Period of Pendulum

12 Key excerpts on "Period of Pendulum"

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  Pendulum in Physics

  • (Author)
  • 2014(Publication Date)
  • The English Press

    (Publisher)

  When given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines. Period of oscillation ________________________ WORLD TECHNOLOGIES ________________________ The period of a pendulum gets longer as the amplitude θ 0 (width of swing) increases ________________________ WORLD TECHNOLOGIES ________________________ The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ 0 , called the amplitude. It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is: where L is the length of the pendulum and g is the local acceleration of gravity. For small swings, the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude . This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time. This formula is strictly accurate only for tiny infinitesimal swings. For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of θ 0 = 23° it is 1% larger than given by (1). The true period cannot be represented by a closed formula but is given by an infinite series: The difference between this true period and the period for small swings (1) above is called the circular error . 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.

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  College Physics

  • Michael Tammaro(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  The period does not depend on the mass of the bob. If this surprises you, consider what happens when the mass is increased. The restoring force increases, but so does the inertia of the system. These two effects offset each other. It is sometimes useful to express the angular frequency ω of a simple pendulum in terms of its length L and the acceleration of gravity g. Because f T 2 2 ω π π = = / , we can use Equation 11.4.2 to write g L ω = (11.4.3) The Physical Pendulum A simple pendulum is an idealization because the mass is assumed to be concentrated at a single point. In a real pendulum, called a physical pendulum, the mass is distributed over a finite volume. Figure 11.4.1 shows a simple pendulum of length d and a physical pendulum. The physical pendulum is a uniform rod of length 2d that pivots about one end. The distance from its center of mass to the pivot point is d. Which pendulum has a longer period, or are they the same? The period of the physical pendulum depends on its moment of inertia I about the pivot point. [The moment of inertia (Section 8.5) is a measure of both the pendulum’s mass I N T E R A C T I V E F E A T U R E Energy in Simple Harmonic Motion | 297 and the distribution of mass about the pivot point.] Omitting the derivation, which is some- what complex, the period of a physical pendulum is given by Equation 11.4.4, T I mgd 2π = (11.4.4) where m is the pendulum’s mass, I is its moment of inertia about an axis through the pivot point, and d is the distance from its center of mass to its pivot point. Now let’s compare the periods of the simple and physical pendulums in Figure 11.4.1.

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  Workshop Physics Activity Guide Module 2

  Mechanics II

  • Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  However, the length of the pendulum has an obvious effect on the period. Design an experiment to quantify the dependence of the period of a pendulum on its length. For example, how does the period change if you double the length of the pendulum? Try to be as accurate as possible with your measurements. Describe your experiment and summarize your data and conclusions. Hint: Be sure to use some dramatically different lengths for your pendulum and measure the time for multiple oscillations to get a good average for the period. f. Does your result from part (e) agree with your prediction for how the length affects the period? Can you predict a mathematical relationship between the period and length of the pendulum? Theoretical Analysis of a Simple Pendulum In our analysis of the mass-spring system, we showed mathematically that if the restoring force is proportional to the object’s displacement, then the object will undergo SHM. In the mass-spring system, the restoring force on the object has the form F spring,x = −kx, where x is the displacement from equilibrium. In the next activity we will derive an equation governing the motion of a simple pen- dulum for small angles. As we will see, the equation we find will be very similar to the equation for the mass-spring system, demonstrating that the pendulum should also undergo SHM. To derive the equation of motion, we first note that the pendulum bob actually rotates about the point where the string is connected to the ceiling (see Fig. 14.4). Thus, we will use the rotational version of Newton’s second law with respect to this pivot point, assuming the z-axis points out of the page (so the counter-clockwise direction is positive). The rotational version of Newton’s second law is then  net,z = I z  z (14.8) where  net,z is the net torque, I z is the rotational inertia, and  z is the rotational acceleration (all about the z-axis). 476 WORKSHOP PHYSICS ACTIVITY GUIDE The situation is shown in Fig.

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  An Elementary Treatise on Theoretical Mechanics

  • Sir James H. Jeans(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  213. The importance of cycloidal motion is as follows. It has been seen that the motion of a simple pendulum is only strictly simple harmonic when the amplitude is so small that it may be treated as infinitesimal. For finite amplitudes the motion is not simple harmonic, and consequently the period is different from that of the simple harmonic motion described when the amplitude is very small. Thus the period depends on the amplitude, so that a clock which beats true seconds when the pendulum swings through one angle will gain or lose as soon as the pendulum is made to swing through any different angle. Variations of amplitude must always occur during the motion of any pendulum, and these cause irregularities in the timekeeping of the clock.

