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3: Systems of Particles

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    • 3.1: Introduction to Systems of Particles
      In a system of particles, there may be very little or no interaction between the particles (as in a loose association of stars separated from each other by large distances) or there may be (as in the brick) strong forces between the particles. Most (perhaps all) of the results to be derived in this chapter for a system of particles apply equally to an apparently solid body such as a brick.
    • 3.2: Moment of Force
      This page covers the concept of torque, defined as the product of force and the perpendicular distance to the line of action. It highlights torque's vector nature through the cross product of position vector and force, and explains how to resolve torque into components in three-dimensional contexts. The page concludes with the dimensions and SI units of torque, specified as N m.
    • 3.3: Moment of Momentum
      Moment of momentum plays a role in rotational motion analogous to the role played by linear momentum in linear motion, and is also called angular momentum. Several choices for expressing angular momentum in SI units are possible; the usual choice is J s (joule seconds).
    • 3.4: Notation
      In this section I am going to suppose that we n particles scattered through three-dimensional space. We shall be deriving some general properties and theorems – and, to the extent that a solid body can be considered to be made up of a system of particles, these properties and theorems will apply equally to a solid body.
    • 3.5: Linear Momentum
      The total momentum of a system of particles equals the total mass times the velocity of the centre of mass.
    • 3.6: Force and Rate of Change of Momentum
      This page examines the connection between the rate of change of total momentum in a particle system and external forces, as outlined by Newton's laws. It introduces a theorem linking momentum change to the sum of external forces, applies it to individual and multiple particles, and includes a corollary. This corollary demonstrates that with no net external forces, linear momentum is conserved, reinforcing the law of conservation of linear momentum.
    • 3.7: Angular Momentum
      This page explains the relationship between a system's angular momentum around its center of mass and an arbitrary origin. It introduces relevant notation for angular momentum, position vector, and linear momentum. The main theorem states that total angular momentum is the sum of the angular momentum relative to the center of mass and an additional term involving the position vector and linear momentum.
    • 3.8: Torque
      This page explains the connection between total torque around the origin and torque around a point C, introducing relevant notation for torques and external forces. It presents a theorem linking these torques, revealing that total torque is the sum of the torque about point C plus an extra term involving the position vector and total external force. This relationship is essential for comprehending torque calculations in physical systems.
    • 3.9: Comparison
      This page compares formulas for angular momentum and torque, presenting general equations for both in terms of center of mass and individual particles. It explains how to calculate these quantities using mass, position, and velocity. Additionally, it introduces components such as total momentum and force, highlighting their interrelationships within rotational dynamics, with an emphasis on the mathematical expressions involved.
    • 3.10: Kinetic energy
      We remind ourselves that we are discussing particles, and that all kinetic energy is translational kinetic energy.
    • 3.11: Torque and Rate of Change of Angular Momentum
      The rate of change of the total angular momentum of a system of particles is equal to the sum of the external torques on the system.  The rate of change of the total angular momentum of a system of particles is equal to the sum of the external torques on the system.
    • 3.12: Torque, Angular Momentum and a Moving Point
      This page discusses the relation \(\dot{\bf L} = \boldsymbol{\tau}\) for a moving point Q within a particle system, indicating that modifications are necessary compared to stationary points. It presents a theorem refining the angular momentum's time derivative \(\dot{\bf L}_{Q}\), which includes torque \(\boldsymbol{\tau}_{Q}\) and a term related to Q's acceleration.
    • 3.13: The Virial Theorem
      The virial Equation tells us whether the cluster is going to disperse or collapse.

    Thumbnail: A hoop of radius a rolling along the ground. (Tatum).

    This page titled 3: Systems of Particles is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.