When oscillations are small (i.e., the pendulum isn't swinging too much),1 we can make a small angle approximation which allows us to derive the following simple formula for the period of a simple pendulum of length \(L\) in a gravitational field of strength \(g\): When oscillations are large, we expect the period to deviate from this simple formula.2, Part I: Determining the Random Error in Time Measurements. As before, have one partner man the stopwatch and the other measure the angle and start the swinging. Pendulums are used to regulate the movement of clocks because the interval of time for each complete oscillation, called the period, is constant. Simple Pendulum is a mass (or bob) on the end of a massless string, which when initially displaced, will swing back and forth under the influence of gravity over its central (lowest) point. ), Begin with a \(15^\circ\) angle from the vertical.4. Thus, we have to measure it. Ď:$�O\������ ����N�ƥ� �,q Use this online simple pendulum calculator to calculate period, length and acceleration … %%EOF It also depends on the amplitude that is the maximum angle that a pendulum can swing form the point zero or … As with simple harmonic oscillators, the period T for a pendulum is nearly independent of amplitude, especially if \(\theta\) is less than about 15°. For I support = kg m 2: m = kg: g = m/s 2: L cm = cm: the period is T = s: Rod pendulum: Circular geometries: Combinations: Index with your meter stick each time. 00:00. This requires some thought: what is your dominant uncertainty here?3. Repeat this measurement for lengths increasing to about 1m in increments of about 10cm, measuring the actual length L (which may not be exactly 10cm, 20cm, etc.!) A pendulum is one of most common items found in households. Thus the period equation is: T = 2π√(L/g) Over here: T= Period in seconds. The equation is exact; that is, it works perfectly, for all angles. SE�q����� �!�I�*8�G�KI)��p�� 'BH��@��jȷo���ɏx�Yz��v���zIy0X-lp�3���qn�t���������yN�*�to�6����t�ŃM&o9����z-�nO2"��t�__�?��9�ؙp#�\�]. The formula for the period T of a pendulum is T = 2π Square root of √ L / g, where L is the length of the pendulum and g is the acceleration due to gravity. ch. In particular, which is most important: the precision in reading the ruler itself, your ability to estimate the location of the center of mass of the hanging object, or your ability to estimate the location of the swing point? Have one partner (Partner 1) hold the stopwatch, and the other partner (Partner 2) hold the pendulum up at a small angle from the vertical (say, \(15^\circ\) or so - again, no need to be precise yet). It depends on the length of the pendulum and the gravity of the place where it is been measured. For the derivation of this formula from first principles, see here. Ideally, your setup should look like this: in which case you should be able to slide your string freely by loosinging the screw a bit, then clamp it at any height. Students are often asked to evaluate the value of the acceleration due to gravity, g, using the equation for the time period of a pendulum. You should just be able to enter "K" and the computer will understand what you mean; if not, try "EllipticK" instead. They swing back and forth periodically, from some maximum angle to the opposite side and back again. It probably varies from person to person! Set the length of the pendulum to 50cm or so.1 You don't have to be precise with the length yet; we're not concerned about the actual values. In the planar case, one can find an explicit formula for the period of a pendulum in terms of elliptic integrals, but that's beyond the scope of what we're doing here. Take the slope of the most linear graph, and enter it into your data table. From this slope, calculate \(g\), and propagate uncertainty from slope to \(g\). Record this on your data table. Since frequency is related to time period as: f=1/T Watch also video For sake of physical simplicity, we assume that the pendulum is "rigid" (is of fixed shape and size as it oscillates) and that the top of the pendulum is fixed (so it's swinging from a single point). The equation is closed form; that is, there are no strange numbers or series of terms to evaluate. It is a device that is commonly found in wall clocks. After a swing or two, Partner 1 should hit the start button at the bottom of the swing. 