en>Scgtrp: grammar

2013-10-12T05:14:33Z

grammar

← Older revision		Revision as of 07:14, 12 October 2013
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	~~Hi there. My name is Garland although it~~ is not the ~~title on my birth certification~~. ~~Bookkeeping~~ is ~~what he does~~. ~~To perform badminton is some thing he really enjoys doing~~. ~~Alabama has always been his house and his family members enjoys it~~.<br><br>~~My web~~ page ~~... [http~~://~~Www~~.~~igomt~~.~~com~~/xe/~~index~~.~~php?document_srl~~=~~7701&mid~~=~~model_church_board http~~://~~Www~~.~~igomt~~.~~com~~/xe/~~index~~.~~php?document_srl~~=~~7701&mid~~=~~model_church_board~~]		A mechanical system is '''scleronomous''' if the equations of constraints do not contain the time as an explicit variable. Such constraints are called '''scleronomic''' constraints.

			==Application==
			:Main article：[[Generalized velocity]]
			In 3-D space, a particle with mass <math>m\,\!</math>, velocity <math>\mathbf{v}\,\!</math> has [[kinetic energy]]
			:<math>T =\frac{1}{2}m v^2 \,\!.</math>

			Velocity is the derivative of position with respect time. Use [[chain rule#Chain rule for several variables\|chain rule for several variables]]:
			:<math>\mathbf{v}=\frac{d\mathbf{r}}{dt}=\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\,\!.</math>

			Therefore,
			:<math>T =\frac{1}{2}m \left(\sum_i\ \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i+\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!.</math>

			Rearranging the terms carefully,<ref name="Herb1980">{{cite book \|last=Goldstein\|first=Herbert\|title=Classical Mechanics\|year=1980\| location=United States of America \| publisher=Addison Wesley\| edition= 3rd\| isbn=0-201-65702-3 \| page=25}}</ref>

			:<math>T =T_0+T_1+T_2\,\!:</math>
			:<math>T_0=\frac{1}{2}m\left(\frac{\partial \mathbf{r}}{\partial t}\right)^2\,\!,</math>
			:<math>T_1=\sum_i\ m\frac{\partial \mathbf{r}}{\partial t}\cdot \frac{\partial \mathbf{r}}{\partial q_i}\dot{q}_i\,\!,</math>
			:<math>T_2=\sum_{i,j}\ \frac{1}{2}m\frac{\partial \mathbf{r}}{\partial q_i}\cdot \frac{\partial \mathbf{r}}{\partial q_j}\dot{q}_i\dot{q}_j\,\!,</math>

			where <math>T_0\,\!</math>, <math>T_1\,\!</math>, <math>T_2\,\!</math> are respectively [[homogeneous function]]s of degree 0, 1, and 2 in generalized velocities. If this system is scleronomous, then the position does not depend explicitly with time:

			:<math>\frac{\partial \mathbf{r}}{\partial t}=0\,\!.</math>
			Therefore, only term <math>T_2\,\!</math> does not vanish:
			:<math>T = T_2\,\!.</math>
			Kinetic energy is a homogeneous function of degree 2 in generalized velocities .

			==Example: 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’s length is a constant. Therefore, this system is scleronomous; it obeys 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> is length of the string.

			[[File:Pendulum02.JPG\|frame\|right\|A simple pendulum with oscillating pivot point]]
			Take a more complicated example. Refer to the next figure at right, Assume the top end of the string is attached to a pivot point undergoing a [[simple harmonic motion]]
			:<math>x_t=x_0\cos\omega t\,\!,</math>

			where <math>x_0\,\!</math> is amplitude, <math>\omega\,\!</math> is angular frequency, and <math>t\,\!</math> is 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 rheonomic constraint
			:<math> \sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!.</math>

			==See also==
			*[[Lagrangian mechanics]]
			*[[Holonomic system]]
			*[[Nonholonomic system]]
			*[[Rheonomous]]

			== References ==
			<references />

			[[Category:Mechanics]]
			[[Category:Classical mechanics]]
			[[Category:Lagrangian mechanics]]

			[[de:Skleronom]]

en>Tabletop: Spell ambigous => ambiguous (13)

2012-07-07T03:09:06Z

Spell ambigous => ambiguous (13)

New page

Hi there. My name is Garland although it is not the title on my birth certification. Bookkeeping is what he does. To perform badminton is some thing he really enjoys doing. Alabama has always been his house and his family members enjoys it.<br><br>My web page ... [http://Www.igomt.com/xe/index.php?document_srl=7701&mid=model_church_board http://Www.igomt.com/xe/index.php?document_srl=7701&mid=model_church_board]

McCarthy Formalism - Revision history

en>Scgtrp: grammar

en>Tabletop: Spell ambigous => ambiguous (13)