每日彙整: 2018-04-03

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【二．五】

如同 SymPy mechanics 程式庫

Masses, Inertias, Particles and Rigid Bodies in Physics/Mechanics

This document will describe how to represent masses and inertias in mechanics and use of the RigidBody and Particle classes.

It is assumed that the reader is familiar with the basics of these topics, such as finding the center of mass for a system of particles, how to manipulate an inertia tensor, and the definition of a particle and rigid body. Any advanced dynamics text can provide a reference for these details.

Mass

The only requirement for a mass is that it needs to be a sympify-able expression. Keep in mind that masses can be time varying.

Particle

Particles are created with the class Particle in mechanics. A Particle object has an associated point and an associated mass which are the only two attributes of the object.:

>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import Symbol
>>> m = Symbol('m')
>>> po = Point('po')
>>> # create a particle container
>>> pa = Particle('pa', po, m)

The associated point contains the position, velocity and acceleration of the particle. mechanics allows one to perform kinematic analysis of points separate from their association with masses.

 

所言，不熟悉力學基本者，恐難以應用也。

作者嘗試起個頭，作此搭橋之舉，然不過掛一漏萬之說而已。

就像問『粒子』是什麼？強調『質量』時，又常稱之為『質點』。只講其核心概念是可視為『點』的『粒子』

Point particle

A point particle (ideal particle[1] or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension: being zero-dimensional, it does not take up space.[2] A point particle is an appropriate representation of any object whose size, shape, and structure is irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object.

※ 參照

Galaxies are so large that stars can be considered particles relative to them

Particle

In the physical sciences, a particle (or corpuscule in older texts) is a small localized object to which can be ascribed several physical or chemical properties such as volume or mass.[1][2] They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic particles like atoms and molecules, to macroscopic particles like powders and other granular materials. Particles can also be used to create scientific models of even larger objects depending on their density, such as humans moving in a crowd or celestial bodies in motion.

The term ‘particle’ is rather general in meaning, and is refined as needed by various scientific fields. Something that is composed of particles may be referred to as being particulate.[3] However, the noun ‘particulate‘ is most frequently used to refer to pollutants in the Earth’s atmosphere, which are a suspension of unconnected particles, rather than a connected particle aggregation.

 

In the theory of gravity, physicists often discuss a point mass, meaning a point particle with a nonzero mass and no other properties or structure. Likewise, in electromagnetism, physicists discuss a point charge, a point particle with a nonzero charge.[3]

Sometimes, due to specific combinations of properties, extended objects behave as point-like even in their immediate vicinity. For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their centers of mass. In Newtonian gravitation and classical electromagnetism, for example, the respective fields outside a spherical object are identical to those of a point particle of equal charge/mass located at the center of the sphere.[4][5]

In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume. For example, the atomic orbit of an electron in the hydrogen atom occupies a volume of ~10−30 m3. There is nevertheless a distinction between elementary particles such as electrons or quarks, which have no known internal structure, versus composite particles such as protons, which do have internal structure: A proton is made of three quarks. Elementary particles are sometimes called “point particles”, but this is in a different sense than discussed above.

Property concentrated at a single point

When a point particle has an additive property, such as mass or charge, concentrated at a single point in space, this can be represented by a Dirac delta function.

Physical point mass

 

An example of a point mass graphed on a grid. The grey mass can be simplified to a point mass (the blackcircle). It becomes practical to represent point mass as small circle, or dot, as an actual point is invisible.

 

Point mass (pointlike mass) is the concept, for example in classical physics, of a physical object (typically matter) that has nonzero mass, and yet explicitly and specifically is (or is being thought of or modeled as) infinitesimal(infinitely small) in its volume or linear dimensions.

Application

A common use for point mass lies in the analysis of the gravitational fields. When analyzing the gravitational forces in a system, it becomes impossible to account for every unit of mass individually. However, a spherically symmetric body affects external objects gravitationally as if all of its mass were concentrated at its center.

 

，怕太虛無飄渺乎？？

還是藉『殼層定理』說明『均勻的球』之重力『作用可以等效』於『球心質點』好耶！！

Shell theorem

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

Isaac Newton proved the shell theorem[1] and stated that:

  1. A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre.
  2. If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object’s location within the shell.

A corollary is that inside a solid sphere of constant density, the gravitational force varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass. This can be seen as follows: take a point within such a sphere, at a distance \displaystyle r from the centre of the sphere. Then you can ignore all the shells of greater radius, according to the shell theorem. So, the remaining mass \displaystyle m is proportional to \displaystyle r^{3} , and the gravitational force exerted on it is proportional to \displaystyle m/r^{2}  , so to \displaystyle r^{3}/r^{2}=r  , so is linear in \displaystyle r .

These results were important to Newton’s analysis of planetary motion; they are not immediately obvious, but they can be proven with calculus. (Alternatively, Gauss’s law for gravity offers a much simpler way to prove the same results.)

In addition to gravity, the shell theorem can also be used to describe the electric field generated by a static spherically symmetric charge density, or similarly for any other phenomenon that follows an inverse square law. The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force. Moreover, the results can be generalized to the case of general ellipsoidal bodies.[2]

Note: As viewed from m, the shaded blue band appears as a thin annulus whose inner and outer diameters converge to R sin θ as dθ vanishes.

 

但是將它當成『剛體』時，卻又不能看作『質點』呦！！？？

所以終究不如聽一堂 MIT 古典力學的公開課來的好哩？？！！

Video Introduction by Prof. Deepto Chakrabarty and Dr. Peter Dourmashkin

Classical Mechanics Course Introduction

 

Course Meeting Times

Lectures: 2 sessions / week, 2 hours / session

Problem Solving: 1 session / week, 1 hour / session

Prerequisites

This course has no prerequisites. 18.01SC Single Variable Calculus is a corequisite.

Course Overview

This first course in the physics curriculum introduces classical mechanics. Historically, a set of core concepts — space, time, mass, force, momentum, torque, and angular momentum — were introduced in classical  mechanics in order to solve the most famous physics problem, the motion of the planets.

The principles of mechanics successfully described many other phenomena encountered in the world. Conservation laws involving energy, momentum and angular momentum provided a second parallel approach to solving many of the same problems. In this course, we will investigate both approaches: Force and conservation laws.

Our goal is to develop a conceptual understanding of the core concepts, a familiarity with the experimental verification of our theoretical laws, and an ability to apply the theoretical framework to describe and predict the motions of bodies.

Textbook

The textbook for this course is “Classical Mechanics: MIT 8.01 Course Notes” (PDF – 67.9MB) by Peter Dourmashkin. Specific readings for each assignment are provided in the Readings section.

Topics Covered