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A compari son of experimental with idealized data will instead demonstrate a cor relation with that of the damped harmonic oscillator equation and reve al the inconsistencies of the simple model with that of reality. The \+ change in period derived will also act as an indicator for the change \+ in angle and thus the amplitude of the pendulum swing with time." }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 259 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 478 "Given an experiment** in which could be seen the physica l consequences of simple harmonic motion in a not-so-simple world, it \+ is speculated that an accurate model would be more than that for a sim ple harmonic oscillator. The pendulum, after all, will not swing fore ver as suggested by the simple harmonic oscillator (SHO) model. By us ing actual data gathered from experimentation, the biases can be surmi sed as an additional factor in the model, that of a dampening constant . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 599 "Si nce it is difficult to measure position as a function of time, we can \+ consider the period of the pendulum as a function of its starting angl e. The derivation* of the period equation from the simple harmonic os cillation equation can be used to show that the simple pendulum does i ndeed act similarly, making the differential equation for simple harmo nic motion an apt beginning model for the experiment. In order to sho w that the data taken in the experimental phase relates to and is mode led with the simple harmonic oscillator differential equation, a deriv ation of the period equation follows." }}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 10 "Derivation" }}{PARA 0 "" 0 "" {TEXT -1 87 "The equation \+ for the force exerted on a mass in simple harmonic motion can be given as\n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "F(x) = -kx" "6#/-%\"FG6#%\"xG, $%#kxG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 270 1 "k" }{TEXT -1 96 " is a constant which enco mpasses properties of the spring (ie. material, elasticity, etc.), and " }{TEXT 271 1 "x" }{TEXT -1 135 " is the position of the object with respect to its equilibrium point. In order for SHO to exist, the fol lowing condition must be met.\n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "F(x) = -dU/dx;" "6#/-%\"FG6#%\"xG,$*&%#dUG\"\"\"%#dxG!\"\"F-" }}{PARA 0 " " 0 "" {TEXT -1 28 "\nIt therefore follows that \n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "U(x) = -kx^2/2" "6#/-%\"UG6#%\"xG,$*&%#kxG\"\"#\"\"#!\" \"F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "w here U(x) is the potential energy of the object due to its position ( " }{TEXT 272 1 "x" }{TEXT -1 157 "). Using Newton's second law of mot ion, we can relate the forces derived above in order to find a differe ntial equation to act as a model for the pendulum.\n" }}{PARA 0 "" 0 " " {XPPEDIT 18 0 "F = ma" "6#/%\"FG%#maG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "F = -kx" "6#/%\"FG, $%#kxG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "Since acceleration is the second derivative of position, we ca n equate the force equations as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "-kx = m(d^2*x/(dt ^2))" "6#/,$%#kxG!\"\"-%\"mG6#*(%\"dG\"\"#%\"xG\"\"\"*$%#dtG\"\"#F&" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "d^2*x/(dt^2) = -kx/m" "6#/*(%\"dG\"\"#%\"xG\"\"\"*$%#dt G\"\"#!\"\",$*&%#kxGF(%\"mGF,F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "This is t he differential equation for SHO. By rearranging this equation, we ca n show that \n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "d^2*x/(dt^2)+kx/m = 0 " "6#/,&*(%\"dG\"\"#%\"xG\"\"\"*$%#dtG\"\"#!\"\"F)*&%#kxGF)%\"mGF-F)\" \"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Si nce a sine or a cosine will satisfy this condition, the position can b e shown to be" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{XPPEDIT 18 0 "x = x 0cos(omega*t+phi)" "6#/%\"xG-%&x0cosG6#,&*&%&omegaG\"\"\"%\"tGF+F+%$ph iGF+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 282 " where x0 is the maximum position of the object in harmonic motion and \+ can also be thought of as the amplitude. By substituting position int o the differential equation above, it can be shown that the period is \+ related to the harmonic oscillation differential equation for a spring .