  We have, however, seen that if a particle describes a cycloid, the period is independent of the amplitude, so that variations of amplitude cannot affect the timekeeping powers of a particle moving in a cycloid.

  FIG.133

  The simplest way of causing a particle to move in a cycloid is, in practice, to suspend it from a fixed point by a string, in such a way that during its motion the string wraps and unwraps itself about two vertical cheeks. If the curve of these cheeks is rightly chosen, the particle can be made to describe a cycloid, and it can easily be shown that the curves of the cheeks must be portions of two cycloids each equal to the cycloid which is to be described by the particle.

      EXAMPLES  

  1. In cycloidal motion prove that the vertical component of the velocity of the particle is greatest when it has described half of its vertical descent.

  2 . A particle oscillates in a cycloid under gravity, the amplitude of the motion being b and the period being τ . Show that its velocity at a time t measured from a position of rest is .

   

  3. A particle of mass m slides on a smooth cycloid, starting from the cusp. Show that the pressure at any point is 2 mg cos ψ , where ψ

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  Principles of Engineering Physics 1

  • Md Nazoor Khan, Simanchala Panigrahi(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  i. The motion of a simple pendulum in vacuum is simple harmonic. Its time period is given by 2 , T g π =   = length of the simple pendulum. ii. Take a cleaned U-shaped glass tube and fix it vertically on a stand. Partially fill it with mercury. Now the levels of the mercury column in both sides are equal. Blow slowly through one end so that the mercury column in the other end rises slightly. When we stop blowing, the mercury column executes simple harmonic motion having time period 2 , 2 T g π =   = length of the mercury column in the U-tube. 8 Principles of Engineering Physics 1 iii. A body dropped into a hole dug through the centre of earth executes simple harmonic motion with time period 2 , R T g π = R = radius of Earth. iv. Place a small block inside a smooth curved surface having radius of curvature ‘ R ’. Move the sphere slightly along the curved surface away from the equilibrium position and then release. It will perform simple harmonic motion with time period 2 . R T g π = v. Place a sphere of radius ‘ r ’ inside a curved surface of radius of curvature ‘ R ’. Move the sphere slightly along the curved surface away from the equilibrium position and then release. It will execute simple harmonic motion with time period 7( ) 7 2 2 5 5 R r R T g g π π − = ≈ if R>>r vi. If a small iron cylinder, partially submerged in mercury vertically, is pressed slightly and then released, it will execute simple harmonic motion with time period 2 . T g ρ π ρ = ′  Here  = length of the cylinder, r = density of the cylinder, r ¢ = density of mercury. Example 1.1 A spring 15 cm in length is fixed to the ceiling. When a body of mass 1 kg is hung at the free end, its length becomes 17 cm. Calculate the spring constant of the spring. Solution The increase in length ‘ x ’ of the spring when a 1kg body is hung = 17 cm – 15 cm = 2 cm = 0.02 m The downward force acting on the spring = weight of the body hung.

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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  In this problem we show that if the path of the pendulum bob is a portion of a cycloid, the pendulum is isochronous, that is, the period of the oscillations is independent of the amplitude of the oscillations. This can be achieved by varying the length of the pendulum string as the pendulum swings back and forth. However, the analysis is easier if we replace the pendulum with the equivalent problem of a bead of mass m sliding without friction on a wire having the shape of a cycloid. Suggestion: Recall that simple harmonic motion (such as a mass on a spring) is isochronous. Write the Lagrangian for the bead on the wire in terms of the arc length s and show that the potential energy must be proportional to s 2 for isochronous motion. Then obtain expressions for x and y and show they can be expressed in the form of the parametric equations of a cycloid, x = ρ(φ + sin φ) and y = ρ(1 − cos φ). See Figure 12.16. Problem 12.22 Our analyses of pendulums have mostly ignored the effects of the environment. For example, the period of a pendulum depends on the local gravitational acceleration, and on the temperature (because the expansion and contraction of the bob changes the length). Furthermore, unless it is situated in a vacuum, a pendulum is affected by the air in various ways. Physicists have obtained formulas to account for the buoyancy of air, for the effect of air being dragged along by the pendulum and the resistance of air on the string and on the bob. Assume the pendulum consists of a massless string and a bob of mass M and volume V . In this problem, we consider the effect of the buoyancy of air which can be treated as an additional force acting opposite to gravity.

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  Mathematics and the Physical World

  • Morris Kline(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Another deduction from (16), which Galileo also made, is that the period is independent of the mass of the pendulum. Whether the bob is made of lead or cork the period will be the same.

