Double pendulum lab report

Double pendulum lab report. Setup Time. The objective of this experiment is to determine the moment of inertia of a biflar pendulum experimentally and then comparing it to the theoretical prediction. This is true only for small angle and therefore small displacement. Measure the length from the. anchor point to the center of the bob. length of the pendulum is doubled then the period will double. Due to low friction and high mechanical transparency offered by QDDs, one of the actuators can be Overall rating of the Experiment. 20 20. 1119/1. (16. He also determined the length of what was called the second’s pendulum, i. 10 min. You should be wearing eye protection for the entire duration of the lab. In principle, the efective length is Le = k2 rm where k is the radius of gyration and rm is the radius of the As the double pendulum loses energy, the transition from chaotic to periodic motion should be graphically dramatic. In deriving these equations, we have neglected air resistance and other forms of friction, and have assumed that we may model our pendulum as a point mass attached to a rigid, massless rod or string. This is given as the letter Dec 19, 2023 · terminal. Measure the gravitational constant gfrom the pendulum data and compare with know values. Therefore the simple pendulum provides a good “test case” for the application of the scientific method. 1. A = tan A = sin A for small A. Report for Experiment # 12 The Simple Pendulum Lab Partner TA: Your Instructor’s Name Experiment Date: 01/19/ Abstract. The bullet collides with and sticks inside of the wood block. The two simulations being run are: pendulum rods, so that the uncoupled pendula P1 and P2 have identical frequencies. (For those This video was sponsored by GoogleWant to see how to try this at home with the Google Assistant? Check out this link: https://youtu. Fig. 2 acceleration due to gravity on the moon is 1 m/s 2. Drag the pendulum to change the starting position. This report shows how to find an approximate of moment of inertia or ‘g’ using a pendulum experiment. be/dil9YvQ-bwQMy “Hey Goo Flag. What makes it simple? A simple pendulum is simple because it is a small mechanical device that alters the amplitude or direction of a force. The word “pendulum” is new Latin, from the Latin ‘pendulus’, meaning ‘hanging’. Our exponent is quite close to the theoretical value of 0. Place the pivot hole of the rod over the knife-edge. This is a simulation of two rigid bodies connected to form a double pendulum. The back-and-forth motion of a swing demonstrates the physics of a pendulum. This may be written in bulletpointform. While a general motion of this system might seem complicated, any motion of this system can be described by the sum of two normal modes. Clamp the pendulum clamp to the top of the Aluminum pole. 𝒍. Figure 1. Torsional Pendulum lab report - Free download as PDF File (. The pendulum consists of a rigid boom with two identical bobs at each end, suspended by a Aug 18, 2023 · The single, double, and triple pendulum has served as an illustrative experimental benchmark system for scientists to study dynamical behavior for more than four centuries. B) Prepare the equipment for the measurement: a. 17335. The nature of the graph shows the physical properties of the compound pendulum. In physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaotic pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. 6. Essentially, this experiment measures the effects of gravity such as the acceleration when variables like the length of the string or the angle that the pendulum is released from is changed. Thus, T =2π√L g T = 2 π L g for the period of a simple pendulum. No matter how closely the motions of the two pendula are started, they eventually must undergo virtually total divergence. When one of the pendula is held at rest and the other is set oscillating, the energy of the oscillating pendulum is transferred to the other pendulum by the spring. They are determined from application of Newton’s Second Law - Equation of Motion. Observe the energy in the system in real-time, and vary the amount of friction. Set the spring to the desired launch Simple pendulum definition. To understand the relationships of the energetics, forces, acceleration, and velocity of an oscillating pendulum, and to determine how the motion of a pendulum depends on the mass, the length of the string, and the acceleration due to gravity. Set the mass of the pendulum as desired. 5, and therefore our data is consistent with the pendulum’s period depending on the square-root of length. The different pendulum physical properties are listed below: Symbol Description Value . In order to determine a value for g, however, we had to graph T2 against l, then consider the gradient of the linear function. To demonstrate Lissajous figures. pdf), Text File (. Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, the strength of gravity, and the amplitude of the swing. 3302 m May 24, 2024 · About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright Dec 19, 2023 · Physics-based simulation of a simple pendulum. Abstract: In the following practical, the time required for ten oscillations of the chosen fixed-mass object (a travel sized hand sanitiser) was obtained by way of a stopwatch on a smartphone. Students are able to find the periodic time for the oscillation of various length of simple pendulum and determine a value for the acceleration due to gravity, g. May 11, 2024 · We’ll take two equal pendulums, coupled by a light spring. In order to make sense of the entire exercise, compile a brief, informal report describing the aims, methods and results; excluding the data, a page should be enough. foundation of this lab. Characterize the small amplitude oscillations of the rod pendulum for di erent supporting positions. It was found that: 1) The oscillation period does not depend on the mass of the pendulum bob. 2) The oscillation period depends on the length of the string, increasing as the square root of the length increases. Home \ experiment. Images. R. A. Also available: source code , documentation and how to customize . Levien. Show any relevant calculations. Arrange the Photogate so the pendulum bob blocks the Photogate’s beam when the bob is at rest. Authors: R. its periodic motion is defined by an acceleration that is proportional to its (Sample Lab Report) Joe Glotz (Partner: Al Thumbs) Physics 2048C Lab #12, Simple Pendulum November 12, 2007 Instructor: Dr. Sze Meng Tan. The dependent variable would be the period time. 230 kg 𝑳. Measure h 2 each time and put the values in the chart. Citations (108) Abstract. The 9. Using the heaviest bob, vary the length of the pendulum. to the force of gravity on each pendulum. If the amplitude of motion of the swinging pendulum is small, in other words the angle at which the pendulum is raised (about 10°-15°), then the pendulum behaves approximately as a simple harmonic motion, and the period (T) of the oscillation of a simple pendulum is given approximately by the following equation: Where l is the length of the Two pendula are coupled by a spring. Determine the period of oscillation, T, for each length by using equation (3) and place the result for each length in your data table. 6) for the period of a simple pendulum. Which means if we graph T against l, we should get a square root function. This result is interesting because of its simplicity. Learn how to design and conduct your own pendulum experiments with this fun and engaging simulation. Tie one end of the string to the hook on the Force Sensor and tie the other end to the pendulum bob. Set the pendulum into oscillation, with amplitude of about five degrees. L is equivalent to the expression 1 h√k2 + h2 , which has a value found graphically as explained in procedure 4 (above). Stop the stopwatch after 50 cycles. Martin, Emma Neary, Joshua Rinaldo, and Olivia Woodman via source content that was edited to the style and standards of the LibreTexts Download Free PDF. 𝒑,𝒍. If the bob moves away from the vertical to some angle θ A double pendulum is a system consisting of a standard pendulum directly attached to another one. You can change parameters in the simulation such as gravity, initial angle of the two bodies. Procedures: Measure and record the mass of each pendulum bob. Figure 1: LARGE CLAMP 2. 6 %âãÏÓ 24 0 obj > endobj 31 0 obj >/Filter/FlateDecode/ID[5CC30276835B164E8F52CACCDE8B7D1D>]/Index[24 17]/Info 23 0 R/Length 56/Prev 78315/Root 25 0 R Aug 24, 2020 · I have a handful of students learning remotely, so I thought I'll allow you the opportunity to see what a very quick intro to a lab looks like. %PDF-1. Measure the period using the stopwatch or period timer. IV. In the picture, there are two extra masses attached to the bottom of the pendulum. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. Nov 1, 1993 · DOI: 10. Description. The pivot of the first pendulum is fixed to a point , and all motion is frictionless. This concept of acceleration due to gravity is explained further in the lab using a simple pendulum. Sample calculation for uncertainty in T2 (first data point): Pendulum Lab Report. (You only have to measure this once, since the starting height will be the same every time. We will write down equations of motion for a single and a double plane pendulum, following Newton’s equations, and using Lagrange’s equations. The period of a simple pendulum is T = 2π√L g. Place the aluminum pole in the groove and tighten up the screw. Introduction: The purpose of this experiment is to understand the relationship between the period and the length of a simple pendulum. Picarro Inc. . It is suspended from a fixed support. , Santa Clara, California, U. You can modify the starting angles and gravity. txt) or read online for free. Objective. Einstein Purpose The purpose of this experiment was to determine whether length, mass and angle of displacement affect the period of a pendulum. comes from gravity. The total force on the spring is equal to 9 * weight on the spring in kilograms. 9: Sample lab report review (Measuring g using a pendulum) is shared under a CC BY-SA 4. A graph is plotted of the period versus the distance of the axis of suspension from one end of the pendulum. A simple pendulum consists of a heavy pendulum bob (of mass M) suspended from a light string. 0 license and was authored, remixed, and/or curated by Ryan D. ) Do six trials of launching the ball into the catcher. For the simple pendulum: T = 2π m k−−−√ = 2π m mg/L− −−−−−√. lum, double pendulum, spring pendulum, Foucault pendulum, Kater's pendulum, and spherical pendulum [1, 2]. What is a simple pendulum? A simple pendulum is an object with a small mass commonly known as the pendulum bob, suspended from a light wire or string. In practice, the movable mass is divided into two or more elements to enable coarse and fine adjustment. The double pendulum consists of two pendulum bobs and two rigid rods. The masses, denoted by M, are points. It stretches the spring to 0. The period, T T, of a pendulum of length L L undergoing simple harmonic motion is given by: T = 2π L g−−√ T = 2 π L g. 2. The following figure shows the model of a double pendulum. This is similar to the development of equations shown for the Moveable Pendulum. Essay, Research paper, Coursework, Discussion Board Post, Questions-Answers, Case Study, Term paper, Powerpoint Presentation 3 Report The lab exercise that you have done was broken down into many small steps, whose relationship to one another may not be obvious. 3. There are many variables we could use to see into, some of them are displacement and angle. (This turns out to be a good approximation. 4. Apparatus. The period is completely independent of other factors, such as mass. above it. Use the phet s The document summarizes an experiment investigating how changing various factors affects the oscillation period of a simple pendulum. Student Manual. The mean position or the equilibrium position of a simple pendulum is the vertical line Feb 27, 2022 · One of our KiwiCo boxes this month was this double pendulum project. 20. Phy133 lab 1 - physics lab, ams midterm, cse 220 hw material, 216 recitation, etc. Using this equation, we can find the period of a pendulum for amplitudes less than about 15º 15º. 8: Test lab show (Measuring guanine after an pendulum) your shared under ampere CC BY-SA 4. Insert the lab post into the table and clamp the pendulum apparatus to this post. The ballistic pendulum should already be set up. A device where its point mass is attached to a light inextensible string and suspended from a fixed support is known as a simple pendulum. After the collision the wood block and bullet swing upward This is a physics-based simulation of a double pendulum whose support point is moveable. Compares two double pendulum simulations that are run simultaneously. This page titled 27. 𝒎. Your order is writtenBefore any paper is delivered to you, it first go through our strict checking process in order to ensure top quality. 4. This system has two natural modes: 1. Using this value for L compute L T2, and find g from the expression: g = 4π2 L T2. 5. The restorative force of a pendulum is the part of The pendulum rods are massless and are of length ‘. Under forcing, even the single pendulum may exhibit chaos A pendulum is a weight hanging from a fixed point so that it swings freely under the combined forces of gravity and momentum. The intent of the experiment is to investigate the motion of one particular kind called the bi lar pendulum. By utilising L, the distance from the position of suspension to the object’s centre, it was Feb 20, 2022 · T = 2π m k−−−√ = 2π m mg/L− −−−−−√. It is easiest to do this by. In Figure 1, the pendulum is depicted as a massless rod of length ` attached to a particle mass m at its end. Figure 2: Pendulum Clamp DATA ANALYSIS 1. 