\n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "d^2*x/(dt^2) = -omega^2*x0cos(om ega*t+phi)" "6#/*(%\"dG\"\"#%\"xG\"\"\"*$%#dtG\"\"#!\"\",$*&%&omegaG\" \"#-%&x0cosG6#,&*&F/F(%\"tGF(F(%$phiGF(F(F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "omega^2 = k/m " "6#/*$%&omegaG\"\"#*&%\"kG\"\"\"%\"mG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x = x0cos(omeg a(t+2*pi/omega)+phi)" "6#/%\"xG-%&x0cosG6#,&-%&omegaG6#,&%\"tG\"\"\"*( \"\"#F.%#piGF.F*!\"\"F.F.%$phiGF." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 39 " stands for the angular frequency, and " }{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 148 " represents phase shift. The quantity of which the cosine is taken is the phase angle of the system. Note \+ that the function repeats itself every \n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "2*pi/omega;" "6#*(\"\"#\"\"\"%#piGF%%&omegaG!\"\"" }{TEXT -1 1 " \n" }}{PARA 0 "" 0 "" {TEXT -1 28 "This means that the period (" } {XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 48 ") is related to the above and can be written as\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "tau = 2*pi/omega" "6#/%$tauG*(\"\"#\"\"\"%#piGF'%&omegaG!\"\"" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Making t he substitution for " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 7 " yields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "tau = 2*pi*sqrt(m/k)" "6#/%$tauG*(\"\"#\"\"\"%#pi GF'-%%sqrtG6#*&%\"mGF'%\"kG!\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "omega;" "6#%&omega G" }{TEXT -1 35 " is also related to the frequency (" }{XPPEDIT 18 0 " nu" "6#%#nuG" }{TEXT -1 15 "), we can write" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "omega = 2*pi*n u" "6#/%&omegaG*(\"\"#\"\"\"%#piGF'%#nuGF'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "omega = 2*pi/tau " "6#/%&omegaG*(\"\"#\"\"\"%#piGF'%$tauG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "The period equation must now be manipulated to show a correlation to the physical dimensions o f the swinging pendulum. By examining a free-body diagram of the pend ulum, it can be derived that " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {XPPEDIT 18 0 "F = -mgsin*theta" "6#/%\"FG,$*&%&mgsinG\"\"\"%&thetaGF( !\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 " This is not simple harmonic motion. By using a small angle approximat ion (sin " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 46 " in radians), the dis placement along the arc (" }{TEXT 276 3 "x=L" }{TEXT -1 1 " " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 67 ") can be said to be a straight line. The displacement is then sin " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 3 " = " }{TEXT 277 7 "x / L, " }{TEXT -1 9 "y ielding " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "F = -mg(x/L)" "6#/%\"FG,$-%#mgG6#*&%\"xG\"\"\"%\"LG!\"\"F-" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 "Simple h armonic motion is when the restoring force is proportional to the disp lacement and is oppositely directed. We can see that, for small angle s, the requirements for simple harmonic motion now hold if we take the constant value " }{TEXT 278 4 "mg/L" }{TEXT -1 7 " to be " }{TEXT 279 1 "k" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "F = -mg*x/L" "6#/%\"FG,$*(%#mgG\"\"\"%\"xGF(%\"LG!\"\"F+" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Remember \+ that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "F = -kx" "6#/%\"FG,$%#kxG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Therefore by substitution ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "k = mg/L" "6#/%\"kG*&%#mgG\"\"\"%\"LG!\"\"" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Also recall that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "tau = 2*pi*sqrt(m/k)" "6#/%$tauG*(\"\"#\"\"\"%#piGF'-%% sqrtG6#*&%\"mGF'%\"kG!\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Therefore, by substituting for the spring const ant (" }{TEXT 274 1 "k" }{TEXT -1 54 "), we eliminate the dependency o f mass on the period (" }{XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 77 " ) of the pendulum. The equation for small angles will then be the fol lowing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "tau = 2*pi*sqrt(L/g)" "6#/%$tauG*(\"\"#\"\"\"%#piGF '-%%sqrtG6#*&%\"LGF'%\"gG!