Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 1 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS In the physcal measurements we often make a seres of measurements of the dependent quantty Y as a functon of some ndependent quantty X and so our observatons consst of pars X, Y. The quanttes X can be assumed to be precse but there are random errors n the measured values of the quanttes Y causng equal uncertanty to each Y value. We know that the functonal dependence between the quanttes X and Y s lnear, e.g. Y a bx, (L.1) where a and b are the ntercept and the slope of a straght lne. Ths s stuaton for example n ths exercse, when the sprng constant s determned by measurng both the elongatons of the sprng and the oscllaton perods of the sprng-weght system as a functon of mass. In both cases the mass of the weght can be assumed to be precse but random errors are affectng on the observed places and oscllaton tmes. In ths Appendx we study two dfferent methods for fttng the straght lne descrbed by Eq. (L.1) to our observatons. In the graphcal method the observed values are frst presented n a sutable graph and then a straght lne smoothly followng the observatons s drawn and the slope b and the ntercept a are determned from the graph. In the least-squares method the most probable values for the slope b and the ntercept a are determned by mnmzng the sum of squares of the dscrepances between the theoretcal and the measured values of the quantty Y. 1. Graphs The followng recommendatons concern the graphs presented n reports: 1. Choosng a paper: If you draw the graph manually, use a mllmeter paper. When drawng wth a computer prnt the graph n a sze large enough. Often an appendx ncludng only the graph and the necessary nformaton on t s the best.. Choosng the scale and markng the ponts: Choose the scale so that drawng s as easy as possble and so that all the necessary detals are separated from the graph. Mark the observed ponts clearly wth a sutable symbol, for example a trangle, a square or a cross.
APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS 3. Dvson and cuttng of the axes: Mark the dvsons and the numbers of the axes clearly. If the observatons are far away from the orgn, the axes should be cut and only the area ncludng observatons should be presented. The cuttng s marked on the axs for example wth two lnes accordng to the model gven below. 4. amng the axes: ame the axes so that the name tells both the quantty and the unt. Use the same symbols n the graph as elsewhere n the report, when referrng to the quanttes. 5. umber and head the graph as follows: Fgure 1. Elongaton of the sprng as a functon of mass. The ttle of the graph can be placed ether above or below. Especally when the report ncludes several fgures they are numbered. When the fgures are numbered the pont s usually used at the end of the ttle and also after the number. The most mportant rules concernng the fgures are seen below n the fgure model. umber and head the fgure. Fg. 1 (a) Place of a sprng part as functon of mass. x 1 (m) ame the axes n the form quantty (unt). Mark the dvson and the numbers clearly on the axes. 1,50 Mark the observatons clearly. Use for nstance trangles, squares or crosses as 1,40 symbols. Choose the scale so that the observatons and the straght lne fll 1,30 almost the whole area of the fgure. 1,0 Draw the straght lne usng a ruler. 1,10 If the lne does not go through the orgn extrapolate t so you can read 1,00 both the ntercepts wth the 0,90 horzontal and vertcal axes. 0,00 0,10 0,0 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 1,10 m (kg)
