The Pendulum. Purpose


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1 The Pedulum Purpose To carry out a example illustratig how physics approaches ad solves problems. The example used here is to explore the differet factors that determie the period of motio of a pedulum. Apparatus Meter stick, stopwatch, strig, weights, ad a stad. I. Prelimiary Discussio The fist step is to ask the questio, i this case what are the quatities which affect the period (or frequecy) of a simple pedulum, ad determie quatitatively how they effect the period. May systems, both atural ad mamade, udergo periodic motio. Some examples iclude the beatig heart, the electros i a radio atea, metroomes, ad atoms i solid materials. Other simple systems also exhibit harmoic motio, such as the motio of a pedulum ad that of a mass attached to a sprig. The motio of all such systems has a period, T, which is determied by the elemets that make up the system. The period is simply the amout of time that the motio takes to complete oe cycle. Sometimes we describe a periodic motio by its frequecy, f, which is simply 1/T. For example, Old Faithful (Yellowstoe Natioal Forest) erupts oce every hour*. Old Faithful s period is oe hour, ad its frequecy is oce per hour. Period is usually measured i the most coveiet uit of time (secods, miutes, hours, etc.), while frequecy is ofte measured i Hertz (1 Hz = 1 cycle/secod) or Mega Hertz (1 MHz = 1000 Hz). I this experimet, we will test the differet elemets of a pedulum to determie which oes affect the period ad frequecy of the system. * Old Faithful's actual period actually varies betwee 35 miutes ad 2 hours. The iterval betwee successive eruptios however ca be quite accurately predicted based upo the duratio of the last eruptio.
2 II. Experimet A. The Pedulum Experimetal Procedure There are three pedulum elemets that we ca easily chage: the suspeded mass, the iitial amplitude, ad the strig's legth. We will test each of the elemets idividually to see which oe(s) affect the period of the pedulum. Record your data directly i a spreadsheet to save writig. Set up your pedulum by hagig a 200g mass from your stad with approximately 60cm of strig. Costruct a table usig your spreadsheet to record your data. Now, accurately measure the legth of the strig. NOTE: The legth is the distace from where the strig first touches the support stad to the ceter of gravity of the suspeded mass. Be very coscious of the chage i ceter of gravity height as you add or subtract weights from the pedulum. Record this legth ad the amout of suspeded mass i your table. We will first test the amplitude's effect o the period. Start with a small iitial amplitude (aroud 510 cm). Measure the amplitude horizotally from where the strig would hag aturally to where you are releasig the mass. Drop the mass from this amplitude ad measure how log it takes for the pedulum to make 20 cycles. HINT: If you release the mass from the left side of the stad, start your stop watch whe the mass completes its first full swig. The time 20 cycles. This will reduce much of the error iheret i tryig to start the stopwatch at the same time as the mass is released. Measure the time for 20 cycles two times ad record the average of the two readigs. Now icrease your amplitude by at least 5cm ad take 2 readigs of the time for 20 cycles agai. IMPORTANT NOTE: If you do ot otice a chage i the measured time, icrease the amplitude agai ad take oe more set of data, completig data for 3 differet amplitudes. If there is a differece i the time readigs, cotiue takig measuremets util you have data for 5 differet amplitudes, otherwise cotiue to the ext step below. Do ot take uecessary data. Now, test how the mass affects the period. This time do the etire experimet, chagig oly the mass. Keep the pedulum at the legth of 60 cm (ote that you may have to adjust your strig legth whe you chage the mass) ad a amplitude of your choosig. Choose weights betwee 20g ad 500g. Record your data for at least 3 differet masses. If you fid that mass chages the period, cotiue util you have data for 5 differet masses; otherwise proceed to the ext step. Next, test how the legth affects the period. For this part, use a pedulum