\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 139 "If, as in the case of this experiment, t he angles are large, the equation for the period must be approximated \+ using the following formula,\n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "tau = 2*pi*sqrt(L/g)*(1+sin(theta/2)^2/(2^2)+1^2*3^2*sin(theta/2)^4/(2^2*4^ 2))" "6#/%$tauG**\"\"#\"\"\"%#piGF'-%%sqrtG6#*&%\"LGF'%\"gG!\"\"F',(\" \"\"F'*&-%$sinG6#*&%&thetaGF'\"\"#F/\"\"#*$\"\"#\"\"#F/F'**\"\"\"\"\"# \"\"$\"\"#-F46#*&F7F'\"\"#F/\"\"%*&\"\"#\"\"#\"\"%\"\"#F/F'F'" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "where the sum inside of the parenthesis is the first three terms of an expansio n." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 275 11 "Definitions" }}{PARA 0 " " 0 "" {TEXT 285 26 "(demonstrate each as well)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 15 "Simple Pendulum" }{TEXT -1 15 "* - This is an " }{TEXT 282 9 "idealized" }{TEXT -1 250 " syste m consisting of a mass suspended by a cord. The mass is considered to be a particle or a point and the cord is considered to be massless an d rigid. The motion of a simple pendulum takes place in one verticle \+ plane, acted upon only by gravity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 281 18 "Physical Pendulum*" }{TEXT -1 149 " - Th is is any system that can be mounted such that it will swing in vertic al plane about an axis passing through the body of the system. It is \+ the " }{TEXT 283 13 "non-idealized" }{TEXT -1 64 " or general represen tation of a simple pendulum (or vice verse)." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 23 "Procedure and Apparatus" }}{PARA 0 "" 0 "" {TEXT -1 921 "In order to best simulate a simple pendulum, the apparatus was set up as follows. Using a heavy bench as the base from which the ap paratus would hang, a long light wire was attached to a pivot point su pported by that bench. The other end was attached to a heavy steel ba ll and allowed to swing freely during the experiment. The mass of the ball was assumed to exist at the end of the wire in order to satisfy \+ conditions for a simple pendulum. The mass, other than its position, \+ has nothing to do with the theoretical period of the pendulum. The le ngth of the pendulum wire was determined by swinging the setup from a \+ very small angle. This was done to avoid the errors inherent in guess ing the actual location of the center of mass, and measuring around th e large spherical shape. A mirror was attached behind a protractor mo unted at the pivot point in order to reduce parallax error while readi ng pendulum angle.\n" }}{PARA 0 "" 0 "" {TEXT -1 842 "Once the apparat us had been assembled as above, and the pendulum was free of obstacles , the procedure was begun. The pendulum was brought out to a predeter mined angle. Without lending any force to the pendulum, the mass was \+ let go. When the pendulum ball reached a mark on the floor (reference ), the timer was started. Each swing through the mark in the same dir ection as the initial swing was counted as one swing. After a predete rmined number of swings have passed, the timer was stopped and the tim e recorded. The final angle was also measured and recorded. The peri od for each swing could then be determined from the time recorded and \+ the number of passes made. Any decay in angle could then be determine d from the initial and final values of angle recorded. This procedure was repeated for each predetermined angle measurement.\n" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 287 18 "Detailed Procedure" }}{PARA 0 "" 0 "" {TEXT -1 884 "Before beginning the experiment, the apparatus (consisti ng of wooden support beams, metal support rods, two heavy benches, a C -clamp, a pivot support, a pivot support pin, a pendulum bob unit cons isting of a guitar string embedded on one end in a one kilogram ball m ass, a protractor, a mirror, a stopwatch, and a micrometer) was assemb led. Considerations were made in order to minimize the amount of error inherent in the experiment due to flexing of the structure, and measu rement technique. In order to establish good technique for timing the pendulum motion, the pendulum was released from an arbitrarily chosen small angle and various swings timed. The range was used to determin e timing accuracy. The mirror behind the protractor is used