Physics
Simple Pendulum
A simple pendulum is a weight suspended from a pivot point that oscillates back and forth under the influence of gravity. The motion of a simple pendulum is periodic and can be described using the principles of harmonic motion. The period of the pendulum, or the time it takes to complete one full oscillation, depends on the length of the pendulum and the acceleration due to gravity.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "Simple Pendulum"
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
The Simple Pendulum A Simple Pendulum is an idealized model of a pendulum that consists of a point mass (called the bob) that is connected to the end of a massless rod or string and oscillates without friction about a fixed pivot point. Video 11.4.1 shows the motion of a Simple Pendulum and illustrates the key variables. The length of the string is L and the mass of the bob is m. The two forces acting on the pendulum bob are gravity, with magnitude mg, and the tension force, with magnitude T. Unlike the mass on a spring, which moves along a straight line, the pendulum bob moves along the arc of a circle of radius L. We define x as the distance along the arc from the equilibrium position, at which x 0 = and the string is vertical. 11.4 THE PENDULUM Learning Objectives The Pendulum | 295 Suppose that the string makes an angle θ to the vertical. The tension force has no component along the arc, but the x component of the weight is given by F mg sin x θ = − This equation is not in the form of Equation 11.1.4, F Cx = − . For small angles (measured in radians), however, x L sinθ θ ≈ = / , and we can write F mg L x x ≈ − (11.4.1) The force F x is the net force along the arc and, since it has the form F Cx = − , the bob will perform simple harmonic motion along the arc (provided that the oscillations are small). Video 11.4.1 A Simple Pendulum consists of a small mass connected to a light string. The restoring force is the component of the weight that is parallel to the arc along which the mass moves. I N T E R A C T I V E F E A T U R E Going Deeper Justification of the Small-Angle Approximation Here we will justify the approximation sinθ θ ≈ for small angles (where θ is measured in radians). The figure shows a circle of radius a and right triangle with one small angle, θ , whose adjacent side is a radius of the circle. Arc Radius h o a θ
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
KEY IDEAS • A Simple Pendulum consists of a rod of negligible mass that pivots about its upper end, with a particle (the bob) attached at its lower end. If the rod swings through only small angles, its motion is approximately simple harmonic motion with a period given by T = 2 √ I mgh , where I is the particle’s rotational inertia about the pivot, m is the particle’s mass, and L is the rod’s length. • If all the mass of a Simple Pendulum is concentrated in the mass m of the particle-like bob, which is at radius L from the pivot point, then the period simplifes to T = 2 √ L g . • A physical pendulum has a more complicated distribution of mass. For small angles of swinging, its motion is simple harmonic motion with a period given by T = 2 √ I mgh , where I is the pendulum’s rotational inertia about the pivot, m is the pendulum’s mass, and h is the distance between the pivot and the pendulum’s centre of mass. We turn now to a class of simple harmonic oscillators in which the springiness is associated with the gravitational force rather than with the elastic properties of a twisted wire or a compressed or stretched spring. The Simple Pendulum If an apple swings on a long thread, does it have simple harmonic motion? If so, what is the period T ? To answer, we consider a Simple Pendulum, which consists of a particle of mass m (called the bob of the pendulum) suspended from one end of an unstretchable, massless string of length L that is fixed at the other end, as in figure 15.14. Pdf_Folio:300 300 Fundamentals of physics FIGURE 15.14 A Simple Pendulum. L m Pivot point The bob is free to swing back and forth in the plane of the figure, to the left and right of a vertical line through the pendulum’s pivot point. The restoring torque The forces acting on the bob are the force T from the string and the gravitational force F g , as shown in figure 15.15 where the string makes an angle with the vertical.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- The English Press(Publisher)