3 made up of a strig ad a small lead weight. Do the etire experimet, chagig oly the legth of the pedulum, usig legths from the smallest you ca achieve startig with these three legths: 4cm, 7cm ad 10cm. Record data for up to 5 differet legths if you otice a chage i the period of the pedulum. You ca use legths up to 100 cm for the other legths, if you eed more tha three legths. B. Experimetal Aalysis First, for each of the average time measuremet you have recorded, calculate the correspodig period, Timefor20cycles T = 20 Geerate a table with the amplitude i oe colum ad the correspodig average period i the ext colum to the right. Similarly, costruct a table for mass ad the correspodig average period, ad a table for legth ad the correspodig average period. Usig the spreadsheet s graphig fuctios, costruct graphs of period vs. amplitude, period vs. mass, ad period vs. strig legth. To graph your data easily make sure the legth, or mass, or amplitude are i a sigle colum, ad your correspodig period T, i the adjacet colum just to the right of it, with o spaces i the colums. If you did ot record your data this way you may easily geerate these adjacet colums by copyig ad pastig the data. Now select (highlight) these two colums. Now click o the chart wizard, third ico from the right o the tool bar which appears as various colored vertical bars. The chart type meu will appear Click o the xy scatter plot. This is the oly optio which will produce a xy plot i Excel. Click o ext. The graph should appear ad if it satisfactory click ext agai ad a meu to allow you to label axis ad title the graph will appear. Label your axes ad title the graph ad click fiish. You may adjust the size ad shape by clickig aywhere iside the border ad draggig the outlie with the poiter at oe of the small squares o the outlie. Make graphs of T (the period) versus each of the three variables, mass, legth ad amplitude. NOTE: I the graphs of period vs. amplitude ad the period vs. mass, you eed to adjust the scale o the y axis (set miimum to zero, maximum to 2) to get a better picture of the relatioship betwee these two variables. Make a pritout of these graphs for your report. It should be quite obvious ow which elemets affect the period of the pedulum. You should ow fid that oe of your graphs does ot give a straight horizotal lie plot ad looks like a curve. From this iformatio, ca we ow develop a algebraic expressio which relates the period to the quatities which cause it to chage? There are a ifiite umber of relatioships that are possible, but fortuately, may simple physical systems obey fairly simple relatioships. For example, the relatioship could be liear ( y = mx + b) where y could be the
4 period ad x the mass. Aother possibility would be a expoetial relatioship, kx y = e. You might recogize this as the radioactive decay law. Eve more commo is the power law behavior, y = Ax. The parameters that affect the period of the pedulum obey the power law relatioship. Power laws are difficult to work with, but there is a easy trick to simplify the procedures. By usig logarithms, we ca tur y = Ax ito log y = log x + log A. This looks very similar to the equatio of a straight lie ( y = mx + b). Therefore, if we costruct a plot of logy vs. log x, the slope of this lie will give us the expoet we are lookig for. (We will ot cocer ourselves with the costat A i this lab. It is a simple exercise to fid oce we have foud the power expoet). Select the graph for the elemet, or elemets, which affected the period of the pedulum. For this elemet which affects the period, we ow say that this elemet obeys a power law relatioship with the period of the pedulum, ad hece ca be aalyzed usig logarithms as show the previous paragraph. So ow, create a plot of Log T vs. log of the period affectig elemet. This will covert the curved plot to a straight lie log/log plot. Iclude a pritout of this graph i your report. Get the slope of the straight lie log/log plot. This slope gives us the expoet i the power law equatio relatig the period T ad the period affectig elemet. Usig the power law equatio y = Ax, get the equatio for the period of the pedulum by substitutig the period T for y, your slope for, ad replace x with a, m, or l (depedig o whether the amplitude, mass, or legth was the
5 elemet that affected the period). Cogratulatios, you have deduced the equatio for the period of a pedulum! III. Discussio Questios 1. Which of the three elemets (iitial amplitude, mass, or strig legth) affected the period of the pedulum? 2. Cosider two pedulums: Pedulum A has strig legth l, Pedulum B has strig legth 2l. Which pedulum has the logest period? Is it twice the period of the other pedulum? 3. Pedulum A has twice the mass of pedulum B. The force due to gravity actig o Pedulum A (2mg) is twice the magitude of the force due to gravity actig o Pedulum B (mg). Why the does't Pedulum A have twice the frequecy (half the period) of Pedulum B? 4. What is the actual equatio for the period of the pedulum (recall from lecture or search o the Web). What does it predict for the expoet ad how does it compare to the value of obtaied i your experimet? What is the value of the costat A i the actual equatio for the period of the pedulum? Origial Versio by Dr. Bill Moulto Modified for PHY2048 by B. Reyes, 4/2/2010
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