0 license the was penned, remixed, and/or curated per Rhonda DICK. Latch the pendulum at the top so it is out of the way. It is generally assumed that the mass of the string is negligible. S. Physics 1 phy 133 l69 simple pendulum: lab report introduction: this lab studies the simple pendulum, which is system that includes an object that is suspended. The right pendulum can be dragged with a spring force by clicking your mouse near it. Use the pendulum to find the value of g on Planet X Sep 2, 2003 · Theoretically, the result should be T = 2. The two pendula swing 180 degrees out of phase. Figure 1: A simple plane pendulum (left) and a double pendulum (right). A) Make a copy of Table 1 in your report to fill in for this part. In contrast, this simulation shows Simple Pendulum Lab Report May 25, 2022 Cycle 2 – Section J. Phys 132 Lab # LAB REPORT, part 1A Procedure: Describe your experimental technique for determining the spring constants in the static experiment. Title: Acceleration due to gravity, g. However, I am a little shaky on this assumption because, from experience, I have seen pendulums gradually decrease in amplitude/movement, and am not sure how that would affect the time keeping. NursingManagementBusiness and EconomicsHealthcare+80. For small oscillations the simple pendulum has Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, the strength of gravity, and the amplitude of the swing. When the length of a pendulum is increased, the frequency (increases, decreases). Pendulum distance from pivot to center of mass (long) 0. Theory/formulas for the physics experiment of torsional pendulum and compound pendulum. It implements a double pendulum platform built using two quasi-direct drive actuators (QDDs). You can change the separation between the pendulums to compare them more easily. LAP REPORT: THE SIMPLE PENDULUM Author: Muhammad Sohaib Alam fContent Page number Abstract 2 Objective 2 Theory 2 Apparatus 5 Procedure 6 Result and Analysis 6 Discussion 12 Conclusion 12 References 12 Page 1 of 11 f1. Theoretical Simple Harmonic Motion. 21. Using your data, make a graph of the period versus the amplitude. Have students sketch the results on the axes provided in the student lab report. Discrepancies in the power and in the leading term are most likely due to precision errors in the measurement of pendulum length. You can even use your phone and a sensor app to record your pendulum's movement and determine its period of oscillation. The two pendula swing in phase with each other. The de ection angles are small, and the equations of motion are linearized: sin ˘= . It is a simple yet chaotic system that demonstrates sensitive dependence on initial conditions and exhibits complex, non-linear behavior. Or click below to set initial conditions: The math behind the simulation is shown below. To determine the frequencies ! 0 and ! 1 of the two normal modes and the exchange frequency ! Lissajous Figures - Sand Pendulum Purpose. This select titled 27. It is assumed that each pendulum oscillates in a plane and has one degree of freedom. We describe an experiment which takes advantage of the surprising ‪Pendulum Lab‬ - PhET Interactive Simulations Double Pendulum. Use the pendulum to find the value of g on Planet X Nt1310 Unit 5 Lab Report 909 Words | 4 Pages. Record the mass in the Data Table in the Lab Report section. In this experiment, you will investigate the factors that affect the speed and duration of a pendulum's swing, also called an oscillating motion. ) For small angles of oscillation, we take the Lagrangian to be. The system of both pendulums has 2 degrees of freedom. b. Press the "Fire" button. i i e 4 3 (14) So, the (10) can be expressed as, 4 2 2 2 3 1 1 1 e M e k e e J e k e rd (15) And the derivative of (14) is obtained as, In this step, for calculating the actual control inputs, a second Lyapunov function based on all errors is defined as follows, By choosing the following controller, q sqref s q d sdref s d f i e k L u f i e k L u C)CHART 1 Make a data chart in your lab report like the one in Chart 1. 5. 2 Newton’s equations The double pendulum consists of two . Procedure A ballistic pendulum is a device used to determine the speed of a bullet. Adjust the length, mass, gravity, and friction to see how they affect the motion and energy. Use the horizontal sliders to adjust the mass of the bullet, mass of the wood block, and the initial velocity of the bullet. Students can change the support point and pendulum masses as well as gravity, damping and other parameters. ! Couple the two pendula with the spring at a distance h= 100 cm, then elevate and release one pendulum to set them swinging and acquire 15-20 seconds of data. KEYWORDS Introduction. Test different variables and measure the results. Also shown are the forces on the bob, which result in a net force of −mg sinθ toward the equilibrium position—that is, a