  Formula (16) reveals the immense usefulness of the pendulum. By knowing the acceleration due to gravity, the quantity 32 in the formula, and the length l of the string to which the bob is attached, one can predict the period of the pendulum and hence measure time. Moreover, by fixing l, one can make the period whatever he wishes. Many years after Galileo studied the pendulum to learn some facts about its motion he designed a practical pendulum clock and had his students construct one. About fifteen years later, Huygens, working independently, constructed one that gained wide acceptance. Of course pendulum clocks do not work well on rolling ships and so the search for a reliable spring clock continued long after Galileo’s and Huygens’ time.

  In our treatment of the pendulum, as well as in that of the motion of the bob on a spring, we used the fact that the force of gravity acting on a mass m is 32m . The quantity 32 is of course the acceleration due to gravity at points near the surface of the earth. We know, however, that the acceleration of masses under the pull of gravitation is not always 32. In fact the acceleration is, more generally, the quantity GM/r 2 —formula (2) of chapter 16—where G is the gravitational constant, M is the mass of the earth, and r is the distance of the mass being accelerated from the center of the earth. Then the acceleration varies with distance from the center of the earth. Hence, if a pendulum clock is adjusted so that it beats seconds at one location, it will not be accurate at another location if the acceleration due to gravity is different there. The pendulum clock may have to be readjusted if it is moved from one location to another.

  It is an ill wind, however, that blows no good. The fact that the period of the pendulum depends upon the acceleration due to gravity was turned to excellent advantage by seventeenth- and eighteenth-century mathematicians and scientists. To follow their work let us first rewrite formula (16). In our derivation of this result we used 32 for the acceleration due to gravity. Let us now reconsider the derivation with the quantity 32 replaced by the symbol g, which is the customary symbol for the acceleration due to the direct action of gravity, as opposed to the symbol a, which is used for any acceleration, whether due to gravity or some other force. Then the derivation would continue to hold as long as g

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  Why Toast Lands Jelly-Side Down

  Zen and the Art of Physics Demonstrations

  • Robert Ehrlich(Author)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  But here is a more interesting test. Make a sim-ple pendulum (mass on a string), of length I = 2/3 meter, whose period is therefore expected to be T = 2vi/l/g = 2Tr^/2/3g. Based on the preceding equa-tion for the period of a physical pendulum, this sim-ple pendulum of length 2/3 meter should have exactly the same period as a physical pendulum of length one meter. All you need to do to verify this prediction is to allow the mass on the string of length 2/3 meter swing side by side with the meter stick on the paper clip, and see whether they remain swinging together. Another interesting variation of this demo is to al-low the meter stick to swing about some point other than one next to one end. How do you suppose the 124 Mechanical Oscillations and Waves period might change if the pivot point were made nearer the center of the stick? The general equation for the period of a physical pendulum with an arbi-trary pivot point can be expressed as T = 2iTy/I/mgd, where / is the moment of inertia, m is the mass, and d is the distance of the pivot to the center of gravity. Notice that this equation predicts an infinite period when d = 0 (pivot at the center of the stick), which is reasonable, because in that case the stick would not swing at all. If the period is infinite for a pivot point right at the center of the stick, you might think that the period would gradually lengthen as the pivot moved from one end toward the center, but surprisingly that is not the case. In fact, the preceding equation can be used to show that as the pivot approaches the cen-ter from one end, the period actually decreases for a while before eventually increasing to infinity. As a re-sult, there is a pivot point on the stick located a sixth of a meter from the center (d = 1/6 meter), which has exactly the same period as a point right next to the end.

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  Science 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.

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  Experiments And Demonstrations In Physics: Bar-ilan Physics Laboratory (2nd Edition)

  Bar-Ilan Physics Laboratory

  • Yaakov Kraftmakher(Author)
  • 2014(Publication Date)
  • World Scientific

    (Publisher)

  Its magnetic field is directed perpendicular to the disc, and the interaction of the induced eddy currents with the field leads to magnetic braking, according to Lenz’s law. With magnetic braking, the decay time of the pendulum can be varied over a wide range. Fig. 1. The driven physical pendulum. 2.5. Driven physical pendulum 95 time (s) time (s) angular position (rad) 8 5 0 1. Free oscillations The experiment includes the following items: (i) the magnetic damping; (ii) the period versus oscillation amplitude; and (iii) the phase portrait of free oscillations. Additional equipment: Rotary motion sensor , driven physical pendulum, permanent magnet, magnetic core, coil. For demonstrating free oscillations , it is sufficient to start the data acquisition and then to displace and release the pendulum. The decay of the oscillations depends on the magnetic damping (Fig. 2). Fig. 2. Free oscillations without and with magnetic damping. Period versus oscillation amplitude. Capstone calculates oscillation amplitudes in definite time intervals with the Calculator / Special / peakamp tool. The tool calculates half the distance between the maximum and minimum values in every time interval. An expression appears in the dialog box. The number (1) is the time range, over which the function is operating. Since the period of the pendulum oscillations is less than 1 s, this time (1 s) is quite acceptable. The place after the number is reserved for indicating the source of data chosen from the dialog box. Here, the source is the Angle measured by the Rotary motion sensor . The expression in the dialog box becomes Calc1 = . The Calculator / Special / period tool calculates the oscillation period. An expression appears in the dialog box. The two first numbers (10,10) are the thresholds to finding the peaks and valleys in the data source.