to reduce parallax error when measuring angle of displacement. Other assumptio ns or observations will be noted throughout the procedure." }}{PARA 0 "" 0 "" {TEXT -1 882 "\n\011In order to find the length of the pendulu m, the pendulum was displaced to five degrees by pulling the mass to t he desired angle and then released. The displacement reading was made using the mirror behind the protractor in order to ensure that the an gle was being seen from a position normal to the protractor. Care was taken that the release of the pendulum bob be as immediate as possibl e. This measure was taken in order to avoid the application or subtra ction of velocity to or from the mass. The timing device was started \+ upon the mass crossing a designated point on the floor within one cycl e from release. The pendulum was allowed to swing freely for one hund red complete cycles. Upon the completion of the last swing, the timer was stopped, and the time recorded. This procedure was repeated five times in order to improve the accuracy of the length determination." }}{PARA 0 "" 0 "" {TEXT -1 747 "\n\011For the remainder of the experim ent, the following procedure was incorporated. The pendulum bob was p ositioned to the desired angle and released using techniques and consi derations mentioned above. The desired angle was one that was between five and forty-five degrees, in five-degree increments. The timer wa s started as before and was stopped after a number of swings had elaps ed. The number of swings was determined by watching the angle travers ed by the pendulum arm. In order to minimize the amount of angular de cay that occurred during the course of the pendulum motion, the number of swings was kept to that which would only allow no more than three \+ degrees of decay. The total time and angular decay were then recorded in Table 1.\n" }}{PARA 0 "" 0 "" {TEXT -1 420 "\011Additional data wa s taken at twenty-five, thirty-five, and forty-five degrees, for fifty cycles, and for different lubricants at the pivot point. The lubrica nts were oil, and powdered graphite. A run was made with no lubricant first, then graphite (at forty-five degrees only). The graphite was \+ wiped off of the pivot support pin. The final runs were made after ap plying 3-in-1 oil to the clean surface of the pin.\n" }}{PARA 0 "" 0 " " {TEXT -1 760 "\011Before it was investigated whether or by how much \+ the pendulum wire stretched at the point of maximum velocity, the micr ometer was calibrated by the measurements of a one-inch and a one-and- a-half inch standard. These two standards were chosen because the mea surement for which the instrument was intended falls between these two values. Each measurement was made by carefully surrounding the stand ard with the micrometer measuring surfaces such that the planes of tho se surfaces were flush with the opposing surfaces of the standard. In this way, damage to both the standard and the micrometer due to misal ignment was avoided. Any damage to the standard, or to the micrometer would cause a corresponding change in any measurements made by the mi crometer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 857 "After the calibration, a stack of plastic protractors, used for s pacers, was made on top of a wooden box below the pendulum's lowest po int. The stack was made such that the pendulum bob and the stack made contact, but did not allow the bob to rest upon the stack. One space r was then removed. The pendulum was then raised to higher angles and allowed to swing passed the stack. The pendulum angle was increased \+ until the pendulum just contacted with the stack again. This contact \+ was confirmed by sound and by the slight motion of the plastic spacer. At this point, the spacer removed at the beginning was measured usin g the micrometer. The one-inch standard was used to fill the gap in t he micrometer, as it only reduced to one inch and the spacer thickness was less than one inch. The resulting final angle and spacer thickne ss was then recorded.\n" }}{PARA 0 "" 0 "" {TEXT -1 220 "In order to e nsure that no length change had occurred during experimentation, a fin al one hundred cycle episode was measured at a small angle (less than \+ five degrees), and the time recorded (Table 1) for later analysis.