Forest UCM EnergyIntPart - Revision history

Foretony: /* Total Mechanical Energy conservation */

2014-09-29T12:39:47Z

Total Mechanical Energy conservation

2014-09-29T12:37:09Z

Total Mechanical Energy conservation

2014-09-29T12:36:00Z

Total work given by one potential

2014-09-29T12:35:07Z

Total work given by one potential

2014-09-29T12:34:36Z

Total work given by one potential

2014-09-29T12:32:33Z

Total work given by one potential

2014-09-29T12:31:14Z

Total work given by one potential

2014-09-29T12:27:52Z

Total work given by one potential

2014-09-29T12:23:29Z

Total work given by one potential

2014-09-29T12:19:30Z

Both forces from same potential

← Older revision		Revision as of 12:39, 29 September 2014
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	: <math>W_{\mbox{total}} = d \vec{r}_1 \cdot \vec{F}_{12} + d \vec{r}_2 \cdot \vec{F}_{21} = dT_1 + dT_2</math>		: <math>W_{\mbox{total}} = d \vec{r}_1 \cdot \vec{F}_{12} + d \vec{r}_2 \cdot \vec{F}_{21} = dT_1 + dT_2</math>
		+	:: <math>= dT = -dU</math>
		+
		+	Thus
		+
		+	:<math>dT + dU = 0 </math>
		+
		+	or
		+	:<math>E_{\mbox{total}} = T_1 + T_2 + dU = </math>constant

	=Elastic Collisions=		=Elastic Collisions=

← Older revision		Revision as of 12:37, 29 September 2014
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	: <math>dT_2 = d \vec{r}_2 \cdot \vec{F}_{21} </math>		: <math>dT_2 = d \vec{r}_2 \cdot \vec{F}_{21} </math>
		+
		+
		+	: <math>W_{\mbox{total}} = d \vec{r}_1 \cdot \vec{F}_{12} + d \vec{r}_2 \cdot \vec{F}_{21} = dT_1 + dT_2</math>

	=Elastic Collisions=		=Elastic Collisions=

@@ Line 73: / Line 73: @@
 ::<math>= -\left ( d \vec{r} \right ) \cdot \left (  \vec{\nabla}U(\vec {r}_1 - \vec{r}_2) \right ) </math>
 ::<math>= -\left ( d \vec{r} \right ) \cdot \left (  \vec{\nabla}U(\vec {r}) \right ) =-dU</math>
 The work done by <math>\vec{F}_{12}</math> as object 1 moves through <math>d\vec{r}_1</math> is given by the work energy theorem as

@@ Line 72: / Line 72: @@
 ::<math>= \left ( d \vec{r}_1  - d \vec{r}_2 \right ) \cdot \left ( - \vec{\nabla}U(\vec {r}_1 ) \right ) </math>
 ::<math>= -\left ( d \vec{r} \right ) \cdot \left (  \vec{\nabla}U(\vec {r}_1 - \vec{r}_2) \right ) </math>
-::<math>= -\left ( d \vec{r} \right ) \cdot \left (  \vec{\nabla}U(\vec {r}) \right ) </math>
+::<math>= -\left ( d \vec{r} \right ) \cdot \left (  \vec{\nabla}U(\vec {r}) \right ) =-dU</math>
 The work done by <math>\vec{F}_{12}</math> as object 1 moves through <math>d\vec{r}_1</math> is given by the work energy theorem as

@@ Line 70: / Line 70: @@
 ::<math>= d \vec{r}_1 \cdot \vec{F}_{12} + d \vec{r}_2 \cdot (-\vec{F}_{12}) </math>
 ::<math>= \left ( d \vec{r}_1  - d \vec{r}_2 \right ) \cdot \vec{F}_{12} </math>
-::<math>= \left ( d \vec{r}_1  - d \vec{r}_2 \right ) \cdot \left ( - \vec{\nabla}U(\vec r) \right ) </math>
+::<math>= \left ( d \vec{r}_1  - d \vec{r}_2 \right ) \cdot \left ( - \vec{\nabla}U(\vec {r}_1 ) \right ) </math>
 The work done by <math>\vec{F}_{12}</math> as object 1 moves through <math>d\vec{r}_1</math> is given by the work energy theorem as

← Older revision		Revision as of 12:31, 29 September 2014
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	This NET work can be determine by taking the derivative of the potential energy		This NET work can be determine by taking the derivative of the potential energy

		+
		+	Proof:
		+
		+	: <math>W_{\mbox{total}} = d \vec{r}_1 \cdot \vec{F}_{12} + d \vec{r}_2 \cdot \vec{F}_{21} </math>
		+	::<math>= d \vec{r}_1 \cdot \vec{F}_{12} + d \vec{r}_2 \cdot (-\vec{F}_{12}) </math>
		+	::<math>= \left ( d \vec{r}_1 - d \vec{r}_2 \right ) \cdot \vec{F}_{12} </math>

	The work done by <math>\vec{F}_{12}</math> as object 1 moves through <math>d\vec{r}_1</math> is given by the work energy theorem as		The work done by <math>\vec{F}_{12}</math> as object 1 moves through <math>d\vec{r}_1</math> is given by the work energy theorem as

← Older revision		Revision as of 12:27, 29 September 2014
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	:the work done by <math>\vec{F}_{21}</math> as object 1 moves through <math>d\vec{r}_2</math>		:the work done by <math>\vec{F}_{21}</math> as object 1 moves through <math>d\vec{r}_2</math>
		+
		+	This NET work can be determine by taking the derivative of the potential energy
		+
		+
		+	The work done by <math>\vec{F}_{12}</math> as object 1 moves through <math>d\vec{r}_1</math> is given by the work energy theorem as
		+
		+	: <math>dT_1 = d \vec{r}_1 \cdot \vec{F}_{12} </math>
		+
		+	similarly for <math>\vec{F}_{21}</math>
		+
		+	: <math>dT_2 = d \vec{r}_2 \cdot \vec{F}_{21} </math>

	=Elastic Collisions=		=Elastic Collisions=

← Older revision		Revision as of 12:23, 29 September 2014
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	==Total work given by one potential==		==Total work given by one potential==

		+
		+	The total work is the sum of
		+
		+	:the work done by <math>\vec{F}_{12}</math> as object 1 moves through <math>d\vec{r}_1</math>
		+
		+	plus
		+
		+	:the work done by <math>\vec{F}_{21}</math> as object 1 moves through <math>d\vec{r}_2</math>

	=Elastic Collisions=		=Elastic Collisions=

← Older revision		Revision as of 12:19, 29 September 2014
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	:<math> \vec{\nabla}_1 U(\vec{r}_1 - \vec{r}_2) = -\vec{\nabla}_2 U(\vec{r}_1 - \vec{r}_2)</math>		:<math> \vec{\nabla}_1 U(\vec{r}_1 - \vec{r}_2) = -\vec{\nabla}_2 U(\vec{r}_1 - \vec{r}_2)</math>
		+
		+
		+	;You can find the net external force on a body in the system once you have the potential for the system

	==Total work given by one potential==		==Total work given by one potential==