Home | Sign Up | Log In | RSSThursday, 2018-12-13, 7:13 AM


Site menu
Section categories
Education [150]
Videos [1]
Music [0]
Chat Box
learn c language learn c language branching and loop learn c language functions learn c language Pointer Basics how web page works Oracle Solaris 10 Oracle Solaris 10 How Web Pages Work Images How Web Pages Work Adding images&gr How Web Pages Work Introduction to How to Install WAMP How to Host Your Own Website for Fr How to Install the Apache Web Serve Atomic Structure The text provides 000 tons of conversations Introduction to How Radio Works Rad often over millions of miles citizens band radio Introduction to How the Radio Spect Can information travel faster than How F-15s Work by Tom Harris Browse How F/A-18s Work by Robert Valdes B How F/A-18s Work by Robert Valdes B Flying Video Game: In the Cockpit I How F/A-22 Raptors Work by Gary Wol 10 Unidentified Sounds That Scienti 10 Things You Didn't Know About Ein Is glass really a liquid? by Laurie How Radio Works by Marshall Brain B How Radio Works by Marshall Brain B 10 Most Terrifying Vehicle Manufact How a Top Fuel Dragster Works by Ch but can Siri meet our nee How Siri Works by Bernadette Johnso How the Tesla Turbine Works by Will Lecture 1: Inflationary Cosmology: Lecture 2: Inflationary Cosmology: Lecture 4: The Kinematics of the Ho Lecture 5: Cosmological Redshift an Lecture 6: The Dynamics of Homogene Lecture 7: The Dynamics of Homogene Lecture 8: The Dynamics of Homogene Lecture 9: The Dynamics of Homogene Lecture 10: Introduction to Non-Euc IBPS Clerical Cadre Exam Pattern De Institute of Banking Personnel Sele Educational Qualifications: A Degre INTERVIEW Candidates who have been official trailer for Mission Imposs How Relativity Connects Electric an in 1738 Kinetic Theory of Gases: A Brief Re Frames of Reference and Newton’s La attempts to measure the UVa Physics 12/1/07 “Moving Clocks Run Slow” pl More Relativity: The Train and the 12/1/07 The Formula If I walk from Adding Velocities: A Walk on the Tr 3/1/2008 The Story So Far: A Brief Mass and Energy Michael Fowler Energy and Momentum in Lorentz Tran How Relativity Connects Electric an Analyzing Waves on a String Michael by f even earlier than the bra Fermat's Principle of Least Time 9/ Hamilton's Principle and Noether's meaning they have Mechanical Similarity and the Viria Hamilton's Equations 9/10/15 A Dyna A New Way to Write the Action Integ Maupertuis came up with a kind of p Maupertuis' Principle: Minimum Acti Canonical Transformations Point Tra Introduction to Liouville's Theorem Adiabatic Invariants and Action-Ang Hyperbolas Michael Fowler Prelimina Mathematics for Orbits: Ellipses Keplerian Orbits Michael Fowler Pre Newton's equations for particle mot Dynamics of Motion in a Central Pot A Vectorial Approach: Hamilton's Eq Elastic Scattering Michael Fowler B Driven Oscillator Michael Fowler (c Dynamics of a One-Dimensional Cryst I usefor the spring constant (is a a mass on a spring Motion in a Rapidly Oscillating Fie Anharmonic Oscillators Michael Fowl in which the distance betw Motion of a Rigid Body: the Inertia Moments of Inertia: Examples Michae Euler's Angles Michael Fowler Intro or more precisely one our analysis of rotational motion h Euler's Equations Michael Fowler In Motion in a Non-inertial Frame of R Ball Rolling on Tilted Turntable Mi live cricket score roll the ball ba

Total online: 1
Guests: 1
Users: 0
Home » 2017 » September » 1 » Mechanical Similarity and the Virial Theorem
1:05 AM
Mechanical Similarity and the Virial Theorem

Mechanical Similarity and the Virial Theorem

Some Examples

Similar triangles are just scaled up (or down) versions of each other, meaning they have the same angles. Scaling means the same thing in a mechanical system: if a planet can go around the sun in a given elliptical orbit, another planet can go in a scaled up version of that ellipse (the sun remaining at the focus). But it will take longer: so we can't just scale the spatial dimensions, to get the same equation of motion we must scale time as well, and not in general by the same factor.

In fact, we can establish the relative scaling of space and time in this instance with very simple dimensional analysis. We know the planet's radial acceleration goes as the inverse square of the distance, so (radial acceleration)x(distance)2 = constant, the dimensionality of this expression is , so , the square of the time of one orbit is proportional to the cube of the size of the orbit. A little more explicitly, the acceleration  so for the same  if we double the orbit size, the equation will be the same but with orbital time up by 

Galileo established that real mechanical systems, such as a person, are not scale invariant. A giant ten times the linear dimensions of a human would break his hip on the first step. The point is that the weight would be up by a factor of 1,000, the bone strength, going as cross sectional area, only by 100.

Mechanical similarity is important is constructing small models of large systems. A particularly important application is to fluid flow, for example in assessing fluid drag forces on a moving ship, plane or car. There are two different types of fluid drag: viscous frictional drag, and inertial drag, the latter caused by the body having to deflect the medium as it moves through. The patterns of flow depend on the relative importance of these two drag forces, this dimensionless ratio, inertial/viscous, is called the Reynolds number. To give meaningful results, airflow speeds around models must be adjusted to give the model the same Reynolds number as the real system.

Lagrangian Treatment

(Here we follow Landau.) Since the equations of motion are generated by minimizing the action, which is an integral of the Lagrangian along a trajectory, the motion won't be affected if the Lagrangian is multiplied by a constant. If the potential energy is a homogeneous function of the coordinates, rescaling would multiply it by a constant factor. If our system consists of particles interacting via such a potential energy, it will be possible to rescale time so that, rescaling both space and time, the Lagrangian is multiplied by an overall constant, so the equations of motion will look the same.

Specifically, if the potential energy is homogeneous of degree , and the spatial coordinates are scaled by a factor

To get the kinetic energy term to scale by the same factor, we take , so  the kinetic energy is scaled by for this to match the potential energy scaling, we must have

For planetary orbits, , so confirming our hand waving derivation above.

For the simple harmonic oscillator,  so  and  What does that mean? Scaling up the orbit does not affect the time -- the oscillation time is always the same.

Falling under gravity:  So doubling the time scale requires quadrupling the length scale to get the scaled motion identical to the original.

The Virial Theorem

For a potential energy homogeneous in the coordinates, of degree say, and spatially bounded motion, there is a simple relation between the time averages of the kinetic energy, , and potential energy,  It's called the virial theorem:



we have


We now average the terms in this equation over a very long time, that is, take


Since we've said the orbits are bounded in space, and we assume also in momentum, the exact differential term contributes


in the limit of infinite time.

So we have the time averaged


and for a potential energy a homogeneous function of degree k in the coordinates, from Euler's theorem:

So, for example, in a simple harmonic oscillator the average kinetic energy equals the average potential energy, and for an inverse-square system, the average kinetic energy is half the average potential energy in magnitude, and of opposite sign (being of course positive).

Category: Education | Views: 237 | Added by: farrel | Tags: Mechanical Similarity and the Viria, meaning they have | Rating: 0.0/0
Total comments: 0
Name *:
Email *:
Code *:
Log In
Entries archive

Copyright eduCampus.tk © 2018
Powered by uCoz