Kater Pendulum. Introduction. It is well-known result that the period T of a simple pendulum is given by. T = 2π

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1 Kater Penulum ntrouction t is well-known result that the perio of a simple penulum is given by π L g where L is the length. n principle, then, a penulum coul be use to measure g, the acceleration of gravity. However, practical ifficulties primarily in measuring the length accurately make it unsatisfactory for high precision measurements. t turns out that a physical penulum typically, a mass suspene from a knife ege can be use for much more accurate measurements. he Kater penulum is such a physical penulum. t is name after its inventor, Captain Henry Kater, a captain in the British army an a Fellow of the Royal Society, who invente it aroun 80. For many years, it was the stanar metho of measuring g to high accuracy. he beauty of this experiment is that it allows us, using relatively simple apparatus, to measure a physical quantity very accurately inee we hope, to within a few parts in 0,000. t works as follows: Consier a physical penulum with two supports that lie along a line through the center of mass, as shown in Figure. center of mass Figure Kater Penulum he penulum can oscillate about a knife ege through either hole. he holes are locate so that the two perios of oscillation are nearly equal. Suppose the istances an are ajuste so that the perios aroun both supports are equal:. hen it can be shown that this perio is the perio of a simple penulum with length, the istance between the supports. hus, if the perio an that istance can both be measure accurately, one can fin the acceleration of gravity: π g g or b g ()

2 heory n this section, we give a etaile erivation of Equation (). he perio of a physical penulum is π Mg where is the moment of inertia about the axis of rotation of the penulum. From the Parallel Axis heorem, it is apparent that c M where c is the moment of inertia about the center of mass. hus for our physical penulum in Figure, we have c M π an π Mg c M Mg But we can always write c in terms of the raius of gyration k, the raius of a cylinrical ring that has the same mass M an the same rotational inertia as the actual object: c Mk Hence the equation for the perios can be written or simplifying, Mk M π an π Mg Mk M Mg k k π an π () g g (Note that by using the raius of gyration, we have eliminate the mass. his step is not necessary, but it makes the rest of the erivation less cumbersome.) f we have ajuste the istances an so that the perios are the same, it follows that k k We bring all of the k terms to the left sie an rearrange as follows: k k t follows immeiately that k k F HG F HG k KJ KJ

3 f we substitute this result into Equation (), we fin Simplifying, we fin the result we were seeking! π an π g π, g g Apparatus Our Kater penulum consists of a inch wie by ¼ inch thick brass bar, as shown in Figure. We have several lengths available. L Figure he vertical line marks the center of mass. a is the istance from the en of the bar to the ege of the first hole. a here are two ways of oing the experiment. You may try either one. As you will see, it takes some time to take the ata for Metho, but the analysis is very simple. By contrast, it oes not take long to take the necessary ata using Metho, but the analysis is much more complex. We aren t sure yet which metho is more accurate. he experiment is new at CSB/SJU, an there are quite possibly systematic errors that we o not yet unerstan. Metho We attach sliing weights (for the moment, large paper clips) to the bar, an measure the perios an about each hole as functions of the position of the weights, measure by the centimeter scale tape on the bar. t will turn out that at some position of the weights, the perios will be nearly equal. his position is usually someplace between the two holes. One must take some preliminary measurements to fin out where that position is. hen, one takes careful measurements of perio vs. weight position for several centimeters on either sie of the position at which the two perios are about equal. hus one has two sets of ata: Perio vs. weight position for each support point. Do a leastsquares fit to each set, an then plot both sets, with fits, on the same graph. he two fit lines will intersect each other on the graph of Perio vs. weight position. hat point of intersection is the 3

4 point at which the two perios are exactly equal, an one can then rea that value of the perio from the graph. Once this work is one, the analysis is straightforwar: From the graph, one fins the perio from the graph an then uses Equation () to fin g. he calculation of the uncertainty is also straightforwar. Metho t is much easier an quicker to attach weights to the bar in such a way that is nearly equal to, an measure both perios accurately. However, the analysis is consierably more involve. At the cost of a little algebra, one can fin an expression for g that epens on both an. We begin by squaring Equation () above to obtain k h k an g h g We solve each of these equations for k an equate, as follows: h h g g k We collect all the terms involving g on the left han sie, as follows: Or, rearranging, we obtain g π 4 c h g his result is perfectly correct. But it is not in a form that is particularly useful in this experiment, where the quantity we can measure irectly is the istance between the supports,. For this reason, we will try to fin quantities A an B that satisfy an equation of the form A B g he algebra takes a few lines, an so we will work it out in Appenix. he result is (4) g ( ) ( ) ake a moment to look at this equation, an remember that we are trying to measure g very accurately. t turns out that casting Equation (3) in this form allows us to o so. he first term in the brackets contains three quantities two times, an, an the istance that we can measure very accurately. he secon term contains two quantities, an, that we know much less accurately; but if an are nearly equal, the secon term will be much smaller (3) 4

