Modeling a falling slinky

Cross, Rod; Wheatland, M. S.

doi:10.1119/1.4750489

Cited by 25 publications

(42 citation statements)

References 12 publications

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“…(2) resembles the numerator of the second central finite difference with respect to position along the bar, and that finite difference is an approximation to the second spatial derivative. Hence these equations resemble the nonhomogeneous wave equation used to model a falling Slinky (or similar spring) in Calkin [1], Aguirregabiria et al [4], and Cross and Wheatland [7]. Eqs.…”

Section: Formulationmentioning

confidence: 94%

“…(4), the dimensional collapse time is t n ¼ ffiffiffiffiffiffiffiffiffi m=k p ½ðN 2 À1Þ=3 1=2 . For the particular 86-coil metal Slinky A in Cross and Wheatland [7], this formula furnishes the approximate dimensional collapse time t n ¼ 0:29 s.…”

Section: Collapse Time Of a Slinkymentioning

confidence: 98%

“…The associated formula derived in Calkin [1] and in Cross and Wheatland [7] with a continuous representation of the Slinky is ½Mξ 3 1 =ð3KÞ 1=2 where M is the total mass of the spring, K is the total stiffness of the spring, and ξ 1 ¼ ðn À n p Þ=n. That formula yields a collapse time of 0.27 s, slightly lower than the approximate one from the discrete model.…”

Section: Collapse Time Of a Slinkymentioning

confidence: 99%

“…In particular, the collapse time of the Slinky is estimated, which is defined as the time from when the top mass is released until all masses are stuck together, and corresponds to the "total collapse time" discussed in Calkin [1] and in Cross and Wheatland [7].…”

Section: Collapse Time Of a Slinkymentioning

confidence: 99%

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Falling vertical chain of oscillators, including collisions, damping, and pretensioning

Plaut

Borum

Holmes

et al. 2015

Journal of Sound and Vibration

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Section: Formulationmentioning

confidence: 94%

Section: Collapse Time Of a Slinkymentioning

confidence: 98%

Section: Collapse Time Of a Slinkymentioning

confidence: 99%

Section: Collapse Time Of a Slinkymentioning

confidence: 99%

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Falling vertical chain of oscillators, including collisions, damping, and pretensioning

Plaut

Borum

Holmes

et al. 2015

Journal of Sound and Vibration

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“…The slinky is a toy of spring, that can walk downstairs. The interesting behavior of a falling slinky was studied by several authors using the wave equation for an elastic wave, which is a partial differential equation [4][5][6]. The bottom particle does not move until the wave of deformation reaches the bottom, because information on the imbalance of force propagates with a finite velocity for an elastic wave.…”

mentioning

confidence: 99%

Shock Waves in Falling Coupled Harmonic Oscillators

Sakaguchi

2013

J. Phys. Soc. Jpn.

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Shock waves propagate in falling coupled harmonic oscillators. The bottom end of coupled harmonic oscillators does not fall downwards until a shock wave reaches the bottom end. The exact solution can be expressed by the Fourier series expansion, and an approximate solution can be expressed by the integral of the Airy function. The width of the shock wave increases slowly in accordance with a power law.Shock waves are generated in compressive fluids. Typical shock waves appear in air compressed by supersonic planes or meteorites. There is a jump in the fields of pressure, temperature, and fluid velocity. The Rankine-Hugoniot relation is satisfied for the jump under normal shock [1]. There have been numerous investigations of shock waves [2]. It is considered that a shock wave is a typical nonlinear wave. The simplest model of a shock wave is the Burgers equation [3]. Nonlinearity and dissipation are essential for shock waves.We consider a linear chain of coupled harmonic oscillators under gravity. It is a linear system and there is no dissipation. It is a typical system of particles considered in a basic course of mechanics. However, there is a nontrivial phenomenon similar to a shock wave in this simple system. A similar phenomenon was discussed in falling elastic bars, using a partial differential equation [4]. We will focus on the effect of the discreteness in this paper.The model equation is written aswhere N is the total number of particles of mass m, k is the spring constant, a is the natural length of the spring, x i is the height of the ith particle, and g denotes the acceleration of gravity. The heights of the bottom and top particles are expressed respectively as x 1 and x N . If x N is fixed to a constant value x N 0 by holding the top particle, the stationary positions of the other particles are determined from the relation x i+1 − x i = a + (mgi/k) aswhere the position of the bottom particle, x 1 , is expressed asWe study the free-fall motion of this system by releasing the top particle with an initial velocity of 0 from the stationary state. There are five parameters, i.e., m, k, a, g, and N , in this system, but N is the only essential parameter. The other parameters can be set to a unit value by changing the scales of x and t. Figure 1 shows the positions of ten particles at t = 0, 1, · · · , 10 for N = 10, k = 1, g = 0.1, m = 1, and a = 1. The top particle falls with a nearly constant velocity. The bottom particle does not move until t ∼ 8 in the gravity field.

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The Simple Harmonic Oscillator

Garrett

2020

Graduate Texts in Physics

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This chapter will introduce a system that is fundamental to our understanding of more physical phenomena than any other. Although the “simple” harmonic oscillator seems to be only the combination of the most mundane components, the formalism developed to explain the behavior of a mass, spring, and damper is used to describe systems that range in size from atoms to oceans. Our investigation goes beyond the “traditional” treatments found in the elementary physics textbooks. For example, the introduction of damping will open a two-way street: a damping element (i.e., a mechanical resistance, Rm) will dissipate the oscillator’s energy, reducing the amplitudes of successive oscillations, but it will also connect the oscillator to the surrounding environment that will return thermal energy to the oscillator. The excitation of a harmonic oscillator by an externally applied force, displacement, or combination of the two will result in a response that is critically dependent upon the relationship between the frequency of excitation and the natural frequency of the oscillator and will introduce the critical concepts of mechanical impedance, resonance, and quality factor. Finally, the harmonic oscillator model will be extended to coupled oscillators that are represented by combinations of several masses and several springs.

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Modeling a falling slinky

Cited by 25 publications

References 12 publications

Falling vertical chain of oscillators, including collisions, damping, and pretensioning

Falling vertical chain of oscillators, including collisions, damping, and pretensioning

Shock Waves in Falling Coupled Harmonic Oscillators

The Simple Harmonic Oscillator

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