Mathematically, for small swings the pendulum approximates a harmonic oscillator, and its motion approximates to simple harmonic motion: Compound pendulum The length L of the ideal Simple Pendulum above, used for calculating the period, is the distance from the pivot point to the center of mass of the bob. For a real pendulum consisting of a swinging rigid body, called a compound pendulum , the length is more difficult to define. A real pendulum swings with the same period as a Simple Pendulum with a length equal to the distance from the pivot point to a point in the pendulum called the center of oscillation . This is located under the center of mass, at a distance called the radius of gyration, that depends on the mass distribution along the pendulum. However, for the usual sort of pendulum in which most of the mass is concentrated in the bob, the center of oscillation is close to the center of mass. ________________________ WORLD TECHNOLOGIES ________________________ Christiaan Huygens proved in 1673 that the pivot point and the center of oscillation are interchangeable. This means if any pendulum is turned upside down and swung from a pivot at the center of oscillation, it will have the same period as before, and the new center of oscillation will be the old pivot point. History One of the earliest known uses of a pendulum was in the 1st century seismometer device of Han Dynasty Chinese scientist Zhang Heng. Its function was to sway and activate one of a series of levers after being disturbed by the tremor of an earthquake far away. Released by a lever, a small ball would fall out of the urn-shaped device into one of eight metal toad's mouths below, at the eight points of the compass, signifying the direction the earthquake was located. Many sources claim that the 10th century Egyptian astronomer Ibn Yunus used a pendulum for time measurement, but this was an error that originated in 1684 with the British historian Edward Bernard.
eBook - PDFWorkshop Physics Activity Guide Module 2
Mechanics II
- Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Name Section Date UNIT 14: SIMPLE HARMONIC MOTION Alaettin YILDIRIM/Shutterstock.com The swing of this antique clock pendulum is one of many examples of what physicists call simple harmonic motion. Due to the pendulum motion, each tick of the clock takes the same amount of time, and so it serves as a reliable timekeeper. And if the clock is running slow or fast, one can adjust the pendulum to compensate. By the end of this unit, you should have an understanding of why a pendulum undergoes simple harmonic motion. 460 WORKSHOP PHYSICS ACTIVITY GUIDE UNIT 14: SIMPLE HARMONIC MOTION OBJECTIVES 1. To learn about the quantities used to describe periodic (and harmonic) motion, such as period, frequency, amplitude, and phase. 2. To understand the basic properties of simple harmonic motion, in which the displacement of an object varies sinusoidally in time, and to experi- mentally measure such motion in different systems. 3. To use Newton’s laws to theoretically explore the factors that influence the motion of both a mass-spring system and a Simple Pendulum. 14.1 OVERVIEW Any motion that repeats itself regularly is known as periodic motion. The pendulum in a grandfather clock, molecules in a crystal, the vibrations of a car after it encounters a pothole on the road, and the rotation of Earth around the sun are examples of periodic motion. In this unit we will be especially interested in a type of periodic motion known as simple harmonic motion, which is often abbreviated as SHM. Simple harmonic motion involves a displacement that changes sinusoidally in time. We will study the behavior of two systems that undergo SHM—an object hanging from a spring (a mass-spring system) and a Simple Pendulum that oscillates at small angles. Pendula and masses on springs are two common examples of periodic systems that oscillate with SHM (or at least approximately SHM).
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
Thus, as the pendulum bob moves to the right, as in Fig. 15-11a, its acceleration to the left increases until the bob stops and Key Ideas ● A Simple Pendulum consists of a rod of negligible mass that pivots about its upper end, with a particle (the bob) attached at its lower end. If the rod swings through only small angles, its motion is approximately simple harmonic motion with a period given by T = 2π √ I mgL (Simple Pendulum), where I is the particle’s rotational inertia about the pivot, m is the particle’s mass, and L is the rod’s length. ● A physical pendulum has a more complicated distribu- tion of mass. For small angles of swinging, its motion is simple harmonic motion with a period given by T = 2π √ I mgh (physical pendulum), where I is the pendulum’s rotational inertia about the pivot, m is the pendulum’s mass, and h is the distance between the pivot and the pendulum’s center of mass. ● Simple harmonic motion corresponds to the projection of uniform circular motion onto a diameter of the circle. Figure 15-11 (a) A Simple Pendulum. (b) The forces acting on the bob are the gravitational force F → g and the force T → from the string. The tangential component F g sin θ of the gravitational force is a restoring force that tends to bring the pendulum back to its central position. θ L θ θ F g sin θ F g cos m s = Lθ L m (a) (b) Pivot point T F g This component merely pulls on the string. This component brings the bob back to center. 426 CHAPTER 15 OSCILLATIONS Figure 15-12 A physical pendulum. The restoring torque is hF g sin θ. When θ = 0, center of mass C hangs directly below pivot point O. begins moving to the left. Then, when it is to the left of the equilibrium posi- tion, its acceleration to the right tends to return it to the right, and so on, as it swings back and forth in SHM. More precisely, the motion of a simple pendu- lum swinging through only small angles is approximately SHM.
