A Hooke's Law experiment is performed in which several elongations are made in the length of the spring, and the resulting restoring forces produced are measured by a force sensor. The data table shown to the right below displays elongation-restoring force measurements for six elongations to which the spring was subjected. The first entry in each data pair is the elongation of the spring x, measured in meters (m). The second entry is the corresponding restoring force developed in the spring F, measured in newtons (N). Note that the first data pair is one which should be known a priori. It would not make sense that a spring would produce a restoring force in the absence of an elongation. Make sure you include this important data ate. pair in other similar experimental situations if appropri- HOOKE'S LAW TEST DATA trial elongation x restoring force F .000 m 0.0 N restoring force F (N) N 034 m 11.3 N 065 m 22.4 N .093 m 33.5 N 42.4 N LO .00 02 04 06 08 10 12 .14 . 107 m elongation x (m) .128 m 47.0 N CO 2 Perform a general linear regression analysis on the data set above, and obtain a regression equation of REGRESSION EQUATION: F = a + bx the form F = a + bx. Display the regression equation in the table to the right, with all coefficients rounded to 3 significant figures and accompanied by appropriate units. Enter the correlation coefficient r for this analysis in the table, rounded to 3 decimal places correlation coefficient r 2 Use the regression equation F(x) obtained above to estimate the restoring force that would be produced ESTIMATION USING F = a + bx by spring elongations of 0.052 m (interpolation) and 0. 164 m (extrapolation). Estimate the elongation needed F'(x =.052 m) to produce a restoring force of 27.1 N. Enter these estimates in the table to the right, each rounded to 3 F'(x = . 164 m) significant figures. Learn to use your calculator to handle all of the math in these estimations. x'(F = 27.1 N)7 Perform an exponential regression analysis on Log the data set above, and obtain a regression equation of the form V = aebt. Display the equation in the table to REGRESSION EQUATION: V = aebt the right, with all coefficients rounded to 3 significant figures. Make sure that each coefficient in the regres- sion equation includes the units associated with it so that the equation is dimensionally correct. Enter the correlation coefficient r for this analysis in the table, rounded to 3 decimal places. correlation coefficient r 8 Use the regression equation V(t) obtained above to estimate: (1) the voltage across the capacitor at ESTIMATION USING V = aebt time 3.75 s (interpolation); (2) the time at which the voltage across the capacitor had decreased to 13.6 V V'(t = 3.75 s) (interpolation); and (3) the time at which the voltage across the capacitor had decreased to 5.0 V (extrapo- t'(V = 13.6 V) lation). Enter these estimates in the table to the right, each rounded to 3 significant figures. t'(V = 5.0 V) 9 The photo-electric effect is a quantum physics phenomenon in which photons incident on the surface incident photo- of certain materials liberate electrons. The kinetic photons electrons energy of the liberated electrons is given by K = -( + C/ 1, where Q (the work function of the material) is the minimum energy required to produce photo-electrons, C is a physical constant, and ^ is the wavelength of the incident photons. Suppose that a photo-electric experi- ment is performed in which six different wavelengths of UV radiation are used to liberate electrons from the surface of some photo-sensitive material. The data table below shows results from this experiment. The first oto-sensitive material entry in each data pair is the wavelength ^ of the source, measured in nanometers (nm). The second entry in each pair is the kinetic energy K of the liberated electrons, measured in electron volts (eV). PHOTO-ELECTRIC EFFECT DATA pair wavelength a K-energy K 221 nm 2.11 eV 240 nm 1.67 ev kinetic energy K (ev) m 268 nm 1.13 eV 285 nm 0.85 eV LO 304 nm 0.57 eV 210 240 270 300 330 360 wavelength ^ (nm) CO 338 nm 0.17 eV10 Perform a hyperbolic regression analysis on the data set above, and obtain a regression equation of the form K = a + b/). Display the regression equation in the REGRESSION EQUATION: K = a + bix WHAT table to the right, with all coefficients rounded to 3 Regr significant figures and accompanied by appropriate led units. Enter the correlation coefficient r for this analysis in the table, rounded to 3 decimal places. correlation coefficient r 11 Use the regression equation above to obtain: (1) a regression-interpreted value for the physical constant C INTERPRETATION in the theoretical relation; and (2) a regression-inter- preted value for the work function Q of the photosensi- physical constant C tive material. Enter these values in the table to the right, each rounded to 3 significant figures. work function Q 12 Use the regression equation to estimate the following quantities: (1) the kinetic energy of photo- ESTIMATION electrons liberated by 319-nm photons; and (2) the wavelength of photons which would produce photo K'(2 = 319 nm) electrons with kinetic energies of 1.48 eV. Enter these estimates in the table to the right, each rounded to 3 2 " (K = 1.48 ev ) significant figures, and with appropriate units. 