Step 1: Define Displacement, Velocity, and Acceleration of Double Pendulum Masses. We therefore see that by measuring the period of a pendulum of known length, we may calculate the gravitational accelerationnearEarth’ssurface. The period of the energy transfer can be predicted and Sep 2, 2003 · Theoretically, the result should be T = 2. Springs - Labs. L = 1 2mℓ2θ˙2 1 + 1 2mℓ2θ˙2 2 − 1 2mgℓθ21 − 1 Pre-Lab Questions 1. This lab's goal was to measure a basic pendulum's period in order to ascertain how its mass and Carleton University Laboratory Report Course #: PHYS 1007 Experiment #: 5 Simple Pendulum Experiment Charlotte Bakker 101102402 Date Performed: November 23, 2018 Date Submitted: November 30, 2018 Lab Period: Friday, PM, L7 Partner: Philippe Beaulieu Station #: 25 TA: Mohamed Page 1! of 14 Abstract The goal of the experiment was to identify the relationship between the period of a pendulum and Dec 19, 2023 · terminal. 0 Abstract: The experiment was carried out to show that the mass of the bob has no effect on the ment is the ideal simple pendulum, whose period is related to its length, L, and the pull of gravity: T = 2π L g 1. Mass of pendulum (long) 0. Figure 1: Schematic of a Kater [1818] pendulum. If you study the derivation of the motion of the pendulum, at some point the angle is assumed to be small so that the angle (measued in radians) is equal to the sine of the angle. 3 the elevator is in free falling the pendulum will no longer feel the gravity, 1404Orders prepared. 01 L 0. Theory: a simple pendulum performs simple harmonic motion i. Using this equation, we can find the period of a pendulum for amplitudes less than about 15º. A simple pendulum uses a bob and a string. Teacher Mackenzie (UK) 8 years ago. Equipment. The two pendula are started into apparently identical oscillations, but their motion soon diverges. The vertical distance between the point of suspension and the center of the bob is called the effective length of the simple pendulum. The glow in the dark surface of the board is activated by the UV light at the end of th 4 Report Checklist Your lab report should include at least the following parts: 1. Start the stopwatch as a pendulum comes to rest at one extreme position. In a given normal mode all LAB 4: Pendulum. 3) For angles less than 10 degrees, the small Sep 12, 2003 · Kater used this device to measure the acceleration of gravity at various locations in England. Real pendulum have an efective length Lethat can be dicult to define and measure. Measure the initial height, h 1, and put it in the chart. Although somewhat idiosyncratic it does provide a reasonable explanation of the chaotic nature of a double pendulum, the algorithms that can be used to simulate it and how they can be verified. Figure 1 Double Pendulum For a two degree-of-freedom system there are two coupled differential equations that govern the motion of the system. Thus, by measuring the period of a pendulum as well as its length, we can determine the value of g g: g = 4π2L T2 g = 4 π 2 L T 2. Notice the anharmonic behavior at large amplitude. This movement of the support point is an input to A double pendulum consists of two pendulums attached end to end. This illustrates the modern meaning of "chaos. Tighten the bolt, after making sure it is resting on the bottom of the hole. A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. Make a table to record the period T as a function of the amplitude A. Nowadays, the conventional pendulum is widely used in engineering, such as energy harvesting and robot design. Projectile Motion Data Sheet - Sheet 1. The bar can be hung from any one of these holes allowing us to change the location of the pivot. A. To make the pendulum, use a piece of string that is about one meter long. Modify gravity, mass or friction (damping). We assumed that the frequency and period of the pendulum depend on the length of the The goal for the first part of the experiment is to measure the period of a pendulum. Also shown are free body diagrams for the forces on each mass. A 6 kg bowling ball is hung from a spring of un-stretched length 0. Lab Based on which station you use, you will either have a long or short pendulum. The bob is filled with sand, displaced from equilibrium and released or gently pushed. Let Kbe a constant. B. Then we can divide the total force by the displacement of and therefore: T = 2π√Mk2 + Mh2 Mgh = 2π√k2 + h2 gh. Single Inverted Pendulum: N=1. Then open the Physical Pendulum lab file. keeping the string attached to the bob but changing the string’s connection to the support. The linear displacement from equilibrium is s, the length of the arc. Theory. Procedure PHY 241 Pre-Lab 10; Lab report 2 - Pendulum Experiment vs Theory How period changes with length; Lab Report 6 - Lab on Conservation of Energy; Lab 4 - Lab 4 report on Fan Carts; Lab 3 Template (empty) PHY 241 Lab Syllabus Fall 2019 A Doubly Suspended Pendulum. We take the spring restoring force to be directly proportional to the angular difference between the pendulums. This project offers an open-source and low-cost kit to test control algorithms for underactuated robots with strongly non-linear dynamics. Each pendulum consists of a bob connected to a massless rigid rod that is only allowed to move along a vertical plane. It's easy to measure the period using the photogate timer. , a pendulum whose half-period is one second. 