Related papers: Deriving Pi from Colliding Blocks
We discuss a method for calculating $ \pi $ using elastic collision between two masses $ M $ and $ m $, with $ X = m/M = 10^{2(1-d)} $ where $ d $ is an integer, and a wall. The total number of collisions between $ M $, $ m $ and the wall…
In this study, we conducted an experiment to estimate $\pi$ using body-to-body and body-to-wall collisions. By geometrically analyzing the system's motion, we first review how the collision count corresponds to the digits of $\pi$. This…
In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number $\pi$. This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of $\pi$ in a base…
Galperin introduced an interesting method to learn the digits of $\pi $ by counting the collisions of two billiard balls and a hard wall. This paper studies two quantum versions of the Galperin billiards. It is shown that the digits of $\pi…
In this paper, we list several interesting structures of cyclotomic polynomials: specifically relations among blocks obtained by suitable partition of cyclotomic polynomials. We present explicit and self-contained proof for all of them,…
We study multiple elastic collisions of a block and a ball against a rigid wall in one dimension. The complete trajectory of the block is solved as an analytic function of time. Near the turning point of the block the force carried by the…
A famous pre-Newtonian formula for $\pi$ is obtained directly from the variational approach to the spectrum of the hydrogen atom in spaces of arbitrary dimensions greater than one, including the physical three dimensions.
We study the fluid squeeze-out from the interface between an elastic solid with a flat surface and a rigid solid with a randomly rough surface. As an application we discuss fluid squeeze-out between a tire tread block and a road surface.…
A computational study of sliding blocks on inclined surfaces is presented. Assuming that the friction coefficient $\mu$ is a function of position, the probability $P(\lambda)$ for the block to slide down over a length $\lambda$ is…
A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.
Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals.…
We consider the response of an adsorbed polymer that is pulled by an AFM within a simple geometric framework. We separately consider the cases of i) fixed polymer-surface contact point, ii) sticky case where the polymer is peeled off from…
We calculate the coefficient of restitution, $\epsilon$, starting from a microscopic model of elastic disks. The theory is shown to agree with the approach of Hertz in the quasistatic limit, but predicts inelastic collisions for finite…
We study the distribution of interfacial separations P(u) at the contact region between two elastic solids with randomly rough surfaces. An analytical expression is derived for P(u) using Persson's theory of contact mechanics, and is…
A method is developed to calculate collision probability in this paper. Based on the encounter geometric features of space objects, it is reasonable to separate the radial orbital motions from that in the cross section for most encounter…
The effects of purely elastic collisions on the dynamics of heavy inertial particles is investigated in a three-dimensional random incompressible flow. It is shown that the statistical properties of inter-particle separations and relative…
We study the average separation between an elastic solid and a hard solid with a nominal flat but randomly rough surface, as a function of the squeezing pressure. We present experimental results for a silicon rubber (PDMS) block with a flat…
The intersection numbers of p-spin curves are computed through correlation functions of Gaussian ensembles of random matrices in an external matrix source. The p-dependence of intersection numbers is determined as polynomial in p; the large…
This article studies statistical estimation of $\pi$ based on the fact that the ratio of the volumes of a $d$-dimensional hypersphere and a $d$-dimensional hypercube is a certain function of $\pi$, and the function depends on the dimension…
Polymetric walls are walls built from bricks in more than one size. Architects and builders want to built polymetric walls that satisfy certain structural and aesthetical constraints. In a recent paper by de Jong, Vinduska, Hans and Post…