eBook - PDFMechanics
Lectures on Theoretical Physics
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
CHAPTER III OSCILLATION PROBLEMS The investigations that are to follow will teach us nothing new about the principles of mechanics. So great, however, is the significance of oscillation processes for physics and engineering that their separate systematic treatment is deemed essential. § 15. The Simple Pendulum The oscillating body is a particle of mass m which is attached to a fixed point 0 by means of a weightless rigid rod of length I ; I is called the length of the pendulum. We may neglect friction at the point of suspension and air resistance, so that the only force acting is that of gravity, with a component— mg sin φ in the direction of increasing φ (cf. Fig. 24). The general equation (11.14) for the guided motion along an arbitrary path gives us, with ν=1φ (circular path), the exact equation (1) ml T? = — mg sin φ. For sufficiently small oscillations, <£<^1, we can put sin φ = φ. With the abbreviation (2) FIG. 24. Simple Pendulum. Com-ponent of gravity along the direction of motion. we then obtain the linear pendulum equation (3) S+-V-0. This is the differential equation of harmonic oscillations as treated in § 3 (4). Apart from the designation for the dependent variable it is identical with Eq. (3.23). The circular frequency ω defined in (3.22) is now given by Eq. (2) above. We therefore have / 4 j 2TT {g* m 0 _/Z*_ 2π T ~~ Ψ ^Ν. 87 88 Oscillation Problems 111.15 Notice that T is independent of the mass ra, which dropped out already in (1). Thus different masses have the same period if the pendulum length I is the same. T is the full period, covering a complete swing to and fro. Sometimes one half of this time is designated as the period of oscillation. Thus one speaks of a seconds pendulum for which T equals one second. Its length is calculated from (4) to be î= —a = 1 meter. To the extent to which Eq. (3) is valid the period of oscillation is indepen-dent also of the amplitude of swing; i.e., small pendulum oscillations are isochronous.
eBook - PDFMechanics
Lectures on Theoretical Physics, Vol. 1
- Arnold Sommerfeld(Author)
- 2016(Publication Date)
- Academic Press(Publisher)
CHAPTER I I I OSCILLATION PROBLEMS The investigations that are to follow will teach us nothing new about the principles of mechanics. So great, however, is the significance of oscillation processes for physics and engineering that their separate systematic treatment is deemed essential. § 15. The Simple Pendulum The oscillating body is a particle of mass m which is attached to a fixed point 0 by means of a weightless rigid rod of length I ; I is called the length of the pendulum. We may neglect friction at the point of suspension and air resistance, so that the only force acting is that of gravity, with a component — mg sin φ in the direction of increasing φ (cf. Fig. 24). The general equation (11.14) for the guided motion along an arbitrary path gives us, with ν = 1φ (circular path), the exact equation (1) ml dïï = -mg sin < For sufficiently small oscillations, φ<^.1, we can put sin φ = φ. With the abbreviation (2) ι FIG. 24. Simple Pendulum. Com-ponent of gravity along the direction of motion. we then obtain the linear pendulum equation (3) ^ + ^ = 0. This is the differential equation of harmonic oscillations as treated in § 3 (4). Apart from the designation for the dependent variable it is identical with Eq. (3.23). The circular frequency ω defined in (3.22) is now given by Eq. (2) above. We therefore have (4) 2π : T ~~ :?)·· T=2TT zu 87 88 Oscillation Problems 111.15 Notice that T is independent of the mass m, which dropped out already in (1). Thus different masses have the same period if the pendulum length I is the same. T is the full period, covering a complete swing to and fro. Sometimes one half of this time is designated as the period of oscillation. Thus one speaks of a seconds pendulum for which T equals one second. Its length is calculated from (4) to be 1= —,* ^ 1 meter. To the extent to which Eq. (3) is valid the period of oscillation is indepen-dent also of the amplitude of swing; i.e., small pendulum oscillations are isochronous.
