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How Short and Light Can a Simple Pendulum Be for Classroom Use?
Published in P. I. C. Teixeira, Offbeat Physics, 2022
The simple pendulum is a trivial experiment commonly done by high-school and university students to determine the local acceleration of gravity. A simple pendulum is ideally described as a point mass suspended by a massless rigid rod from some pivot point, about which it is allowed to oscillate. Due to its beauty and simplicity, the simple pendulum is frequently addressed in introductory physics textbooks [1] and has been intensively studied in the literature [2–7]. Nevertheless, most reported studies deal with the extension of the small-angle approximation to large-angle oscillations [4–7], and very little attention has been given to the physical characteristics of a real classroom pendulum [8]. In fact, a real simple pendulum is often constructed using a metal sphere suspended on a string. In order to treat this pendulum as ideal, the metal sphere must have a small radius and a large mass compared to the length and mass of the string from which it is suspended. When the aim is to determine the acceleration of gravity, what should the minimum length of the string and the minimum mass of the sphere be to treat a real pendulum as ideal? The purpose of this paper is to answer this question. We, therefore, compare the period of oscillation of an ideal pendulum with that of a more realistic pendulum composed of a rigid sphere and a rigid slender rod. We determine the relative error in calculating the acceleration of gravity if the simple pendulum period is used instead of the “real” pendulum period.
Terrestrial Locomotion
Published in Malcolm S. Gordon, Reinhard Blickhan, John O. Dabiri, John J. Videler, Animal Locomotion, 2017
For ωlTstep = 4.058, the swing leg has again the rotational velocity of φ˙swing. This implies that the step duration amounts to 0.646 the swing period Tl. In other words, the step duration that is half the stride period exceeds the half-period of the pendulum by a factor of 1.29. It should be noted that in the region where the small angle solution of the pendulum is valid, the pendulum period does not depend on deflection amplitude. Within a limited range (for human-sized walkers <3 m/s), this mechanism allows walking at different speeds by changing the amplitude of the pendulum.
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Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[general] Galileo Galilei (1564–1642) in 1581 observed that a candelabrum in the cathedral in Pisa was pulled and released to reveal a swing motion that appeared to have a steady constant rhythm, the period of the swing. He supposedly used his heart beat as a time-keeping mechanism, because (stop-) watches were not available, only “hour” time pieces. Galilei was studying medicine at the time and the use of his pulse beat may have been an obvious choice (order of second [s]). The period (T) of a pendulum is directly proportional to the square root of the length of the pendulum (ℓ) and is independent of the mass attached, for relatively small angular displacement T≈2π(ℓ/g), where g is the gravitational acceleration (see Figure G.8).
Motion of a simple pendulum: a digital technology approach
Published in International Journal of Mathematical Education in Science and Technology, 2021
Antonio Rivera-Figueroa, Isaías Lima-Zempoalteca
A pendulum consists of a weight, which is a body of point mass m, attached to a non-deformable rod of negligible mass and length a (Figure 1). The pendulum is in a stable equilibrium state when the mass hangs from the rod in vertical position, that is, when the mass is at its lowest possible point. When the body moves from its equilibrium position to a position at which the rod forms an angle with the vertical (Figure 1), from where it is released from rest, it will move due to the force of gravity, describing an oscillatory motion along the arc of a circle on a vertical plane. An oscillation or cycle is the motion of the pendulum from its resting position until it reaches this position again for the first time. During an oscillation, the pendulum departs from its resting position, passes through its equilibrium position, continues its motion until stopping, and returns, passing for a second time through its equilibrium position, to reach the position from where the motion was initiated. The time taken by the pendulum to perform this cycle is called the oscillation period.
Dynamics of carving runs in alpine skiing. II.Centrifugal pendulum with a retractable leg
Published in Sports Biomechanics, 2022
Similarly to the basic model, the pendulum period increases with the length of its leg (Komissarov, 2020). In application to skiing this means that, given the same equipment, shorter skiers will tend to make shorter turns compared to taller skiers. To compensate for this, they may be forced to go for more extreme inclination, which increases the turn duration. In agreement with this general period dependence on the pendulum length, we find that contraction of the pendulum leg near the vertical position reduces the oscillation period compared to the case where it remains unchanged (the same as in the extreme position). In skiing, this implies that leg flexion in transition between turns makes them shorter, both in duration and in length.