5 than the first, an so the uncertainties in an will turn out to matter less. he result will be a very accurate measurement! Proceure (both methos) ake a few minutes to look over the apparatus. he penulum is suspene from a knife ege mounte rigily in a bench clamp. We measure the perio using a photogate etector connecte to the Pasco Science Workshop software. Your instructor will show you how this system works. Play aroun with it for a few minutes until you are comfortable with it. here are a number of places where systematic error (as oppose to ranom error) can creep into this experiment. n an experiment esigne for high accuracy, it is important to look for an eliminate as many sources of systematic error as possible. t is equally important to monitor your ata carefully, with rough graphs, while you are taking it. Recoring a bunch of numbers, an waiting until later to see what the graphs or calculations look like, is a recipe for isaster! Here are some things to watch out for:. t is important that the knife ege support be accurately level. Use a level to check. f the knife ege is not level, place small pieces of paper or foil uner the knife ege or one of the clamps until you get it as nearly level as you can.. he penulum shoul oscillate back an forth in a plane. Be sure it is not wobbling or moving from sie to sie. t s not a ba iea to start it oscillating, an then let it go for a few minutes, before you measure the perio. 3. he plane in which penulum oscillates shoul be perpenicular to the knife ege. 4. he perio of a penulum epens slightly on the amplitue of the oscillation. he equations we use, therefore, are strictly vali only in the limit of very small amplitue. Consequently, keep the amplitue as small as possible. f time permits, see if you can etect the small increase in perio as the amplitue is increase. 5. he experiment is sensitive to vibration. For example, try banging on the table, or even blowing gently on the penulum, while you measuring the perio. he effect is striking. Look for ways to eliminate vibration as much as possible. For example, place the computer mouse on a ifferent surface, so that pressing it to start the experiment will not introuce a source of vibration. here are at least two other possibilities for systematic error: the quartz clock in the Pasco interface box that is use to measure the perio, an the micrometers an calipers that we use to measure istances. t turns out that many sources of systematic error have the effect of lengthening the perio. hus, if systematic errors are present, the value of g will ten to be systematically low. he proceure is as follows: For Metho :. Make some initial measurements to fin the point at which the perios are approximately equal. 5

6 . For several centimeters on either sie of this point, make measurements of perio vs. clamp position, at about 0.5 cm to cm. intervals, for both supports. Be sure to mount the clamps symmetrically on the bar. For each measurement, recor the average of 6 or 8 perios an the stanar eviation, an calculate the stanar eviation of the mean. For later analysis, it can be convenient to recor these measurements irectly in the Linfit spreasheet. 3. For each ata set, make a graph an o a least-squares fit. A fit to a straight line will probably be aequate, but in some cases, the ata may show enough curvature that one shoul fit to a quaratic. he point where the two curves intersect gives the perio neee for Equation (). 4. Measure (the istance between the two supports). t is important to make this measurement as accurately as possible. Your instructor will show you how to o these measurements accurately. Please hanle the calipers an micrometers carefully! For Metho :. Make an initial measurement of the two perios, fairly roughly.. Mount two paper clamps on the penulum. he iea here is to fin, by trial an error, a location that makes the ifference in the perios as small as possible. Be sure to mount the clamps symmetrically on the bar. A goo choice to start is a position between hole an the center of the bar. Once you have foun an optimum position, recor it in your lab notebook. 3. With the penulum oscillating about one support, take about 00 measurements of the perio. Recor the average perio an the stanar eviation, an calculate the stanar eviation of the mean. 4. Repeat for the secon support. 5. Measure the istances a (the istance from the en of the bar to the first hole), (the istance between the two supports), an L (the length of the penulum) as accurately as you can. Your instructor will show you how to o these measurements accurately. Please hanle the calipers an micrometers carefully! hey are sensitive, an moerately expensive. Both methos:. One can set the Pasco software to stop taking ata after a set elapse time thus, one can start timing the penulum, an the program will stop itself, an list each reaing, the number of oscillations, the mean value, an the stanar eviation. Your instructor will show you how this feature of the program works.. Notice that the Pasco program calculates the average an the stanar eviation, but oes not calculate the stanar eviation of the mean the appropriate uncertainty for the average perio! You will nee to recor the number of perios measure, an calculate the stanar eviation of the mean. t is worth noting that the stanar eviation the uncertainty in each perio oes not change appreciably, no matter how many measurements one takes. 3. t can be helpful to ajust y axis of the graph in the Pasco program so that you can see the variation in perio graphically while the ata are being recore. Click on the y axis to o so. 4. he micrometer an calipers claim an accuracy of 0.00 inch (or 0.0 mm). One shoul, of course, regar such claims with a grain of salt, an look for ways of ouble-checking! 6