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 3. Graphcal fttng of a straght lne When the observatons are presented n a sutable coordnate system a straght lne can be ftted to the observatons usng graphcal smoothng e.g. by drawng the straght lne so that t follows as well as possble the observed values. The graphcally smoothed straght lne does not nclude all the observed ponts, but as a rule there are equal number of observatons above and below the straght lne. In the example below the slope of the lne s calculated wth the ad of the graph. The ponts used n the determnaton of the slope are chosen from the both ends of the lne, e.g. the area of the graph s taken nto account as wdely as possble. otce that the ponts are chosen from the lne not from the measurements. Observe the x-values of the chosen ponts (e.g. n the example graph the m-values) and read the correspondng y-values from the graph. Calculate the dfferences x and y of these x- and y-values and determne the slope of the lne by calculatng ther quotent y/x (or y/m n the example). 0,4 y (m) m = (0.96-0.06) kg 0,3 0, y = (0.3500-0.05) m 0,1 b 1 =y/m = 0.375 m/ 0.90 kg = 0.3639 m/kg 0 0,00 0,0 0,40 0,60 0,80 1,00 1,0 Fgure 1b) Elongaton of the sprng as a functon of mass. m (kg)
4 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS 3. Least-squares method 3.1 Least-squares fttng usng tables When the oscllaton perods of the sprng-weght system are measured as a functon of mass the dependence of the square T of the oscllaton perod on the mass s accordng to the Eq. (.7) gven n the exercse nstructon of the form T 4 m 3k j 4 k m or T a b m, where k s the sprng constant and m j s the mass of the sprng. Our am s to determne the most probable values for the parameters a and b wth our observatons. Then, the values of the sprng constant and the mass of the sprng can also be calculated. ow there are two values correspondng to each ndependent quantty measurements: The observed value Y and the theoretcal value from Eq. (L.1) X used n the teor y for whch we get y teor a bx. (L.) We can now calculate the sum Q of the squares of the devatons between the theoretcal and the observed values. Accordng to Eq. (L.) ths sum s Q teor ( y Y ) ( a bx Y 1 1 ). (L.3) In the least-squares method we thnk that the most probable values of parameters a and b wth the least uncertantes are found when the sum Q s at mnmum. For mnmzng the sum Q we frst calculate the partal dervatves of Q wth respect to the parameters a and b. Then, we set these dervatves equal to zero and so we obtan Q ( a bx Y ) 0 a b X Y 0 a 1 1 1. (L.4) Q ( a bx Y ) X 0 a X b X X Y 0 b 1 1 1 1 For example by solvng b from the upper equaton and by substtutng ths expresson to the lower equaton the value of the parameter a can be calculated. Then, for
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 5 obtanng the value of b the value of a can be substtuted to the expresson of the parameter b. By ths means the expressons a X Y X X Y 1 1 1 1 X ( X ) 1 1 X Y X Y 1 1 1 b X ( X ) 1 1. (L.5) are obtaned. Example.1 In an exercse the sprng constant and the mass of a sprng were determned by suspendng 10 weghts whch all have a mass of 100 g one by one to the sprng. The sprng-weght system was set to oscllate and the oscllaton tme of ten subsequent perods was measured three tmes. The measurement results are lsted n Table 1 below. Determne the ntercept a and the slope b of the straght lne presented wth Eq. (.7) by usng the least-squares method and calculate the sprng constant and the mass of the sprng. Soluton: The mean values of the three measured oscllaton perods are frst calculated and they are gven n the rghtmost column of Table 1 below. The perod of one oscllaton are acheved by dvdng the mean values by ten. However, for the leastsquares straght lne of Eq. (.7) we need the squares of the perods. These are gven n Table. Table 1. Measured perods of ten oscllatons wth ther mean values. m (kg) 10T1 (s) 10T (s) 10T3 (s) 10Tmean (s) 0.1 5.48 5.50 5.45 5.476667 0. 7.9 7.4 7.6 7.63333 0.3 8.67 8.71 8.65 8.676667 0.4 9.9 9.85 9.89 9.886667 0.5 10.93 10.96 10.99 10.96000 0.6 11.91 11.98 11.95 11.94667 0.7 1.83 1.86 1.87 1.85333 0.8 13.71 13.69 13.74 13.71333 0.9 14.51 14.53 14.49 14.51000 1.0 15.7 15.6 15.8 15.7000