\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 262 16 "Expected Results" }}{PARA 0 "" 0 "" {TEXT -1 309 "At this point, we know about the theoretical and physical components of the pendulum. What then do we expect to find \+ when we compare the experimental periods with those of theory? Let us take into consideration each of the major biases inherent in the expe riment and the associated effects upon the period. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT 263 14 "Air Resistance" }}{PARA 0 "" 0 "" {TEXT -1 313 "Certainly, the object in motion will, in a room setting, be acted upon by air resistance. The only ways in which to eliminate the effe cts of air resistance are to swing an actual particle (thereby reducin g the surface area exposed), or swing the macroscopic object in a vacu um (taking away the resisting medium)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 302 "The air will resist the motion of th e pendulum. This will in turn cause the pendulum to slow. Since the \+ distance covered is assumed to be constant at this point, the conseque nce of less speed is that it will take more time to cover the same dis tance. The period can then be expected to grow longer. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 264 21 "Friction at the Pivot" }}{PARA 0 "" 0 " " {TEXT -1 284 "Here is another example of real life getting in the wa y. The pivot point at the top of the pendulum system has friction eve n when lubricated. The friction of the pivot point is another resisti ng force. It can therefore be expected that it too would cause the pe riod to grow longer." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 265 13 "Buoyant Force" }}{PARA 0 "" 0 "" {TEXT -1 433 "The buoyant force is a force a cting upward on an object (keeping it afloat... or trying to anyway). \+ It is equivalent to the weight of the liquid (like water, air, etc.) \+ displaced by the object. Though this force exists and would cause a c hange in the mass of the pendulum, it would not affect the period of t he pendulum. The period of a pendulum, unlike a spring's harmonic mot ion, is not dependent upon the mass (see derivation)." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 266 29 "Decay in Angular Displacement" }}{PARA 0 "" 0 "" {TEXT -1 591 "The decay in the angle through which the pendu lum swings is a result of those resisting forces discussed above. If \+ there are no resisting forces, there is no decay. The decay in angula r displacement is seen as a decay in the amplitude of the harmonic mot ion, but hold the phone! The lowered amplitude is not the only effect . The lowered amplitude means that the average angle is less than tha t originally displaced. The lower angle means less distance to travel and therefore less time to do so (see theory). It can be expected th at the decay will cause the period to lessen somewhat. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 286 21 "So What Do We Expect?" }}{PARA 0 "" 0 " " {TEXT -1 276 "The resulting periods should follow the general trend \+ of the theoretical computations. The amplitude should lower over time . The period should be some combination of lowering and raising due t o the decay in angle and resisting forces. Let us take a look at some real life. " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 60 "Animation for the Pendulum Wavefunction (under construction)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "Animation for the decrease in amplitude of the pendulum \+ wavefunction. As the phase angle increases, the period changes (the a mplitude changes as well if take out the w squared).." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "d^2*x/(dt^2) = -omega^2* x0cos(omega*t+phi)" "6#/*(%\"dG\"\"#%\"xG\"\"\"*$%#dtG\"\"#!\"\",$*&%& omegaG\"\"#-%&x0cosG6#,&*&F/F(%\"tGF(F(%$phiGF(F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart;with(plots):eq:=diff(x(t),t$2)=w^ 2*cos(w*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/-%%diffG6$-%\"xG 6#%\"tG-%\"$G6$F,\"\"#*&)%\"wGF0\"\"\"-%$cosG6#*&F3\"\"\"F,F9F9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "de:=dsolve(\{eq,x(0)=0,D(x)( 0)=0\},x(t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%\"xG6#%\"tG ,&-%$cosG6#*&%\"wG\"\"\"F)F0!\"\"F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fin:=unapply(rhs(de),t,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$finGR6$%\"tG%\"wG6\"6$%)operatorG%&arrowGF),&-%$cosG 6#*&9%\"\"\"9$F3!