7 Data Analysis Metho From a careful analysis of your graphs, fin the point at which the perios are equal. hen, use Equation () to calculate g an the uncertainty in g. Metho Refer to Figure, an see if you can calculate approximate values of an, given your measure values of L an a. hese values shoul be measure from the center of mass. However, it turns out that the two holes o not cause the center of mass to be shifte from the center of the bar by more than a half a millimeter or so. t s easy to confirm this point by balancing the penulum on one sie, on the knife ege support. hus, if you calculate an relative to the center of the bar instea of the center of mass, you will know - to an accuracy of about mm. t turns out that this low accuracy measurement will not significantly affect the accuracy with which g is measure! You can then calculate /g using Equation (4). he uncertainty analysis is a bit complicate, so we will go through it carefully in Appenix. he iea is to fin the uncertainty in /g from Equation (4), an then use that result to fin the uncertainty in g. See the Appenix for etails. Be sure you unerstan why the large uncertainty in - oes not affect the accuracy with which g is measure, as long as the two perios are nearly equal. Both Methos t is important to retain as much accuracy as possible in using either Equation () or Equation (4) to calculate g. f you are using a calculator, it is worth using storage registers to store intermeiate results, so that no accuracy is lost in reentering numbers. You can also use a spreasheet, or a program like Mathca or Mathematica to o the calculations. Conclusions Report your value of g, with uncertainty. Discuss any systematic uncertainties or other problems with the experiment that you think nee further refinement. t might also be interesting to compare your value of g here with the value you foun in the free fall experiment. 7

8 APPENDX Derivation of Equation (4) We want to erive Equation (4) above that is, we want to fin the quantities A an B that satisfy A B f we convert the right-han sie of this equation to a common enominator, we obtain b g b g A B Now, in the numerator of the right-han sie, collect all of the terms in an : b g b g A B B A f we now compare the numerators of the left an right sies, we see at once that A B B A hus we have a pair of simultaneous equations that we can solve for A an B. One fins A B (5) Substituting this result into Equation (5), we obtain his result leas immeiately to Equation (4) above. b g b g 8

9 Appenix Calculation of the uncertainty in /g We begin by writing Equation (4) in the form P Q g where P an Q b g b g But the quantity is measure as a single istance, with a single uncertainty. Likewise, one can consier - as a single quantity, known to within a millimeter or so. (Note that in any case, one cannot treat an as inepenent as one increases, the other necessarily ecreases.) Consequently, it is convenient to efine D an Hence, our expressions for P an Q become P an Q D Notice that all the quantities in P are known very accurately; however, Q is known less accurately, both because the numerator is a ifference, an because is known only to within a millimeter or so. t is for this reason that one ajusts the perios to make Q as small as possible. he error calculation is more complicate than can easily be one using the methos in Appenix A of the Physics 9 laboratory manual. t can nevertheless be shown that an that δp P δq Q he uncertainty in /g is therefore F HG F HG F KJ HG KJ F H G δ δ δd D F KJ HG K J KJ F H G K J δ δ δ δ F HG g KJ an it is an easy matter to fin the uncertainty in g. δp δq 9

10 Expecte Value of g at Collegeville, MN Latitue formula: if g is measure in milligals ( milligal 0-5 m/sec ) where h g ge( βsin θ) ge B R ge is the mean acceleration of gravity at the equator, 978,049 milligals θ is the latitue h is the altitue above sea level R is the raius of the earth β is a constant, B is the Bourget anomaly One fins g shoul be about m/sec 0