SR/Piston - Revision history

Joel2: /* Hydrostatic Balance */

2026-03-28T23:30:15Z

Hydrostatic Balance

← Older revision		Revision as of 19:30, 28 March 2026
Line 358:		Line 358:
	<math>K_c\rho_c^{6/5} \biggl\{1 + 2\cdot 3^4~ \biggl[		<math>K_c\rho_c^{6/5} \biggl\{1 + 2\cdot 3^4~ \biggl[
	\frac{1}{6(3 + \xi^2)^3}		\frac{1}{6(3 + \xi^2)^3}
	\biggr]_0^{\xi_i} \biggl\}</math> ~~  DONE~~		\biggr]_0^{\xi_i} \biggl\}</math>
	</td>		</td>
	</tr>		</tr>

	~~<!--~~
	~~<tr>~~
	~~<td align="right"> </td>~~
	~~<td align="center"><math>=</math></td>~~
	~~<td align="left">~~
	~~<math>K_c\rho_c^{6/5} \biggl\{1 - 3^3~ \biggl[~~
	~~\frac{1}{(3 + \xi_{\xi_i}^2)^3} - \frac{1}{3^3}~~
	~~\biggr] \biggl\}</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right"> </td>~~
	~~<td align="center"><math>=</math></td>~~
	~~<td align="left">~~
	~~<math>K_c\rho_c^{6/5} \biggl[ 2 -~~
	~~~ \biggl(1 + \frac{\xi_i^2}{3} \biggr)^{-3}\biggr]</math>~~
	~~</td>~~
	~~</tr>~~
	~~-->~~

	<tr>		<tr>
	<td align="right"> </td>		<td align="right"> </td>
Line 398:		Line 376:
	<td align="left">		<td align="left">
	<math>K_c\rho_c^{6/5} \biggl\{1 + \biggl[		<math>K_c\rho_c^{6/5} \biggl\{1 + \biggl[
	\frac{3^2}{(3 + \xi_i^2)^3} - 1		\frac{3^3}{(3 + \xi_i^2)^3} - 1
	\biggr] \biggl\}</math>		\biggr] \biggl\}</math>
	</td>		</td>
Line 407:		Line 385:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>K_c\rho_c^{6/5} \biggl[		<math>K_c\rho_c^{6/5} \biggl(1 + \frac{\xi_i^2}{3} \biggr)^{-3}</math>
	\frac{3^2}{(3 + \~~xi_i^2~~)^3}
	~~\biggr]~~</math>
	</td>		</td>
	</tr>		</tr>

	~~<tr>~~		</table>
	~~<td align="right"> ~~</~~td>~~
	~~<td align="center"><math>=</math></td>~~
	~~<td align="left">~~
	~~<math>K_c\rho_c^{6/5}~~
	~~\frac{1}{3}\biggl(1 + \frac{\xi_i^2}{3} \biggr)^{-3}</math>~~
	~~</td>~~
	~~</tr~~>

	<~~/table>~~		<!--
	This result should be compared with,		This result should be compared with,

Line 438:		Line 407:

	March 13, 2026:  <font color="red">This difference needs to be resolved!!!</font>		March 13, 2026:  <font color="red">This difference needs to be resolved!!!</font>
			-->

	=See Also=		=See Also=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2: /* Hydrostatic Balance */

2026-03-28T23:19:19Z

Hydrostatic Balance

← Older revision		Revision as of 19:19, 28 March 2026
Line 310:		Line 310:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>\int_0^{r_i} \mathrm{RHS} \cdot dr</math>		<math>-~\int_0^{r_i} \mathrm{RHS} \cdot dr</math>
	</td>		</td>
	</tr>		</tr>
Line 318:		Line 318:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>\frac{K_c^{1 / 2}}{G^{1 / 2}\rho_c^{2 / 5}}\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} ~ \int_0^{\xi_i} \mathrm{RHS} \cdot d\xi</math>		<math>-~\frac{K_c^{1 / 2}}{G^{1 / 2}\rho_c^{2 / 5}}\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} ~ \int_0^{\xi_i} \mathrm{RHS} \cdot d\xi</math>
	</td>		</td>
	</tr>		</tr>
Line 326:		Line 326:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>P_c + \frac{K_c^{1 / 2}}{G^{1 / 2}\rho_c^{2 / 5}}\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} ~ \int_0^{\xi_i}		<math>P_c - \frac{K_c^{1 / 2}}{G^{1 / 2}\rho_c^{2 / 5}}\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} ~ \int_0^{\xi_i}