0 INTRODUCTION The periodic motion exhibited by a simple pendulum is harmonic only for small-angle oscillations, for which there is a well-known period formula.1 Beyond this limit, the equation of motion is nonlinear, which makes difficult the mathematical description of the oscilla- 16-6. The pendulum period formula, T, is fairly simple: T = (L / g)1/2, where g is the acceleration due to gravity and L is the length of the string attached to the bob (or the mass). Hovering over these bubbles will make a footnote pop up. 385 0 obj <>/Filter/FlateDecode/ID[<79654DF92204ACB57AF16128000E97D9>]/Index[359 47]/Info 358 0 R/Length 117/Prev 256446/Root 360 0 R/Size 406/Type/XRef/W[1 3 1]>>stream Simple pendulum frequency formula. endstream endobj startxref Through these measurements and a theoretical formula that pendula should follow, we will calculate the value of \(g\), the acceleration due to gravity. The dimensions of this quantity is a unit of time, such as seconds, hours or days. Through these measurements and a theoretical formula that pendula should follow, we will calculate the value of \(g\), the acceleration due to gravity. Take the uncertainty in \(10T\) as from Part I, and calculate the period \(T\) and the uncertainty in \(T\) (just as you did for Part II). We also typically assume that the pendulum is planar (swings in only the \(xy\) plane with no \(z\) motion), although with the approximations we will make in this lab, that assumption is not strictly necessary. 359 0 obj <> endobj Note Wolfram|Alpha assumes angles are entered in radians by default, although if you say "sin(90 degrees)" explicitly then it will give you the correct results. $XA"l@���� b��l��$���� b)��� ��Q&?FS�LF&&?�L!L]�+^2�0�0��RW�a���6�QT�.67��XM@~J��g`���3� ���6 � �8� ������� � DP: This maximizes your timing precision in determining the period. This article will throw light on this particular device. These can be important subtleties, advanced material, historical asides, hints for questions, etc. For each angle, take the uncertainty in \(10T\) to be given by your uncertainty from Part I. Begin by setting the length to 10cm or so. The period is completely independent of other factors, such as mass and the maximum displacement. Calculate \(T^2\) and \(L^2\). Release (be careful to give it as little initial velocity as possible! The period is not dependent upon the mass, since in standard geometries the moment of inertia is proportional to the mass. Now, we're going to do the same measurement - time for ten swings - for different angles. If our solution is the zeroth-order expansion, we can consider the first-order expansion, and get that relative error times some derivative term (which we assume to be of order of magnitude 1). Then, answer the question on the data table, using the appropriate uncertainty for the comparison you are doing. This result is interesting because of its simplicity. This is the time it takes the pendulum to complete one full swing - from one side to the other and back again (or, equivalently, from the middle, up to one side, down past the middle and up the other side, and back down to the middle again). For each partner, calculate the mean of all five measurements. Record the time to undergo ten swings in your data table as Trial 1. If you are missing the metal clamp entirely, wind/unwind the rope around the screws to change length instead. Estimate your uncertainty in your measurement of this length. See the Guide to Making and Using Plots, as well as the Lab Report Expectations for more details. Your clamp may be on the outside instead; that's fine. (If you don't believe this, feel free to test, and compare the uncertainty of the two methods yourself! The approximations we make are that the size of the object is much smaller than the length of the string and that the object is much heavier than the string. ), and measure ten periods. Part III: Determining the Dependence of Period on Pendulum Length \(L\) and Measuring \(g\). Thus, running this kind of check is important, to detect those! Part II: Determining the Dependence of Period on Angle \(\theta\). endstream endobj 360 0 obj <>/Metadata 32 0 R/Outlines 84 0 R/PageLabels 355 0 R/PageLayout/OneColumn/Pages 357 0 R/PieceInfo<>>>/StructTreeRoot 115 0 R/Type/Catalog>> endobj 361 0 obj <>/Font<>/ProcSet[/PDF/Text]>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 362 0 obj <>stream
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