\"\"F3F3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate(fin(t,w),t=0..70,w=0.1..0.5,frames=20,colour=red,numpo ints=200);" }}{PARA 13 "" 1 "" {GLPLOT2D 353 217 217 {PLOTDATA 2 "6%-% (ANIMATEG667#-%'CURVESG6$7dw7$\"\"!F,7$$\"+Sze 7$$\"+ze')>C))*F97$$\"+qRzehu-\"!#;7$ $\"+;JiC\\F=$\"1=#QcU'G)=\"F\\p7$$\"+5>Qw_F=$\"1))Q1QO,g8F\\p7$$\"+.29 GcF=$\"1W_Xq.VU:F\\p7$$\"+(\\**)zfF=$\"1*y5T$4JNF\\p7$$\"+&3n ZF\\p7$$F.Rp&F\\p7$$\"+?S!3;\"Fjt$\"1;9=xs(R,'F\\p7$$\"+**)zf> \"Fjt$\"1rU$)GH)*QjF\\p7$$\"+zd:J7Fjt$\"1L.SX!=&omF\\p7$$\"+e;Lm7Fjt$ \"1#Q>6#\\<-qF\\p7$$\"+Qv],8Fjt$\"1>$H2vS&RtF\\p7$$\"+S\")>!o(F\\p7$$\"+'Hf=P\"Fjt$\"1uU6BcsB!)F\\p7$$FBFjt$\"1vE;#=)pp$)F \\p7$$\"+b5@U9Fjt$\"1<5$\\x(o<()F\\p7$$\"+NpQx9Fjt$\"1Qf]jQEn!*F\\p7$$ \"+9Gc7:Fjt$\"11y8ZR*zT*F\\p7$$\"+$pQxa\"Fjt$\"1)4Pt4W%p(*F\\p7$$\"+tX \"He\"Fjt$\"1?PZ\\z675!#:7$$\"+_/4=;Fjt$\"1Qfq'\\ws/\"F[z7$$\"+KjE`;Fj t$\"1]9v_lP#3\"F[z7$$\"+6AW)o\"Fjt$\"10vH\"put6\"F[z7$$\"+!4=Os\"Fjt$ \"12_77wA_6F[z7$$FGFjt$\"1$=\"o%>#*o=\"F[z7$$\"+\\)pRz\"Fjt$\"1zpU^bK@ 7F[z7$$\"+Hd9H=Fjt$\"1k,!43&[b7F[z7$$\"+3;Kk=Fjt$\"1)[@/_G$*G\"F[z7$$ \"+([(\\**=Fjt$\"11HQ)*R\"GK\"F[z7$$\"+nLnM>Fjt$\"1_<0'3+fN\"F[z7$$\"+ Y#\\)p>Fjt$\"1%)p#)[ea)Q\"F[z7$$\"+E^-0?Fjt$\"1(3%)o*3r?9F[z7$$\"+05?S ?Fjt$\"1hf9NaN_9F[z7$$\"+%)oPv?Fjt$\"1!*z^7.W$[\"F[z7$$FMFjt$\"1`jKqq# R^\"F[z7$$\"+V'Gd9#Fjt$\"1u'**)*)zxV:F[z7$$\"+BX!4=#Fjt$\"1B^BRh&Hd\"F [z7$$\"+-/3;AFjt$\"1S7p=aU,;F[z7$$\"+\"Gc7D#Fjt$\"1]4l01:H;F[z7$$\"+h@ V'G#Fjt$\"17&*4)R(4c;F[z7$$\"+S!3;K#Fjt$\"1![jqXKAo\"F[z7$$\"+?RycBFjt $\"1!=eyWBvq\"F[z7$$\"+*zf>R#Fjt$\"1kE4!3R>t\"F[z7$$\"+yc8FCFjt$\"1ZEQ Y\"\\av\"F[z7$$FRFjt$\"1PNsfX-yRyc#Fjt$\"1vzOj&p)R=F[z7$$\"+v],. 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It looks like a couple of \+ pendulums swinging, with displacement along the y-axis (so the pendulu m is sideways). As the p-angle is doubled, so too is the period (?) f or one oscillation. Try both at the same time!!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "d^2*x/(dt^2) = -omega^2*x0cos(ome ga*t+phi)" "6#/*(%\"dG\"\"#%\"xG\"\"\"*$%#dtG\"\"#!\"\",$*&%&omegaG\" \"#-%&x0cosG6#,&*&F/F(%\"tGF(F(%$phiGF(F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq2:=diff(x(t),t$2)=-0.01^2*cos(0.01*t+p):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "de:=dsolve(\{eq2,x(0)=0,D(x) (0)=0\},x(t)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fin:=una pply(rhs(de),t,p):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "anima te(fin(t,p),t=0..100,p=0..6.3,frames=20,colour=blue,numpoints=200);" } }{PARA 13 "" 1 "" {GLPLOT2D 399 140 140 {PLOTDATA 2 "6%-%(ANIMATEG667# -%'CURVESG6$7dw7$\"\"!F,7$$\"+Gc7D]!#5$!1.x,A7$$\"+^-05?F7$!1w$p.*H3??F?7 $$\"+9Gc7DF7$!1^J4%*)>j:$F?7$$\"+x`2:IF7$!1[]NRa*\\a%F?7$$\"+SzeQw_&F7$!1d!z-C]t_\"Fhn7$$\"+a2:Ig 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ggl7$Fe]n$!11X'4>NN5$Fggl7$Fj]n$!1s?8v`gRJFggl7$F_^n$!1eMf*pTe<$Fggl7$ Fd^n$!1%\\\\m?VA@$Fggl7$Fi^n$!1a)HW$*3)[KFggl7$F^_n$!1)H[!G4%Fggl7$Fjen$!1e#4R#H(G8%Fggl7$F_ fn$!1!H([x43tTFggl7$Fcfn$!1mOeWFggl7$Ffhn$!1*y5r)el*\\%Fggl7$F[in$!1+sE ea2TXFgglF`in-%+AXESLABELSG6$%\"tG%!G-%%VIEWG6$;F,$\"$+\"F,%(DEFAULTG " 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 267 33 "Experimental Results and Analysis" }} {PARA 0 "" 0 "" {TEXT -1 130 "The experimental results can be summed u p with the following comparative plot of theoretical and experimental \+ period verses angle." }}{SECT 1 {PARA 4 "" 0 "" {TEXT 295 12 "Results \+ Plot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 300 " The following plot shows the individual values for the experimental pe riod (red), the corresponding theoretical periods (blue), and the rang e between which the experimental values are known (an error range). T his range (maroon) is given to illustrate the proximity of the data po ints to one another." }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 "Plot comm ands" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "experimental:=\{[5,1.745], [10,1.748],[15,1.751],[20,1.756],[25,1.766],[30,1.77],[35,1.78],[40,1. 