouldn t be expressed s percentge,since the mesurement error probbly would hve been bout the sme number of degrees regrdless of the vlue of b or C But the sines don t chnge s much. Sine (87.2 degrees) = Sine (88.1 degrees) = Sine (86.9 degrees) = So here the vrition is bout ± in the sine. This is like 3 significnt figures. So we might expect tht the nswer should hve only 3 significnt figures. We would be better off expressing it s 1.58x10 3 cm. You will ssess extreme vlues possible with your set of mesurements. They my be further off thn you would expect from the number of significnt figures. Tke nother look t the formul for the distnce Mesuring Distnce Using Prllx 4

5 distnce sinb bseline The bigger the vlues of sine B nd the bseline, the bigger the distnce tht will result. The smller the vlue of sine A, the bigger the distnce (since we divide by sine A). So choosing the lrgest sine B nd bseline nd the smllest sine A will result in the lrgest vlue of the distnce. Conversely, the smllest sine B, smllest bseline nd the lrgest sine A result in the lrgest distnce. The tble tht follows helps you to orgnize your work to compute these extreme vlues of the distnces. Use your mesurement dt to compute the distnce to ech body. Record your work in the following tble. Be sure to write the units for ech of your quntities, Distnce Formul- choose mx or min Di st Bseline Length Angle A Distnce vlues to fit formul # Sine B Sine A Middle bseline* 1 Middle(sine B) /Middle(sine A) Mx bseline* Mx(sine B) / Min(sine A) Min bseline * Min(sine B) / Mx(sine A) Middle bseline* Middle(sine B) Middle(sine A) Mx bseline* Mx(sine B) / Min(sine A) Min bseline* Min(sine B) /Mx(sine A) Middle bseline* Middle(sine B) /Middle(sine A) Mx bseline* Mx(sine B) / Min(sine A) Min bseline* Min(sine B) /Mx(sine A) Mesuring Distnce Using Prllx 5

6 1) Which of your distnce mesurements hs the gretest percentge of uncertinty? (hint, per cent error (vlue officil vlue) 100 officil vlue Tret the Middle vlue for ech distnce s the officil. Negtive vlues do mtter, since tht mens vlue is smller thn expected. But negtive vlue is considered comprble to positive one. E.g. A vlue with -10% error hs lrger error thn one with +9%) 2) If you tried to mesure nother object which is even more distnt using the sme equipment which you were lredy using, would your nswer become more or less ccurte? Distnce #1 #2 #3 Best Estimte (the middle vlue from the previous tble) Plus (the difference between the best estimte nd the lrgest) or Minus mount (the difference between the best estimte nd the smllest) 3) How ccurtely do you think tht you could mesure the ngles in this experiment? (Hint:: Look t the vrition mong the repeted mesurements of the sme ngle B or C. The vrition is due to your personl mesurement error. Clculte the difference between lrgest nd smllest mesurement of the sme quntity. If you find hlf the difference, tht is mesurement of the uncertinty inthe mesured vlue. You hve 6 different B nd C ngles nd ll of them re mesured the sme wy. Tht is like hving 6 different smples of your mesurement process. Look t ll these uncertinties nd choose vlue tht encompsses the generl run of vlues.) How ccurtely could you determine the vlue for the ngle A, the ngle t the distnt object (the one which you did not mesure, but which you computed from the other two)? ( number of degrees plese!) How ccurtely could you mesure the length of the bseline? ( number of centimeters plese!) Which of these fctors, the bseline, ngle A, or ngle B is the lrgest contributor to the uncertinty in the distnce which you found? 4) Suppose you wnt to mesure the distnce to distnt light pole. Would you get more ccurte vlue using 25 foot bseline or one with 10 foot bseline? Assume tht you cn get the bseline lid out with no trouble nd tht you re using the sme equipment tht you used for the rest of this experiment. Extr Credit The Greeks wnted to know the distnce to the Moon nd to the strs. Suppose tht they hd sttioned two persons on opposite sides of the Erth, nd both of the persons observed the Moon ginst the strs t the sme time. Drw sketch of how two people would set up this mesurement so tht they would be s fr prt s possible. Look up how fr wy the moon is. How lrge would ngle A be ( numericl vlue plese)? Do you think the Greeks could hve observed this chnge? Why or why not? Wht difficulties would there hve been in mesuring the distnce to the Moon with this method? Mesuring Distnce Using Prllx 6

7 Sine Tble Wht if the ngle is MORE thn 90 o? The vlue of the sine strts to decrese from its high of 1. The wy tht it works is tht sine x = sine (180 o -x). For exmple sine 95 o =sine(180 o -95 o )=sin 85 o. Your scientific clcultor will find sines nd relieves the problem of wht to do with the tenths of degree). Check t lest one vlue to be certin tht the clcultor is in the correct mode. Chep clcultors hve no problem. Some clcultors hve severl modes. Be certin to use DEGREES mode. Angle, Degrees Sine A Angle, Degrees Sine A Angle, Degrees Sine A Mesuring Distnce Using Prllx 7