1: A double pendulum. Objective easy to study. First begin by insuring the rotary motion sensor is plugged into the dig/sonic 1 port. Jun 26, 2016 · METHOD: An experimental pendulum is suspended successively about several axes at different points along its length and the period about each axis is observed. Measure the period for five different lengths of string. Mar 17, 2023 · Kater’s pendulum is a type of reversible pendulum used to determine the value of the acceleration due to gravity (g) at a specific location. " . (Sample Lab Report) Joe Glotz (Partner: Al Thumbs) Physics 2048C Lab #12, Simple Pendulum November 12, 2007 Instructor: Dr. The investigation of the biflar pendulum includes determining the accuracy of the moment of inertia inferred from the Dec 19, 2023 · The math for this simulation is similar to the regular Double Pendulum with immobile support point, except there are extra terms carried forward into the equations for the support point velocity and acceleration. This experiment uses a uniform metallic bar with holes/slots cut down the middle at regular intervals. The double pendulum is a classic example in physics that involves the study of a system consisting of two interconnected pendulums. A laptop Lab Handout Physics App for simulation. e. Characterize the oscillations of pendulum at large amplitudes. The standard Double Pendulum simulation shows an idealized situation with massless rods that have point masses at their ends. Measure the length of the pendulum and use Equation (7) to calculate the period of For angles less than about 15º 15º, the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator. The single pendulum consists of a solid rod (a rectangular piece, in our case) attached to an oscillating pivot, about which the rod is free to rotate, as shown in Figure 1. Some experiments to try: Try dragging the anchor point by clicking near it with your mouse. The force on pendulum ain the direction of motion due to the coupling of the rubber band is K‘( a b), and the force on pendulum bdue to the The simple pendulum equation was given to us as T = 2 π l g. 5 m. The report can be found — warts and all — in PDF format here. You can vary friction and the strength of gravity. Use the pendulum to find the value of g on Planet X Explore the physics of pendulums with this interactive simulation. Use the pendulum to find the value of g on planet X. View PDF. Find the spring constant of this spring. Although you may ask your TA for help, each lab group is responsible to decide how to take and analyze data for the relevant quantities on their own in the second half of the lab. Ehgott (Created) 2/12 04/07/01 T. The pendulum system exhibits a wide range of interesting behaviors, from simple harmonic motion in the single pendulum to chaotic dynamics in multi-arm pendulums. For the simple pendulum: T =2π√m k = 2π√ m mg L T = 2 π m k = 2 π m m g L. Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It was invented by Captain Henry Kater in 1817, and it remains an important scientific instrument to this day. pendulum when it is displaced 5°, 10°, 15°, 20°, 25°, 30°, 40°, 50°, and 60° from its equilibrium position. 7 m as shown. Use the pendulum to find the value of g on Planet X May 7, 2009 · The double pendulum was also the subject of my lab report for the cycle. (3 pts) k = N/m 2. Pendulum on heavy stand, platform and black paper, fine sand. Martin, Emma Neary, Joshua Reinaldo, and Oliver Woodmen via data content that was edited the which choose and user regarding the LibreTexts device; a detail edit site Introduction: The pendulum experiment is used to explore the movement of an object that is suspended from a string. Therefore, to Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, the strength of gravity, and the amplitude of the swing. Hao (Revised) 08/07/16 Experiment 9: Compound Pendulum A compound pendulum (also known as a physical pendulum) consists of a rigid body oscillating about a pivot. hh pe oi lj xc mu pf om vh oc