80],[45,1.81]\}: theoretical:=\{[5,1.745],[10,1.748],[15,1.752],[20,1 .758],[25,1.766],[30,1.775],[35,1.786],[40,1.799],[45,1.814]\}: errorh igh:=\{[5,1.75],[10,1.753],[15,1.756],[20,1.761],[25,1.773],[30,1.78], [35,1.79],[40,1.81],[45,1.82]\}: errorlow:=\{[5,1.74],[10,1.743],[15,1 .746],[20,1.751],[25,1.759],[30,1.76],[35,1.77],[40,1.79],[45,1.80]\}: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "with(plots): explot:=pointplot (experimental,colour=red,labels=[\"angle (degrees)\",\"period (seconds )\"],title=\"Comparison of Theory and Experimental Pendulum Periods (w ith Error Zone)\"): thplot:=pointplot(theoretical,colour=blue): highpl ot:=pointplot(errorhigh,colour=maroon,style=line): lowplot:=pointplot( errorlow,colour=maroon,style=line): display(\{explot,thplot,highplot,l owplot\});" }}}{PARA 13 "" 1 "" {TEXT -1 0 "" }{GLPLOT2D 536 324 324 {PLOTDATA 2 "6(-%'POINTSG6,7$$\"\"&\"\"!$\"%X%Fen-% &STYLEG6#%%LINEG-F^p6%7+7$F'$\"$v\"Fao7$F.$\"%` " 0 "" {MPLTEXT 1 0 168 "decay:=\{[5,0.5],[10,1],[15,1],[20,3.5],[25,6.0],[30 ,10],[35,12.5],[40,15],[45,15]\}: errhigh:=\{[5,1],[10,1.5],[15,1.5],[ 20,4]\}: errlow:=\{[5,0],[10,0.5],[15,0.5],[20,3]\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "dplot:=pointplot(decay,colour=blue,title=\"Ampl itude Decay verses Displacement\",labels=[\"angle (degrees)\",\"decay \+ (degrees)\"]): ehplot:=pointplot(errhigh,style=line,colour=red): elplo t:=pointplot(errlow,style=line,colour=red): display(\{d plot,ehplot,elplot\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 " " 1 "" {GLPLOT2D 458 265 265 {PLOTDATA 2 "6'-%'CURVESG6%7&7$$\"\"&\"\" !$\"\"\"F*7$$\"#5F*$\"#:!\"\"7$$F1F*F07$$\"#?F*$\"\"%F*-%'COLOURG6&%$R GBG$\"*++++\"!\")F*F*-%&STYLEG6#%%LINEG-%'POINTSG6,7$F($F)F27$F.F+7$F4 F+7$F6$\"#NF27$$\"#DF*$\"#gF27$$\"#IF*F.7$$FNF*$\"$D\"F27$$\"#SF*F47$$ \"#XF*F4-F;6&F=F*F*F>-F$6%7&7$F(F*7$F.FI7$F4FI7$F6$\"\"$F*F:FA-%+AXESL ABELSG6$Q0angle~(degrees)6\"Q0decay~(degrees)Fjo-%&TITLEG6#QDAmplitude ~Decay~verses~DisplacementFjo" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 299 14 "Error Analysis" }} {PARA 0 "" 0 "" {TEXT -1 82 "Error analysis was conducted in a standar d fashion using the following guidelines." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "- The uncertainty for summed (or s ubtracted) values " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "z = x+y" "6#/%\"zG,&%\"xG\"\"\"%\"yGF'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "was cal culated using" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "delta*z = delta*x+delta*y" "6#/*&%&delta G\"\"\"%\"zGF&,&*&F%F&%\"xGF&F&*&F%F&%\"yGF&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "- The uncertainty for mul tiplied (or divided) values " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "z = xy" "6#/%\"zG%#xyG" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "was calcu lated using fractional uncertainties, as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "delta*z/ z = delta*x/x+delta*y/y" "6#/*(%&deltaG\"\"\"%\"zGF&F'!\"\",&*(F%F&%\" xGF&F+F(F&*(F%F&%\"yGF&F-F(F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "- The uncertainty for averaged values " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "t = (t1+t2+t3)/3" "6#/%\"tG*&,(%#t1G\"\"\"%#t2GF(%#t3GF (F(\"\"$!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "was said to have improved over their individual uncertai nties by the square root of the number of data points averaged." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "delta*t = (delta*t1+delta*t2+delta*t3)/(3*sqrt(3))" "6# /*&%&deltaG\"\"\"%\"tGF&*&,(*&F%F&%#t1GF&F&*&F%F&%#t2GF&F&*&F%F&%#t3GF &F&F&*&\"\"$F&-%%sqrtG6#\"\"$F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 362 "The results of the experiment point \+ to a result very similar to the one predicted. The amplitude decrease d due to the frictional factors in the apparatus. The decrease in amp litude caused a decrease in the period enough to counter the increases expected from the air resistance (and pivot friction) within the rang e of uncertainties in the experimental period." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Due to the counteractive effects of the biases, a detailed analysis of the damping coefficient was not performed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 210 "A final point of interest is that, with a simplificati on of the above procedures, the simple pendulum can be used to determi ne within a reasonable error, the value of gravitational acceleration \+ at any location." }}{SECT 1 {PARA 4 "" 0 "" {TEXT 268 8 "Raw Data" }} {PARA 0 "" 0 "" {TEXT -1 14 "The value for " }{TEXT 292 26 "gravitatio nal acceleration" }{TEXT -1 4 " is " }{TEXT 291 8 "979.565 " }{TEXT -1 3 "+/-" }{TEXT 289 6 " 0.005" }{TEXT -1 1 " " }{TEXT 288 2 "cm" } {TEXT -1 1 "/" }{XPPEDIT 18 0 "s^2" "6#*$%\"sG\"\"#" }{TEXT -1 2 ".