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A mechanical system is '''rheonomous''' if its equations of [[Constraint (classical mechanics)|constraints]] contain the time as an explicit variable.<ref name=Herb1980>{{cite book |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |title=Classical Mechanics |year=1980 |location=United States of America |publisher=Addison Wesley |edition=2nd |isbn=0-201-02918-9 |page=12 |quote=Constraints are further classified according as the equations of constraint contain the time as an explicit variable (rheonomous) or are not explicitly dependent on time (scleronomous).}}</ref><ref name=spiegel1994>{{cite book |last=Spiegel |first=Murray R. |title=Theory and Problems of THEORETICAL MECHANICS with an Introduction to Lagrange's Equations and Hamiltonian Theory |year=1994 |series=Schaum's Outline Series |publisher=McGraw Hill |isbn=0-07-060232-8 |page=283 |quote=In many mechanical systems of importance the time ''t'' does not enter explicitly in the equations (''2'') or (''3''). Such systems are sometimes called ''scleronomic''. In others, as for example those involving moving constraints, the time ''t'' does enter explicitly. Such systems are called ''rheonomic''.}}</ref> Such constraints are called '''rheonomic constraints'''. The opposite of rheonomous is [[scleronomous]].<ref name=Herb1980/><ref name=spiegel1994/> | |||
==Example: simple 2D pendulum== | |||
[[File:SimplePendulum01.JPG|frame|right|A simple pendulum]] | |||
As shown at right, a simple [[pendulum]] is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string has a constant length. Therefore this system is scleronomous; it obeys the scleronomic constraint | |||
: <math> \sqrt{x^2+y^2} - L=0\,\!</math>, | |||
where <math>(x,\ y)\,\!</math> is the position of the weight and <math>L\,\!</math> the length of the string. | |||
[[File:Pendulum02.JPG|frame|right|A simple pendulum with oscillating pivot point]] | |||
The situation changes if the pivot point is moving, e.g. undergoing a [[simple harmonic motion]] | |||
:<math>x_t=x_0\cos\omega t\,\!</math>, | |||
where <math>x_0\,\!</math> is the amplitude, <math>\omega\,\!</math> the angular frequency, and <math>t\,\!</math> time. | |||
Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore this system is rheonomous; it obeys the rheonomic constraint | |||
:<math> \sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!</math>. | |||
==See also== | |||
*[[Lagrangian mechanics]] | |||
*[[Holonomic constraints]] | |||
==References== | |||
<references /> | |||
[[Category:Mechanics]] | |||
[[Category:Classical mechanics]] | |||
[[Category:Lagrangian mechanics]] | |||
Revision as of 19:32, 1 November 2013
A mechanical system is rheonomous if its equations of constraints contain the time as an explicit variable.[1][2] Such constraints are called rheonomic constraints. The opposite of rheonomous is scleronomous.[1][2]
Example: simple 2D pendulum
As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string has a constant length. Therefore this system is scleronomous; it obeys the scleronomic constraint
- ,
where is the position of the weight and the length of the string.
The situation changes if the pivot point is moving, e.g. undergoing a simple harmonic motion
- ,
where is the amplitude, the angular frequency, and time.
Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore this system is rheonomous; it obeys the rheonomic constraint
- .
See also
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 2.0 2.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534