eBook - PDFAnalytical Mechanics
Solutions to Problems in Classical Physics
- Ioan Merches, Daniel Radu(Authors)
- 2014(Publication Date)
- CRC Press(Publisher)
CHAPTER III THE Simple Pendulum PROBLEM The aim of this section is to solve the fundamental problem of me-chanics for a constrained system, namely the Simple Pendulum, by means of six different methods connected to: (i) classical (Newtonian) approach; (ii) Lagrange equations of the first kind; (iii) Lagrange equa-tions of the second kind; (iv) Hamilton’s canonical equations; (v) the Hamilton-Jacobi formalism, and (vi) the action-angle formalism. This way, we shall put into evidence the resemblances and differences be-tween these formalisms, on the one side, and show the generality and potency of the analytical formalism, as compared to the classical one, on the other. The physical system we are going to study is represented by a material point (particle), suspended by a massless rod, constrained to move without friction on a circle in a vertical plane, under the influence of a uniform and homogeneous gravitational field. It is our purpose to study the motion of such a system only in case of free, harmonic, and non-amortized oscillations, performing motions of an arbitrary amplitude. The study of the most general case (non-linear damped and -eventually -forced oscillations) implies knowledge of notions like: bifurcation points, strange attractors, Lyapunov coefficients, etc., which overpass our approach. Our discussion stands for an application of the various methods offered by analytical mechanics which can be used for solving a problem. III.1. Classical (Newtonian) formalism According to this approach, the known elements are: the mass of the body, the acting forces (including the constraint forces), and the initial conditions compatible with the constraints. The reader is asked to find the equation of motion and the elements/characteristics of the 47 motion: trajectory, period, frequency, etc. As well-known, determi-nation of the solution is based on the second Newtonian low ( lex se-cunda ), which furnishes the differential equation of motion.
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
Therefore, such a pendulum could be a useful measure of time. The other important observation in this equation is that it embraces the gravitational acceleration g. This makes the Simple Pendulum of a dual benefit. Measuring the period T can help determine the gravitational acceleration, g, which is a measure of the strength of the earth’s gravitational field. Thus, Equation 9.37 has a more subtle implication that a Simple Pendulum may be used to measure the strength of the gravitational field of any planet on which such an experiment could be carried out. For example, the period of a Simple Pendulum on the Moon relative to the pendulum’s period on the Earth would be of the value T T g g . M E E M = (9.38) EXAMPLE 9.5 Determine (a) the period, (b) the frequency, and (c) the angular frequency of a Simple Pendulum of length ℓ = 0.40 m. S OLUT ION a. From Equation 9.37, T 2 g = π H5129 / , which upon substituting for ℓ = 0.400 m and g = 9.80 m/s 2 gives T 2 0.400 m 9.80 m s 1.27 s. 2 = π / = b. The frequency f = 1/T. Thus, f = 1/1.27 s = 0.787 s − 1 . c. From Equation 9.11, the angular frequency is ω = 2 π f. Thus, ω π = 2 (rad)(0.787 s ) 4.95 rad s. 1 / -= An analysis parallel to that described for a mass-spring system in Section 9.3.2 can be presented for a Simple Pendulum. The main difference is that the displacement of the pendulum is angular, while in the mass-spring system, it is linear. Thus, for a Simple Pendulum, the angular displacement 189 Simple Harmonic Motion © 2010 Taylor & Francis Group, LLC from equilibrium is θ and its amplitude is the largest value, θ o , through which the pendulum’s bob is pulled to the side before it is released. The periodic function expressing θ as a function of t can take the form θ = θ o cos ( ω o t), (9.39) where ω o is the natural angular frequency of the pendulum and ω o = 2 π f o , (9.40) where f o is the natural linear frequency of the pendulum’s SHM.