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  Simulations of Oscillatory Systems

  with Award-Winning Software, Physics of Oscillations

  • Eugene I. Butikov(Author)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  For the oscillatory motion under consideration, the representative point in the phase plane generates a closed path during one cycle, passing along both branches of the separatrix. The pendulum goes twice around almost the whole circle, cov-ering it in both directions. On the other hand, executing rotation, the pendulum makes one circle during a cycle of revolutions, and the representative point passes along one branch of the separatrix (upper or lower, depending on the direction of rotation). To explain why the period of these oscillations is twice the period of corresponding revolutions, we must show that the motion of the pendulum with energy E = E max − Δ E from ϕ = 0 up to the extreme point requires the same time as the motion with the energy E = E max + Δ E from ϕ = 0 up to the inverted vertical position. The major part of each of both motions under consideration occurs very nearly along the same path in the phase plane, namely, along the separatrix from the initial point ϕ = 0 , ˙ ϕ ≈ 2 ω 0 up to some angle ϕ 0 whose value is close to π . 176 CHAPTER 7. OSCILLATIONS OF THE RIGID PENDULUM We choose this value ϕ 0 arbitrarily. To calculate the time interval required for this part of the motion, we can assume that the motion (in both cases) occurs exactly along the separatrix, and take advantage of the corresponding analytical solution, expressed by Eq. (7.12). Assuming ϕ ( t ) in Eq. (7.12) to be equal to ϕ 0 , we can find the time t 0 during which the pendulum moves from the equilibrium position ϕ = 0 up to the angle ϕ 0 (for both cases): ω 0 t 0 = − ln tan π − ϕ 0 4 = − ln tan α 0 4 , (7.29) where we introduced the notation α 0 = π − ϕ 0 for the angle that the pendulum at ϕ = ϕ 0 forms with the upward vertical line. When ϕ 0 is close to π , the angle α 0 is small, so that in Eq. (7.29) we can assume tan( α 0 / 4) ≈ α 0 / 4 . Therefore ω 0 t 0 ≈ ln(4 /α 0 ) .

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  Science Teaching

  The Contribution of History and Philosophy of Science, 20th Anniversary Revised and Expanded Edition

  • Michael R. Matthews(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  This does no harm and can do some good, but there is nothing noteworthy here. Everything about the rich history of the pendulum has been stripped out: no mention of Galileo, Huygens, Newton, Hooke, universal gravitation, timekeeping, clocks, length standards, longitude, shape of the Earth or conservation laws. No connection intimated between science, technology and society; no sense of participation in a scientific tradition. Nothing. Teachers are not even told to talk about these great scientists and their pendulum-based discoveries.

  And in this set piece, nationally distributed, ‘model pendulum lesson’ teachers and students are told to plot period against length on a graph. This is a task with only minimally useful outcomes: such a graph provides a scatter of points that merely establishes a trend. After having done this, 17–18-year-old students could have been so easily asked to plot period against the square root of length. When this is done, a straight line is obtained from the data, not a scatter of points. As discussed above, the physical phenomenon is revealed by mathematical manipulation. Period is seen, as it was by Galileo and Huygens, not just to vary with length, but to vary directly with the square root of length; the conclusion from the data moves from inconclusive T ∝ L to conclusive T = k√L. The model lesson tells teachers to ‘avoid introducing the formal pendulum equation, because the laboratory activity is not designed to verify this known relationship’ (NRC 2006, p. 129). Final-year students in Japan, Korea, Singapore and a good deal of the rest of the world have no such problem, and US students deserve better than a dumbed-down, HPS-free curriculum.

  The graph of period against square root of length shows, in a manageable way, the dramatic impact of mathematics on physics; without the mathematical notion of square root, we see qualitative trends; utilising the square root, we see a precise, quantitative relationship. Further, this precise relationship will allow the pendulum to be connected with free fall, where distance of fall varies as the square of time. All of this is missed in the Lab Report

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