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Applications of Integration to Find Centers of Mass

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Rotational equilibrium provides a natural framework for defining the center of mass of a system. For a plank balanced on a pivot with two unequal masses, equilibrium is achieved when the net torque about the pivot is zero. Torque is defined as the product of a force and its perpendicular distance from the pivot. When the torques due to all forces cancel, the pivot coincides with the center of mass of the system.For a system composed of several discrete point masses, the center of mass lies at...
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The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
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The center of gravity of a body is an imaginary point where the body's total weight is assumed to be concentrated, and the body is perfectly balanced. The center of the mass of a body is a point at which the whole of the mass of the body appears to be concentrated. If the acceleration due to gravity, g, has the same value at all points on a body, its center of gravity is identical to its center of mass. The center of gravity of homogeneous bodies such as a sphere, cube, or rectangular plate...
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Center of Mass00:59

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The center of mass is the point at which the total mass of an object can be said to be concentrated. It is a fundamental principle in mechanics and physics that applies to all objects regardless of their shape or size. The center of gravity is the point at which an object’s weight appears to be concentrated and can be used to balance the object perfectly.
The knowledge of the center of mass can also help us to describe and predict the motion of objects. For example, when a ball is thrown...
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Significance of Center of Mass01:12

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The center of mass of an object is defined as the mass-weighted average position of all the particles that comprise the object. The significance of the center of mass of an object can be seen by looking at its dynamics. The time derivative of the center of mass gives its velocity, assuming that the object's mass remains constant over time. Furthermore, the total linear momentum of an object can be seen as the linear momentum of a single particle of the object's total mass moving with...
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Any object that obeys Newton's second law of motion is made up of a large number of infinitesimally small particles. Objects in motion can be as simple as atoms or as complex as gymnasts performing in the Olympics. The motion of such objects is described about a point called the center of mass of the object. The center of mass of an object is a point that acts as if the whole mass is concentrated at that point. The center of mass of an object with a large number of infinitesimally small...
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Finding Communities by Their Centers.

Yan Chen1, Pei Zhao1, Ping Li1

  • 1Center for Intelligent and Networked Systems, School of Computer Science, Southwest Petroleum University, Chengdu 610500, P.R. China.

Scientific Reports
|April 8, 2016
PubMed
Summary

This study introduces a novel method for detecting network communities and their centers by analyzing local structures. The efficient linear algorithm excels in identifying community organization and centers across diverse networks.

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Area of Science:

  • Network Science
  • Data Mining
  • Computational Social Science

Background:

  • Community detection is crucial across multiple scientific disciplines.
  • Existing algorithms often focus on community composition, neglecting internal structures.
  • Understanding local community structures can enhance detection accuracy.

Purpose of the Study:

  • To develop a method for simultaneously uncovering communities and their centers.
  • To leverage local network structures for improved community detection.
  • To propose a novel metric, 'community centrality', for identifying community cores.

Main Methods:

  • Introduced 'community centrality' to quantify the likelihood of a node being a community center.
  • Employed a diffusion process to propagate center likelihoods throughout the network.
  • Developed an efficient linear-time algorithm for simultaneous community and center detection.

Main Results:

  • The proposed approach effectively identifies both communities and their central nodes.
  • Demonstrated superior performance on various synthetic and real-world network datasets.
  • Particularly effective for networks with sparse connections between community centers.

Conclusions:

  • Local community structures are vital for accurate community detection.
  • The 'community centrality' metric and diffusion process offer a robust detection framework.
  • This efficient algorithm advances the field of network community analysis.