\n " }}{PARA 0 "" 0 "" {TEXT 293 8 "Table 1 " }{TEXT -1 2 "- " }{TEXT 294 96 "Total Time and Angular Displacement Decay as a Function of Dis placement and the Number of Swings" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 290 41 "degree swings time deca y" }}{PARA 256 "" 0 "" {TEXT -1 43 "5.0 100 174.49 \+ 0.5" }}{PARA 257 "" 0 "" {TEXT -1 42 "10.0 100 1 74.77 1.0" }}{PARA 258 "" 0 "" {TEXT -1 42 "15.0 100 \+ 175.09 1.0" }}{PARA 259 "" 0 "" {TEXT -1 42 "20.0 \+ 100 175.62 3.5" }}{PARA 260 "" 0 "" {TEXT -1 44 "25.0 \+ 50 88.31 3.0" }}{PARA 261 "" 0 "" {TEXT -1 44 "30.0 20 35.48 2.0" }}{PARA 262 "" 0 "" {TEXT -1 44 "35.0 20 35.66 2.5" }}{PARA 263 "" 0 "" {TEXT -1 44 "40.0 20 35.94 3.0 " }}{PARA 264 "" 0 "" {TEXT -1 44 "45.0 20 36.25 \+ 3.0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 269 12 "Calculations" }}{PARA 0 "" 0 "" {TEXT -1 125 "The calcul ation for the length of the pendulum wire was obtained by first findin g the average time of five small angle runs.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "averagetime(t) = (t1+t2+t3+t4+t5)/5" "6# /-%,averagetimeG6#%\"tG*&,,%#t1G\"\"\"%#t2GF+%#t3GF+%#t4GF+%#t5GF+F+\" \"&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "averagetime(t) = (174.50*s+174.45*s+174.49*s+174. 43*s+174.49*s)/5" "6#/-%,averagetimeG6#%\"tG*&,,*&$\"&]u\"!\"#\"\"\"% \"sGF.F.*&$\"&Xu\"!\"#F.F/F.F.*&$\"&\\u\"!\"#F.F/F.F.*&$\"&Vu\"!\"#F.F /F.F.*&$\"&\\u\"!\"#F.F/F.F.F.\"\"&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "averagetime(t) = \+ 174.47*s" "6#/-%,averagetimeG6#%\"tG*&$\"&Zu\"!\"#\"\"\"%\"sGF," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Then with this time, the period for the small angle run was determined by divid ing the time (" }{TEXT 297 1 "t" }{TEXT -1 39 ") by the number of swin gs for one run (" }{TEXT 296 1 "N" }{TEXT -1 3 ").\n" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{XPPEDIT 18 0 "tau = t/N" "6#/%$tauG*&%\"tG\"\"\"%\"N G!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{XPPEDIT 18 0 "tau = 174.47*s/100" "6#/%$tauG*($\"&Zu\"!\"#\"\"\"% \"sGF)\"$+\"!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "tau = 1.7447*s" "6#/%$tauG*&$\"&Zu\"!\"% \"\"\"%\"sGF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "With the period (" }{XPPEDIT 18 0 "tau" "6#%$tauG" } {TEXT -1 57 "), the length was then found using the following formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "tau = 2*pi*sqrt(l/g)" "6#/%$tauG*(\"\"#\"\"\"%#piGF'-%% sqrtG6#*&%\"lGF'%\"gG!\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 34 "\nRearran ging to solve for length (" }{TEXT 298 1 "l" }{TEXT -1 24 ") yields th e following.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "l = (t au/(2*pi))^2*g" "6#/%\"lG*&*&%$tauG\"\"\"*&\"\"#F(%#piGF(!\"\"\"\"#%\" gGF(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "l = (1.7447*s/(2*pi))^2*9.79565*m/(s^2);" "6#/%\"lG*** ($\"&Zu\"!\"%\"\"\"%\"sGF**&\"\"#F*%#piGF*!\"\"\"\"#$\"'l&z*!\"&F*%\"m GF**$F+\"\"#F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "l = 75.5*cm" "6#/%\"lG*&$\"$b(!\"\"\"\" \"%#cmGF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "The periods for each new displacement angle during the experiment \+ were found using the equation\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "tau = t/N" "6#/%$tauG*&%\"tG\"\"\"%\"NG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "as before. Fo r comparison, however, the theoretical periods were found using the pe riod equation that follows to " }{XPPEDIT 18 0 "sin^4" "6#*$%$sinG\"\" %" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "t au = 2*pi*sqrt(L/g)*(1+sin(theta/2)^2/(2^2)+1^2*3^2*sin(theta/2)^4/(2^ 2*4^2))" "6#/%$tauG**\"\"#\"\"\"%#piGF'-%%sqrtG6#*&%\"LGF'%\"gG!\"\"F' ,(\"\"\"F'*&-%$sinG6#*&%&thetaGF'\"\"#F/\"\"#*$\"\"#\"\"#F/F'**\"\"\" \"\"#\"\"$\"\"#-F46#*&F7F'\"\"#F/\"\"%*&\"\"#\"\"#\"\"%\"\"#F/F'F'" }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT 284 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 231 "* Physics, 4th edition, Volume 1, Resnick, Halliday, and Krane. pp324-330.\n** Dr Joe Pollock, Dept of Physics and Astro nomy\n*** Information Circular 18 of the North Carolina Department of \+ Conservation and Development (1963).\n" }}}}{MARK "1 0 0" 8 } {VIEWOPTS 1 1 0 1 1 1803 }