6 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS In order to obtan the ntercept a and the slope b from Eq. (.5) we need the sums m, m, T, mt, n whch the ndex goes from 1 to 10. In ths case m corresponds to X and T corresponds to Y n Eq. (.5). The calculaton of these sums s done wth the ad of Table, whose leftmost column ncludes the ndces and the second and the thrd column from the left nclude the masses and ther squares, respectvely. The correspondng calculated squares of the perods are gven n the fourth column from the left and the rghtmost column ncludes the products of the masses and the squares of perods. In the lowest row of Table the necessary sums are gven. Table. Calculaton of the sums needed for the least-squares ft. m (kg) m (kg ) Tmean (s ) mtmean (kgs ) 1 0.1 0.01 0.99939 0.09994 0. 0.04 0.57560 0.10551 3 0.3 0.09 0.75845 0.5854 4 0.4 0.16 0.97746 0.390985 5 0.5 0.5 1.0116 0.600608 6 0.6 0.36 1.478 0.856337 7 0.7 0.49 1.6508 1.156457 8 0.8 0.64 1.880555 1.504444 9 0.9 0.81.105401 1.894861 10 1.0 1.00.33179.33179 Sums 5.5 3.85 13.156017 9.096781 When the sums gven n Table are substtuted n Eq. (L.5) we obtan a ja m T m m ( m ) 50.650665 kg s 38.5 kg m T 50.0396 kg 30.5 kg 3.85 kg s 13.156017 s 0.618370 kg 8.5 kg s 10 3.85 kg 5.5 kg 9.096781 kgs (5.5 kg) 0.074954 s 0.0750 s m T m T 10 9.096781 kgs 5.5 kg 13.156017 s b m ( m ) 10 3.85 kg - (5.5 kg). 90.967810 kgs - 7.358094 kgs 18.609717 kgs s s.5573.6 38.5 kg 30.5 kg 8.5 kg kg kg By usng the slope b gven above we get for the sprng constant k
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 7 s b.5573 kg 4 k 4 k b 4.5573 s kg kgm 17.5 s m 17.5 m. By usng the slope b together wth the ntercept a the mass of sprng calculated m j can be a 4 m 3k j 4 m j bm 4 3 3 b j m j 3a b 3 0.074954 s.5573 s kg 0.0997 kg 99.7 g. 3. Least-squares ft wth Excel owadays, the drawngs and least-squares fts are usually done wth a sutable computer program. In Fg. above an example of Excel drawng representng the observed squares of the perods as a functon of mass as well as the least-squares straght lne s show. T 3 (s ),5 Fgure. Square of the perod as a functon of mass. 1,5 Mark the observatons clearly. 1 0,5 0 m (kg) -0, 0 0, 0,4 0,6 0,8 1 1, -0,5-1 Extrapolate the lne f necessary so that you can observe the ntercepts. Instructons for fttng the least-squares straght lne wth Excel: 1) Regster your observatons to a sutable Excel table so that you have columns ncludng both for the ndependent X values and the dependent Y values. For example, n the
8 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS oscllaton perod measurements above you can begn by regsterng the observed perods and use Excel when calculatng the mean values of three measurements, the perods and the squares T of them. ) Draw the observed X, Y ponts (or here m, T ponts) n an Excel chart and edt the chart so that t s of a sutable sze and ncludes all the detals mentoned n Fg.1 (e.g. the ttle of the graph, the ttles of the axes etc.) 3) Insert the least-squares straght lne to the graph for example by pontng one of the observed ponts wth the rght mouse button and by choosng nsert trendlne from the openng wndow. The program then suggests the alternatve lnear n the followng wndow. If the cross s marked to the alternatve Dsplay equaton on chart the equaton of the straght lne s shown besde the lne. 4) Ft the straght lne to the observatons usng the least-squares method as follows: - Mark for example below the Excel table ncludng the observatons tmes cells and wrte n the upper leftmost cell =LIEST(. When the leftmost bracket has been wrtten Excel starts to ask the parameters. - The whole LIEST-command s of the form =LIEST(y-values; x-values: TRUE; TRUE). It ncludes four parameters separated wth semcolons. - The observed y-values and x-values are marked from the observaton table wth the mouse. - The frst parameter wth a value TRUE can be gnored and so there are only two semcolons successvely n the command. The value TRUE can also be gven wth a number 1. If ths parameter s TRUE also the ntercept s calculated, f t s UTRUE (or 0) the straght lne goes through an orgn and the ntercept s zero. - When the last parameter between the brackets has the value TRUE (or 1) the uncertantes of the slope and the ntercept are calculated. - When you have gven the parameters, press CTRL-SHIFT-ETER. ow the tmes cells nclude n the upper left cell the slope and n the upper rght cell the ntercept. Below them ther uncertantes are seen.