determination of acceleration due to gravity by compound pendulum

The corresponding value of \(g\) for each of these trials was calculated. In this experiment, we measured \(g\) by measuring the period of a pendulum of a known length. A 3/4" square 18" long 4 steel bar is supplied for this purpose. Using the small angle approximation gives an approximate solution for small angles, \[\frac{d^{2} \theta}{dt^{2}} = - \frac{g}{L} \theta \ldotp \label{15.17}\], Because this equation has the same form as the equation for SHM, the solution is easy to find. In this experiment the value of g, acceleration due gravity by means of compound pendulum is obtained and it is 988.384 cm per sec 2 with an error of 0.752%. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. This looks very similar to the equation of motion for the SHM \(\frac{d^{2} x}{dt^{2}}\) = \(\frac{k}{m}\)x, where the period was found to be T = 2\(\pi \sqrt{\frac{m}{k}}\). We built the pendulum with a length \(L=1.0000\pm 0.0005\text{m}\) that was measured with a ruler with \(1\text{mm}\) graduations (thus a negligible uncertainty in \(L\)). >> Theory A pendulum exhibits simple harmonic motion (SHM), which allowed us to measure the gravitational constant by measuring the period of the pendulum. Plug in the values for T and L where T = 2.5 s and L = 0.25 m g = 1.6 m/s 2 Answer: The Moon's acceleration due to gravity is 1.6 m/s 2. (PDF) To Determine The Value of g Acceleration due to gravity by means of a compound pendulum Home Acceleration To Determine The Value of g Acceleration due to gravity by. The angular frequency is, \[\omega = \sqrt{\frac{mgL}{I}} \ldotp \label{15.20}\], \[T = 2 \pi \sqrt{\frac{I}{mgL}} \ldotp \label{15.21}\]. Aim . To determine the value of g,acceleration due to gravity by - YouTube Substitute each set of period (T) and length (L) from the test data table into the equation, and calculate g. So in this case for four data sets, you will get 4 values of g. Then take an average value of the four g values found. Describe how the motion of the pendulums will differ if the bobs are both displaced by 12. Experiment-4(Compound pendulum) - E4-Name of the experiment - Studocu Sorry, preview is currently unavailable. << Compound Pendulum- Symmetric - Amrita Vishwa Vidyapeetham Length . We are asked to find the length of the physical pendulum with a known mass. endobj The restoring torque can be modeled as being proportional to the angle: The variable kappa (\(\kappa\)) is known as the torsion constant of the wire or string. The units for the torsion constant are [\(\kappa\)] = N m = (kg m/s2)m = kg m2/s2 and the units for the moment of inertial are [I] = kg m2, which show that the unit for the period is the second. With the simple pendulum, the force of gravity acts on the center of the pendulum bob. Enter the email address you signed up with and we'll email you a reset link. The aim for this experiment is to determine the acceleration due to gravity using a pendulum bob. Retort stand, boss head, and clamp, string and mass bob, Stopwatch, rulerif(typeof ez_ad_units != 'undefined'){ez_ad_units.push([[300,250],'physicsteacher_in-box-4','ezslot_5',148,'0','0'])};__ez_fad_position('div-gpt-ad-physicsteacher_in-box-4-0'); Record the data in the table below following the instructions in the section above. By adding a second knife-edge pivot and two adjustable masses to the physical pendulum described in the Physical Pendulum demo, the value of g can be determined to 0.2% precision. 