T-II-2 - Revision history

Groupe 01 13: /* Potential Energy */

2011-11-15T14:49:52Z

Potential Energy

← Older revision		Revision as of 16:49, 15 November 2011
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	As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:		As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:
	<center><math>V_0\, {\mathcal{V}(r,t)} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>		<center><math>V_0\, {\mathcal{V}(\vec r,t)} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>
	where <math>\mathcal V(r,t)</math>is the density of potential energy.		where <math>\mathcal V(\vec r,t)</math>is the density of potential energy.

	Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,		Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,
	<center><math>P V^\gamma= </math> constant</center>		<center><math>P V^\gamma= </math> constant</center>
	where <math>\gamma</math> the ratio of specific heats (i.e., the ratio of the gas's specific heat at constant pressure to its specific heat at constant volume). This ratio is approximately <math>1.4 </math> for ordinary air.		where <math>\gamma</math> the ratio of specific heats (i.e., the ratio of the gas's specific heat at constant pressure to its specific heat at constant volume). This ratio is approximately <math>1.4 </math> for ordinary air.
	* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V(r,t)</math> writes		* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V(\vec r,t)</math> writes
	<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>		<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>
	* Using the continuity equation of the gas, <math>\partial_t \rho=-\rho_0 \vec{\nabla} \cdot \partial_t \vec{u}(\vec{r},t)</math>, show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>		* Using the continuity equation of the gas, <math>\partial_t \rho=-\rho_0 \vec{\nabla} \cdot \partial_t \vec{u}(\vec{r},t)</math>, show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>

Groupe 01 13: /* Potential Energy */

2011-11-15T14:48:20Z

Potential Energy

← Older revision		Revision as of 16:48, 15 November 2011
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	* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V(r,t)</math> writes		* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V(r,t)</math> writes
	<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>		<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>
	* Using the continuity equation of the gas, <math>\~~partial~~ \rho=-\rho_0 \vec{\nabla} \cdot \partial_t \vec{u}(\vec{r},t)</math>, show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>		* Using the continuity equation of the gas, <math>\partial_t \rho=-\rho_0 \vec{\nabla} \cdot \partial_t \vec{u}(\vec{r},t)</math>, show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>

	===Lagrangian description===		===Lagrangian description===

Groupe 01 13: /* Potential Energy */

2011-11-15T14:47:57Z

Potential Energy

← Older revision		Revision as of 16:47, 15 November 2011
Line 72:		Line 72:
	* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V(r,t)</math> writes		* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V(r,t)</math> writes
	<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>		<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>
	* Using the continuity equation of the gas show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>		* Using the continuity equation of the gas, <math>\partial \rho=-\rho_0 \vec{\nabla} \cdot \partial_t \vec{u}(\vec{r},t)</math>, show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>

	===Lagrangian description===		===Lagrangian description===

Groupe 01 13: /* Elasticity */

2011-11-15T14:45:54Z

Elasticity

← Older revision		Revision as of 16:45, 15 November 2011
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	Step one: the discrete model		Step one: the discrete model
	* Write the elastic energy of a chain of N identical springs a a function of their position <math> y_1, y_2,\ldots,y_N </math> (See figure).		* Write the elastic energy of a chain of N identical springs as a function of their position <math> y_1, y_2,\ldots,y_N </math> (See figure).
	* Determine the elastic constant of each spring in order to have a a string with an elastic constant <math> k </math>		* Determine the elastic constant of each spring in order to have a string with an elastic constant <math> k </math>

	Step two: The continuum limit		Step two: The continuum limit

Groupe 01 13: /* Potential Energy */

2011-11-15T14:44:50Z

Potential Energy

← Older revision		Revision as of 16:44, 15 November 2011
Line 64:		Line 64:

	As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:		As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:
	<center><math>V_0 {\mathcal{V}} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>		<center><math>V_0\, {\mathcal{V}(r,t)} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>
	where <math>\mathcal V</math>is the density of potential energy.		where <math>\mathcal V(r,t)</math>is the density of potential energy.

	Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,		Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,
	<center><math>P V^\gamma= </math> constant</center>		<center><math>P V^\gamma= </math> constant</center>
	where <math>\gamma</math> the ratio of specific heats (i.e., the ratio of the gas's specific heat at constant pressure to its specific heat at constant volume). This ratio is approximately <math>1.4 </math> for ordinary air.		where <math>\gamma</math> the ratio of specific heats (i.e., the ratio of the gas's specific heat at constant pressure to its specific heat at constant volume). This ratio is approximately <math>1.4 </math> for ordinary air.
	* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V</math> writes		* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V(r,t)</math> writes
	<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>		<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>
	* Using the continuity equation of the gas show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>		* Using the continuity equation of the gas show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>

Groupe 01 13: /* Potential Energy */

2011-11-15T14:43:48Z

Potential Energy

← Older revision		Revision as of 16:43, 15 November 2011
Line 64:		Line 64:

	As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:		As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:
	<center><math>V_0 {\~~cal~~{V}} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>		<center><math>V_0 {\mathcal{V}} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>
			where <math>\mathcal V</math>is the density of potential energy.

	Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,		Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,
	<center><math>P V^\gamma= </math> constant</center>		<center><math>P V^\gamma= </math> constant</center>
	where <math>\gamma</math> the ratio of specific heats (i.e., the ratio of the gas's specific heat at constant pressure to its specific heat at constant volume). This ratio is approximately <math>1.4 </math> for ordinary air.		where <math>\gamma</math> the ratio of specific heats (i.e., the ratio of the gas's specific heat at constant pressure to its specific heat at constant volume). This ratio is approximately <math>1.4 </math> for ordinary air.
	* Show that the total potential energy writes		* Show that the total potential energy, <math>W=\int \, d \vec{r} \, \mathcal V</math> writes
	<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>		<center><math> W= P_0 \int \, d \vec{r} \, \left( \sigma + \frac{\gamma}{2} \sigma^2\right) </math> </center>
	* Using the continuity equation of the gas show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>		* Using the continuity equation of the gas show that <math> \sigma(\vec{r},t)= - \vec{\nabla} \cdot \vec{u}(\vec{r},t) </math> and write the potential energy as a function of <math> \vec{u}(\vec{r},t) </math>

Groupe 01 13: /* Potential Energy */

2011-11-15T14:42:27Z

Potential Energy

← Older revision		Revision as of 16:42, 15 November 2011
Line 64:		Line 64:

	As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:		As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:
	<center><math>V_0 {\cal V} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>		<center><math>V_0 {\cal{V}} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>

	Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,		Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,

Groupe 01 13: /* Potential Energy */

2011-11-15T14:42:14Z

Potential Energy

← Older revision		Revision as of 16:42, 15 November 2011
Line 64:		Line 64:

	As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:		As a consequence of these fluctuations, the volume which contains a given mass <math>M </math> expands from the initial value <math>V_0 </math> to a new value <math>V_0 +\delta V</math>. The pressure inside the domain also fluctuates <math>P=P_0+\ldots </math>. The variation of potential energy is given by the work associated to the volume expansion:
	<center><math> - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>		<center><math>V_0 {\cal V} - \int_{V_0}^{V_0+\delta V} \, P\, d \vec{r}=-P_0 \delta V- \frac{1}{2} \left( \frac{\partial P }{\partial V}\right)_0 \delta V^2+\ldots </math> </center>

	Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,		Generally speaking, a sound wave in an ideal gas oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any temperature perturbations generated by the wave. Under these circumstances, the gas obeys the adiabatic gas law,

Groupe 01 13: /* Dynamics */

2011-11-15T14:40:43Z

Dynamics

← Older revision		Revision as of 16:40, 15 November 2011
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	<center><math> \int_{t_1}^{t_2} d t \int_0^1 d x \, \mathcal{L} (x,t) </math></center>		<center><math> \int_{t_1}^{t_2} d t \int_0^1 d x \, \mathcal{L} (x,t) </math></center>
	where <math> {\mathcal L}(x,t) </math> is the Lagrangian denisty		where <math> {\mathcal L}(x,t) </math> is the Lagrangian denisty
	* Write explicitly <math> {\mathcal L}(x,t) </math> ~~in the small oscillation approximation~~		* Write explicitly <math> {\mathcal L}(x,t) </math> .
	* Write the equation of motion of the chain using the minimum action principle. This equation is the D'Almbert equation.		* Write the equation of motion of the chain using the minimum action principle. This equation is the D'Almbert equation.

Groupe 01 13: /* Gravity */

2011-11-15T14:40:13Z

Gravity

← Older revision		Revision as of 16:40, 15 November 2011
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	===Gravity===		===Gravity===
	We write now the gravitational potential as a functional of <math>y(x)</math>. We will use two different methods. We suppose that the total mass of the line in <math>M</math>.		We write now the gravitational potential as a functional of <math>y(x)</math>. We will use two different methods. We suppose that the total mass of the line in <math>M</math>.

	~~Method 1:~~

	~~In general the gravitational potential can be written as~~
	~~<center> <math> E_{g} = g \int_0^1 dx\, \sqrt{1+ (\partial_x y)^2}\, \rho(x)\, y(x).</math></center>~~
	~~For an elastic string the mass density <math>\rho(x)</math>is not uniform in <math>x</math>. Suppose that each spring has a mass $M/N$ and no mass is accumulated between springs.~~
	* Compute <math>\rho_i</math> for each spring and take the continuum limit.

	~~Method 2:~~

	* Suppose that springs are massless and <math>N</math> identical masses are located at positions <math>y_1,y_2,\ldots</math>		* Suppose that springs are massless and <math>N</math> identical masses are located at positions <math>y_1,y_2,\ldots</math>