determination of acceleration due to gravity by compound pendulum

The following data for each trial and corresponding value of $g$ are shown in the table below. A 3/4" square 18" long 4 steel bar is supplied for this purpose. /Parent 2 0 R We expect that we can measure the time for $20$ oscillations with an uncertainty of $0.5\text{s}$. The vertical pendulum, such as that developed by ONERA, 12 uses gravity to generate a restoring torque; therefore, it has a fast response to thrust due to the larger stiffness. The aim for this experiment is to determine the acceleration due to gravity using a pendulum bob. Indeed, the reversible pendulum measurement by Khnen and Furtwngler 5 in 1906 was adopted as the standard for a world gravity network until 1968. 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 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. Use a stopwatch to record the time for 10 complete oscillations. This research work is meant to investigate the acceleration due to gravity "g" using the simple pendulum method in four difference locations in Katagum Local Government Area of Bauchi State. Continue with Recommended Cookies, if(typeof ez_ad_units != 'undefined'){ez_ad_units.push([[728,90],'physicsteacher_in-box-3','ezslot_8',647,'0','0'])};__ez_fad_position('div-gpt-ad-physicsteacher_in-box-3-0');This post is on Physics Lab work for performing a first-hand investigation to determine a value of acceleration due to gravity (g) using pendulum motion. Change the length of the string to 0.8 m, and then repeat step 3. We transcribed the measurements from the cell-phone into a Jupyter Notebook. To determine the radius of gyration about an axis through the centre of gravity for the compound pendulum. For example, it's hard to estimate where exactly the center of the mass is. To browse Academia.edu and the wider internet faster and more securely, please take a few seconds toupgrade your browser. We can solve T = 2$\pi$L g for g, assuming only that the angle of deflection is less than 15. Like the force constant of the system of a block and a spring, the larger the torsion constant, the shorter the period. To view the purposes they believe they have legitimate interest for, or to object to this data processing use the vendor list link below. Best on the results findings, it showed that the Rafin Tambari has the highest value of acceleration due to gravity which is (10.2 m/s 2). 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. This experiment is discussed extensively in order to provide an example of how students should approach experiments and how experimental data should be processed. Any object can oscillate like a pendulum. This is consistent with the fact that our measured periods are systematically higher. <>stream Repeat step 4, changing the length of the string to 0.6 m and then to 0.4 m. Use appropriate formulae to find the period of the pendulum and the value of g (see below). We also worry that we were not able to accurately measure the angle from which the pendulum was released, as we did not use a protractor. /F1 6 0 R By timing 100 or more swings, the period can be determined to an accuracy of fractions of a millisecond. 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 first need to find the moment of inertia. [] or not rated [], Copyright 2023 The President and Fellows of Harvard College, Harvard Natural Sciences Lecture Demonstrations, Newton's Second Law, Gravity and Friction Forces, Simple Harmonic (and non-harmonic) Motion. Assuming the oscillations have a frequency of 0.50 Hz, design a pendulum that consists of a long beam, of constant density, with a mass of 100 metric tons and a pivot point at one end of the beam. The time period is determined by fixing the knife-edge in each hole. Surprisingly, the size of the swing does not have much effect on the time per swing . Using a simple pendulum, the value of g can be determined by measuring the length L and the period T. The value of T can be obtained with considerable precision by simply timing a large number of swings, but comparable precision in the length of the pendulum is not so easy. 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. We don't put any weight on the last significant figure and this translates to 45.533 cm.5 F. Khnen and P. Furtwngler, Veroff Press Geodat Inst 27, 397 (1906). Useful for B.Sc., B.Tech Students. Which is a negotiable amount of error but it needs to be justified properly. Acceleration due to gravity 'g' by Bar Pendulum OBJECT: To determine the value of acceleration due to gravity and radius of gyration using bar pendulum. Aim . A pendulum exhibits simple harmonic motion (SHM), which allowed us to measure the gravitational constant by measuring the period of the pendulum. Length . Apparatus used: Bar pendulum, stop watch and meter scale. We have described a simple pendulum as a point mass and a string. We suspect that by using $20$ oscillations, the pendulum slowed down due to friction, and this resulted in a deviation from simple harmonic motion. 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 $\theta$ $\theta$. The distance of each hole from the center of gravity is measured. Find more Mechanics Practical Files on this Link https://alllabexperiments.com/phy_pract_files/mech/, Watch this Experiment on YouTube https://www.youtube.com/watch?v=RVDTgyj3wfw, Watch the most important viva questions on Bar Pendulum https://www.youtube.com/watch?v=7vUer4JwC5w&t=3s, Please support us by donating, Have a good day, Finally found the solution of all my problems,the best website for copying lab experiments.thanks for help, Your email address will not be published. As for the simple pendulum, the restoring force of the physical pendulum is the force of gravity. 1 Pre-lab: A student should read the lab manual and have a clear idea about the objective, time frame, and outcomes of the lab. 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. II Solucionario, The LTP Experiment on LISA Pathfinder: Operational Definition of TT Gauge in Space, Solucionario de Fsica Universitaria I, 12a ed, Fsica Para Ingenieria y Ciencias Ohanian 3ed Solucionario. In the experiment, the bar was pivoted at a distanice of Sem from the centre of gravity. We are asked to find g given the period T and the length L of a pendulum. Consider a coffee mug hanging on a hook in the pantry. 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. Therefore, the period of the torsional pendulum can be found using, \[T = 2 \pi \sqrt{\frac{I}{\kappa}} \ldotp \label{15.22}\]. A physical pendulum is any object whose oscillations are similar to those of the simple pendulum, but cannot be modeled as a point mass on a string, and the mass distribution must be included into the equation of motion. This removes the reaction time uncertainty at the expense of adding a black-box complication to an otherwise simple experiment. /F9 30 0 R /F4 15 0 R Release the bob. 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. Click on the lower end of the pendulum, drag it to one side through a small angle and release it. 3 0 obj Both are suspended from small wires secured to the ceiling of a room. The solution to this differential equation involves advanced calculus, and is beyond the scope of this text. 