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Which is a negotiable amount of error but it needs to be justified properly. In the experiment the acceleration due to gravity was measured using the rigid pendulum method. /Filter /FlateDecode This has a relative difference of \(22\)% with the accepted value and our measured value is not consistent with the accepted value. iron rod, as rigidity is important. The period is completely independent of other factors, such as mass and the maximum displacement. See Full PDF We are asked to find g given the period T and the length L of a pendulum. stream Using the small angle approximation and rearranging: \[\begin{split} I \alpha & = -L (mg) \theta; \\ I \frac{d^{2} \theta}{dt^{2}} & = -L (mg) \theta; \\ \frac{d^{2} \theta}{dt^{2}} & = - \left(\dfrac{mgL}{I}\right) \theta \ldotp \end{split}\], Once again, the equation says that the second time derivative of the position (in this case, the angle) equals minus a constant \(\left( \dfrac{mgL}{I}\right)\) times the position. Pendulum 2 has a bob with a mass of 100 kg. /F10 33 0 R Recall that the torque is equal to \(\vec{\tau} = \vec{r} \times \vec{F}\). Kater's pendulum, stopwatch, meter scale and knife edges. The length of the pendulum has a large effect on the time for a complete swing. Object: To determine the acceleration due to gravity (g) by means of a compound pendulum. We suspect that by using \(20\) oscillations, the pendulum slowed down due to friction, and this resulted in a deviation from simple harmonic motion. Therefore, the period of the torsional pendulum can be found using, \[T = 2 \pi \sqrt{\frac{I}{\kappa}} \ldotp \label{15.22}\]. A digital wristwatch or large analog timer 3 is used to verify the period. The Italian scientist Galileo first noted (c. 1583) the constancy of a pendulum's period by comparing the movement of a swinging lamp in a Pisa cathedral with his pulse rate. PDF Experiment 9: Compound Pendulum - GitHub Pages By timing 100 or more swings, the period can be determined to an accuracy of fractions of a millisecond. Two knife-edge pivot points and two adjustable masses are positioned on the rod so that the period of swing is the same from either edge. Accessibility StatementFor more information contact us atinfo@libretexts.org. /F11 36 0 R Anupam M (NIT graduate) is the founder-blogger of this site. Thus you get the value of g in your lab setup. You can download the paper by clicking the button above. Our final measured value of \(g\) is \((7.65\pm 0.378)\text{m/s}^{2}\). A . [Caution: Students are suggested to consult Lab instructors & teachers before proceeding to avoid any kind of hazard.]. Note that for a simple pendulum, the moment of inertia is I = \(\int\)r2dm = mL2 and the period reduces to T = 2\(\pi \sqrt{\frac{L}{g}}\). The time period is determined by fixing the knife-edge in each hole. The mass of the string is assumed to be negligible as compared to the mass of the bob. The magnitude of the torque is equal to the length of the radius arm times the tangential component of the force applied, |\(\tau\)| = rFsin\(\theta\). This page titled 27.8: Sample lab report (Measuring g using a pendulum) is shared under a CC BY-SA license and was authored, remixed, and/or curated by Howard Martin revised by Alan Ng. Pendulums are in common usage. We repeated this measurement five times. %PDF-1.5 Find the positions before and mark them on the rod.To determine the period, measure the total time of 100 swings of the pendulum. Acceleration due to gravity by using Bar Pendulum | Compound Pendulum Change the length of the string to 0.8 m, and then repeat step 3. The period for one oscillation, based on our value of \(L\) and the accepted value for \(g\), is expected to be \(T=2.0\text{s}\). This is consistent with the fact that our measured periods are systematically higher. Step. 16.4 The Simple Pendulum - College Physics 2e | OpenStax A rod has a length of l = 0.30 m and a mass of 4.00 kg. The distance of each hole from the center of gravity is measured. This removes the reaction time uncertainty at the expense of adding a black-box complication to an otherwise simple experiment. In extreme conditions, skyscrapers can sway up to two meters with a frequency of up to 20.00 Hz due to high winds or seismic activity. Pendulum 1 has a bob with a mass of 10 kg. To Determine The Value of g Acceleration due to gravity by means of a This correspond to a relative difference of \(22\)% with the accepted value (\(9.8\text{m/s}^{2}\)), and our result is not consistent with the accepted value. The angle \(\theta\) describes the position of the pendulum. Which is a negotiable amount of error but it needs to be justified properly. All of our measured values were systematically lower than expected, as our measured periods were all systematically higher than the \(2.0\text{s}\) that we expected from our prediction. Note the dependence of T on g. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity, as in the following example. This way, the pendulum could be dropped from a near-perfect \(90^{\circ}\) rather than a rough estimate. To determine g, the acceleration of gravity at a particular location.. There are many ways to reduce the oscillations, including modifying the shape of the skyscrapers, using multiple physical pendulums, and using tuned-mass dampers. 27: Guidelines for lab related activities, Book: Introductory Physics - Building Models to Describe Our World (Martin et al. In the experiment the acceleration due to gravity was measured using the rigid pendulum method. xZnF}7G2d3db`K^Id>)_&%4LuNUWWW5=^L~^|~(IN:;e.o$yd%eR# Kc?8)F0_Ms reqO:.#+ULna&7dR\Yy|dk'OCYIQ660AgnCUFs|uK9yPlHjr]}UM\jvK)T8{RJ%Z+ZRW+YzTX6WgnmWQQs+;$!D>Dpll]HxuC0%X/3KU{AaLKKVQ j!uw$(0ik. But note that for small angles (less than 15), sin \(\theta\) and \(\theta\) differ by less than 1%, so we can use the small angle approximation sin \(\theta\) \(\theta\). In this video, Bar Pendulum Experiment is explained with calculatio. The relative uncertainty on our measured value of \(g\) is \(4.9\)% and the relative difference with the accepted value of \(9.8\text{m/s}^{2}\) is \(22\)%, well above our relative uncertainty. (ii) To determine radius of gyration about an axis through the center of gravity for the compound pendulum. ], ICSE, CBSE class 9 physics problems from Simple Pendulum chapter with solution, How to Determine g in laboratory | Value of acceleration due to gravity -, Simple Harmonic Motion of a Simple Pendulum, velocity of the pendulum bob at the equilibrium position, Transfers between kinetic & potential energy in a simple pendulum, Numerical problem worksheet based on the time period of pendulum, Acceleration, velocity, and displacement of projectile at different points of its trajectory, Satellite & Circular Motion & understanding of Geostationary Satellite. /F1 6 0 R /Type /Page 4 2/T 2. >> University Physics I - Mechanics, Sound, Oscillations, and Waves (OpenStax), { "15.01:_Prelude_to_Oscillations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.02:_Simple_Harmonic_Motion" : "property get [Map 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"program:openstax", "licenseversion:40", "source@https://openstax.org/details/books/university-physics-volume-1" ], https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_University_Physics_(OpenStax)%2FBook%253A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)%2F15%253A_Oscillations%2F15.05%253A_Pendulums, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Example \(\PageIndex{1}\): Measuring Acceleration due to Gravity by the Period of a Pendulum, Example \(\PageIndex{2}\): Reducing the Swaying of a Skyscraper, Example \(\PageIndex{3}\): Measuring the Torsion Constant of a String, 15.4: Comparing Simple Harmonic Motion and Circular Motion, source@https://openstax.org/details/books/university-physics-volume-1, State the forces that act on a simple pendulum, Determine the angular frequency, frequency, and period of a simple pendulum in terms of the length of the pendulum and the acceleration due to gravity, Define the period for a physical pendulum, Define the period for a torsional pendulum, Square T = 2\(\pi \sqrt{\frac{L}{g}}\) and solve for g: $$g = 4 \pi^{2} \frac{L}{T^{2}} ldotp$$, Substitute known values into the new equation: $$g = 4 \pi^{2} \frac{0.75000\; m}{(1.7357\; s)^{2}} \ldotp$$, Calculate to find g: $$g = 9.8281\; m/s^{2} \ldotp$$, Use the parallel axis theorem to find the moment of inertia about the point of rotation: $$I = I_{CM} + \frac{L^{2}}{4} M = \frac{1}{12} ML^{2} + \frac{1}{4} ML^{2} = \frac{1}{3} ML^{2} \ldotp$$, The period of a physical pendulum has a period of T = 2\(\pi \sqrt{\frac{I}{mgL}}\).

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