- Joseph C. Amato, Enrique J. Galvez(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
A body undergoing simple SHM, or sinusoidal oscillation , experiences a linear restoring force proportional to its displacement x from its stable equilibrium position: F x ( t ) = − kx ( t ). In either case, the motion of the body is periodic and can be described by trigonometric functions. For SHM, x ( t ) = A cos( ω t + θ 0 ) or equivalently x t A t ( ) sin( ) = + ′ ω θ 0 , where A is the amplitude of the oscillation and ω is the angular frequency. The initial phase angle θ θ π 0 0 2 ( / = ′ + ) is determined by the position x 0 and velocity v 0 of the body at t = 0. Periodic systems that are not inherently sinusoidal (such as the pendulum) behave like simple harmonic oscillators in the limit of small displace -ments from equilibrium. The mathematics of the mass–spring system or the pendulum is applicable to all sinusoidal oscilla -tors, on all time and length scales, throughout the universe. Whenever a system’s differential equation of motion takes the form a x = d 2 x / dt 2 = − C 2 x , the system undergoes SHM, x ( t ) = A cos( ω t + θ 0 ) with angular frequency ω = C . For the one-body mass–spring system, ω = k m / , while for the pendulum ω = g l / . For the two-body mass–spring oscillator, the equations of motion can be combined to form a new differential equation that predicts SHM for the reduced mass μ = m 1 m 2 /( m 1 + m 2 ) with frequency ω µ = k / . This pro -cedure can be extended to systems with much greater complexity. The sensation of weightlessness led Einstein to propose his Equivalence Principle, and from it, the General Theory of Relativity. Using the principle, we found that a gravitational field deflects light, and also that it causes clocks (time) at different altitudes to advance at different rates, an effect clearly seen in the orbiting atomic clocks of the Global Positional System. In the Principia , Isaac Newton took great pains to define his scientific terms precisely—with several notable exceptions.
eBook - PDFScience Teaching
The Contribution of History and Philosophy of Science, 20th Anniversary Revised and Expanded Edition
- Michael R. Matthews(Author)
- 2014(Publication Date)
- Routledge(Publisher)
An isochronic pendulum is one in which the period of the first swing is equal to that of all subsequent swings: this implies perpetual motion. We know that any pendulum, when let swing, will very soon come to a halt: the period of the last swing will be by no means the same as the first. Furthermore, it was plain to see that cork and lead pendulums have a slightly different frequency, and that large-amplitude swings do take somewhat longer than small-amplitude swings for the same pendulum length. All of this was pointed out to Galileo, and he was reminded of Aristotle’s basic methodological claim that the evidence of the senses is to be preferred over other evidence in developing an understanding of the world. The fundamental laws of classical mechanics are not verified in experience; further, their direct verification is fundamentally impossible. Herbert Butterfield (1900–1979) conveys something of the problem that Galileo and Newton had in forging their new science: 11 They were discussing not real bodies as we actually observe them in the real world, but geometrical bodies moving in a world without resistance and without gravity History and Philosophy: Pendulum Motion 227 – moving in that boundless emptiness of Euclidean space which Aristotle had regarded as unthinkable. In the long run, therefore, we have to recognise that here was a problem of a fundamental nature, and it could not be solved by close observation within the framework of the older system of ideas – it required a transposition in the mind. (Butterfield 1949/1957, p. 5) An objectivist, non-empiricist account of science stresses that the transposi- tion in the mind is really the creation of a new theoretical object or system. Even for Galileo, the pendulum seemed to stop at the top of its swing; it was only in his theory, not his perceptual mind, that it continued in smooth motion.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