13 Regression equations can serve as useful predict tors of quantities, but only within the context of what EQUATION LIMITATIONS makes physical sense. Use the regression equation obtained to estimate the kinetic energy of photo- K'(2 = 450 nm) electrons liberated by 450-nm photons. Enter this estimate in the table to the right, rounded to 3 signifi- Is the estimate reasonable? Why or why not? cant figures, and with appropriate units. In the space provided briefly comment about this estimate in terms of its reasonability. - my4 Sometimes a parabolic regression analysis may be a better choice than a linear one. Perform a para- bolic regression analysis on the data set above, and obtain a regression equation of the form T = a + bx + REGRESSION EQUATION: F = a + bx + cox 7 perform an the data set abo cx2. Display the regression equation in the table to the the form V = a right, with all coefficients rounded to 3 significant the right, with figures and accompanied by appropriate units. Enter figures. Ma the correlation coefficient r for this analysis in the sion equa table, rounded to 3 decimal places. correlation coefficient r that the correla roun 5 Use the regression equation obtained above to estimate the tensional force that would be produced by a spring elongation of .045 m. Estimate the elongation ESTIMATION USING F = a + bx + cx2 needed to produce a tensional force of .325 N. Enter F'(x = . 045 m) these estimates in the table to the right, each rounded to 3 significant figures. Learn how to use your calcula- X1'(F = .325 N) tor to handle all of the math in doing these estima- tions, since it will involve negligible rounding error. X2"(F = .325 N) 6 A capacitor is a circuit element that stores electric charge when a voltage is dropped across it. In I some electrical circuits a capacitor is discharged, SSIPATION causing the voltage across it to decrease over time. ASS Suppose that an experiment is performed in which a LASS A capacitor in a circuit is initially charged to some LASS A nichicon voltage. At some time (called t = 0 seconds) a switch is opened, allowing the capacitor to discharge, with the result being that the voltage across the capacitor nichicon MAX 100RF 450y 100RF 450 decreases in time. The data table shown below dis- plays time-voltage measurements for the discharging 6135 capacitor in this experiment. The first entry in each PLA data pair shown is the time t after the switch was ALL OTHERS opened, measured in seconds (s). The second entry is the corresponding voltage across the capacitor V, measured in volts (V) DISCHARGING CAPACITOR DATA 8 pair time t voltage V 0.1 S 49.8 V 40 2.1 S 30.8 V voltage V (V) 5.0 S 25.4 V 8.2 S 13.4 V LO 11.4 S 9.2 V N time t (s 14.3 S 8.2 VREGRESSION ANALYSIS: INTRODUCTION MacBook Pro WHAT IS REGRESSION ANALYSIS? Regression analysis is a statistical technique in which paired data is summarized by fitting a functional form to it that attempts to desch describe a mathematical relation be- tween the two variables involved. This technique is espe- cially useful when there is an obvious relation between the two variables. HOW GOOD IS A FIT? Maybe a better question is "How poor is a fit?" The poor d4 = y4 - f(X4) - ( Xpryn) O ness of the fit of a functional form to a data set is sug f( X4 ) * V y = f ( x ) gested by the deviations of the measured values of the (X2, )2) O ordinate in the data set from the values as predicted by ( X 1 , )1 ) the functional form (y; - f(x;)). The larger these deviations (X3, )/3) are, in general, the poorer the fit is WHAT CRITERION IS USED? The criterion used to obtain the "best-fitting" equation is X4 that the sum of the squares of the deviations of the ordi- nate values of the data pairs (from what is predicted by HOW IS THE FIT QUANTIFIED? the regression equation) be a minimum. That is, the func- The "goodness" of the fit of a functional form to a data set tional form that minimizes this "sum of squares" is con- can be quantified by a coefficient called the correlation n. . sidered, by definition, to be the "best-fitting" equation. coefficient (r). This is a dimensionless real number ranging between -1 and +1. The magnitude of this number is an index of the goodness of fit, with 1.0 being a perfect fit (all Elyi - f(x;)12 has a minimum value data points lie on the regression curve) and 0 suggesting no correlation between the two variables involved. SHARP EL-506W REGRESSION ANALYSIS OBTAINING A REGRESSION EQUATION SHARP EL-506W 2ndF MODE clears statistical registers of calculator Stat 1 MODE sets calculator to work in statistical mode GO ... selects regression equation functional form enter x STO enter y M+ enters data (repeat until all entered) 2ndF ON/C RCL or ) or x obtains coefficients a or b or c ALPHA MODE obtains correlation coefficient r (n.a. for parabolic) RCL SETUP DEL sin COS tan dx M logarithmic: y = a + binx hyp linear: y = a + bx power law: y = axb X2 x 3 log In cnst VX parabolic: y = a + bx + cx2 inverse: y = a + b/x p ab/c DMS RCL STO M+ exponential: y = aebx ESTIMATION WITH THE EQUATION X -1- obtains regression-estimated y-value enter x-value 2ndF + obtains regression-estimated x-value enter y-value 2ndF Exp obtains second root (X2) of a parabolic estimation 2ndF O