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. What It Shows An important application of the pendulum is the determination of the value of the acceleration due to gravity. The period for this arrangement can be proved 2 to be the same as that of a simple pendulum whose length L is the distance between the two knife edges. Use the moment of inertia to solve for the length L: $$\begin{split} T & = 2 \pi \sqrt{\frac{I}{mgL}} = 2 \pi \sqrt{\frac{\frac{1}{3} ML^{2}}{MgL}} = 2 \pi \sqrt{\frac{L}{3g}}; \\ L & = 3g \left(\dfrac{T}{2 \pi}\right)^{2} = 3 (9.8\; m/s^{2}) \left(\dfrac{2\; s}{2 \pi}\right)^{2} = 2.98\; m \ldotp \end{split}$$, This length L is from the center of mass to the axis of rotation, which is half the length of the pendulum. As in the Physical Pendulumdemo, the pendulum knife-edge support is a U-shaped piece of aluminum that is clamped onto a standard lab bench rod. 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 MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.03:_Energy_in_Simple_Harmonic_Motion" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.04:_Comparing_Simple_Harmonic_Motion_and_Circular_Motion" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.05:_Pendulums" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.06:_Damped_Oscillations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.07:_Forced_Oscillations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.E:_Oscillations_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15.S:_Oscillations_(Summary)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Units_and_Measurement" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Vectors" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Motion_Along_a_Straight_Line" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Motion_in_Two_and_Three_Dimensions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Newton\'s_Laws_of_Motion" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Applications_of_Newton\'s_Laws" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Work_and_Kinetic_Energy" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Potential_Energy_and_Conservation_of_Energy" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Linear_Momentum_and_Collisions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Fixed-Axis_Rotation__Introduction" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:__Angular_Momentum" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Static_Equilibrium_and_Elasticity" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Gravitation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Fluid_Mechanics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Oscillations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_Waves" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17:_Sound" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "18:_Answer_Key_to_Selected_Problems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "Pendulums", "authorname:openstax", "simple pendulum", "physical pendulum", "torsional pendulum", "license:ccby", "showtoc:no", "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$\pi \sqrt{\frac{I}{mgL}}$. Academia.edu no longer supports Internet Explorer. The solution is, \[\theta (t) = \Theta \cos (\omega t + \phi),\], where $\Theta$ is the maximum angular displacement. We thus expect to measure one oscillation with an uncertainty of $0.025\text{s}$ (about $1$% relative uncertainty on the period). The period, T, of a pendulum of length L undergoing simple harmonic motion is given by: T = 2 L g /F5 18 0 R Additionally, a protractor could be taped to the top of the pendulum stand, with the ruler taped to the protractor. /F2 9 0 R This Link provides the handwritten practical file of the above mentioned experiment (with readings) in the readable pdf format. This was calculated using the mean of the values of g from the last column and the corresponding standard deviation. Recall that the torque is equal to $\vec{\tau} = \vec{r} \times \vec{F}$. /Filter /FlateDecode 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 pendulum will begin to oscillate from side to side. /F3 12 0 R Rather than measure the distance between the two knife edges, it is easier to adjust them to a predetermined distance. As with simple harmonic oscillators, the period T for a pendulum is nearly independent of amplitude, especially if $\theta$ is less than about 15. , How to Calculate Acceleration Due to Gravity Using a Pendulum, Free Printable Periodic Tables (PDF and PNG), Periodic Table with Charges - 118 Elements. 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. Learning Objectives 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 Pendulums are in common usage. The period is completely independent of other factors, such as mass and the maximum displacement. 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%. >> For small displacements, a pendulum is a simple harmonic oscillator. The consent submitted will only be used for data processing originating from this website. Kater's pendulum, stopwatch, meter scale and knife edges. The magnitude of the torque is equal to the length of the radius arm times the tangential component of the force applied, |$\tau$| = rFsin$\theta$. Save my name, email, and website in this browser for the next time I comment. Recall from Fixed-Axis Rotation on rotation that the net torque is equal to the moment of inertia I = $\int$r2 dm times the angular acceleration $\alpha$, where $\alpha = \frac{d^{2} \theta}{dt^{2}}: \[I \alpha = \tau_{net} = L (-mg) \sin \theta \ldotp\]. The locations are; Rafin Tambari, Garin Arab, College of Education Azare and Township Stadium Azare. The demonstration has historical importance because this used to be the way to measure g before the advent of "falling rule" and "interferometry" methods. Even simple pendulum clocks can be finely adjusted and remain accurate. When a physical pendulum is hanging from a point but is free to rotate, it rotates because of the torque applied at the CM, produced by the component of the objects weight that acts tangent to the motion of the CM. /Type /Page In order to minimize the uncertainty in the period, we measured the time for the pendulum to make \(20$ oscillations, and divided that time by $20$. 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. Our final measured value of $g$ is $(7.65\pm 0.378)\text{m/s}^{2}$. Formula: Pendulums are in common usage. In the experiment the acceleration due to gravity was measured using the rigid pendulum method. 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. We repeated this measurement five times. %PDF-1.5 The results showed that the value of acceleration due to gravity "g" is not constant; it varies from place to place. Theory A simple pendulum may be described ideally as a point mass suspended by a massless string from some point about which it is allowed to swing back and forth in a place.
Replacement For Bud Factor X, East Greenbush School Tax Bills, Frazer Smith Rose Parade, Articles D

determination of acceleration due to gravity by compound pendulum 2023