	G^{1 / 2} K_c^{1 / 2} \rho_c^{8/5} \biggl(\frac{2^3\pi}{3}\biggr)^{1 / 2}		G^{1 / 2} K_c^{1 / 2} \rho_c^{8/5} \biggl(\frac{2^3\pi}{3}\biggr)^{1 / 2}
	\biggl[1 + \frac{\xi^2}{3}\biggr]^{-4} ~\xi		\biggl[1 + \frac{\xi^2}{3}\biggr]^{-4} ~\xi
Line 337:		Line 337:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>K_c\rho_c^{6/5} + 2K_c\rho_c^{6/5}~ \int_0^{\xi_i}		<math>K_c\rho_c^{6/5} - 2K_c\rho_c^{6/5}~ \int_0^{\xi_i}
	\biggl[1 + \frac{\xi^2}{3}\biggr]^{-4} ~\xi		\biggl[1 + \frac{\xi^2}{3}\biggr]^{-4} ~\xi
	\cdot d\xi</math>		\cdot d\xi</math>
Line 347:		Line 347:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>K_c\rho_c^{6/5} \biggl\{1 + 2\cdot 3^4~ \int_0^{\xi_i}		<math>K_c\rho_c^{6/5} \biggl\{1 - 2\cdot 3^4~ \int_0^{\xi_i}
	(3+\xi^2)^{-4} \xi\cdot d\xi\biggr\}</math>		(3+\xi^2)^{-4} \xi\cdot d\xi\biggr\}</math>
	</td>		</td>
Line 356:		Line 356:
	<td align="center"><math>=</math></td>		<td align="center"><math>=</math></td>
	<td align="left">		<td align="left">
	<math>K_c\rho_c^{6/5} \biggl\{1 - 2\cdot 3^4~ \biggl[		<math>K_c\rho_c^{6/5} \biggl\{1 + 2\cdot 3^4~ \biggl[
	\frac{1}{6(3 + \xi^2)^3}		\frac{1}{6(3 + \xi^2)^3}
	\biggr]_0^{\xi_i} \biggl\}</math>		\biggr]_0^{\xi_i} \biggl\}</math>   DONE
	</td>		</td>
	</tr>		</tr>

			<!--
	<tr>		<tr>
	<td align="right"> </td>		<td align="right"> </td>
Line 380:		Line 381:
	</td>		</td>
	</tr>		</tr>
			-->

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>K_c\rho_c^{6/5} \biggl\{1 + \biggl[
			\frac{2\cdot 3^4~}{6(3 + \xi_i^2)^3} - \frac{2\cdot 3^4~}{6(3 )^3}
			\biggr] \biggl\}</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>K_c\rho_c^{6/5} \biggl\{1 + \biggl[
			\frac{3^2}{(3 + \xi_i^2)^3} - 1
			\biggr] \biggl\}</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>K_c\rho_c^{6/5} \biggl[
			\frac{3^2}{(3 + \xi_i^2)^3}
			\biggr]</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>K_c\rho_c^{6/5}
			\frac{1}{3}\biggl(1 + \frac{\xi_i^2}{3} \biggr)^{-3}</math>
			</td>
			</tr>

	</table>		</table>
	This result should be compared with,		This result should be compared with,

Joel2: /* Hydrostatic Balance */

2026-03-13T20:14:23Z

Hydrostatic Balance

← Older revision		Revision as of 16:14, 13 March 2026
Line 395:		Line 395:
	</table>		</table>

	<font color="red">This difference needs to be resolved!!!</font>		March 13, 2026:  <font color="red">This difference needs to be resolved!!!</font>

	=See Also=		=See Also=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2: /* Hydrostatic Balance */

2026-03-13T20:12:29Z

Hydrostatic Balance

← Older revision		Revision as of 16:12, 13 March 2026
Line 394:		Line 394:
	</tr>		</tr>
	</table>		</table>

			<font color="red">This difference needs to be resolved!!!</font>

	=See Also=		=See Also=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2: /* Hydrostatic Balance */

2026-03-13T20:00:57Z

Hydrostatic Balance

@@ Line 235: / Line 235: @@
    </td>
 </tr>
 <tr>
@@ Line 389: / Line 378: @@
 <math>K_c\rho_c^{6/5} \biggl[ 2 -
 ~ \biggl(1 + \frac{\xi_i^2}{3} \biggr)^{-3}\biggr]</math>
    </td>
 </tr>

Joel2: /* Hydrostatic Balance */

2026-03-13T19:54:59Z

Hydrostatic Balance

@@ Line 370: / Line 370: @@
 \frac{1}{6(3 + \xi^2)^3}
 \biggr]_0^{\xi_i} \biggl\}</math>
    </td>
 </tr>

Joel2: /* Hydrostatic Balance */

2026-03-13T19:41:58Z

Hydrostatic Balance

@@ Line 360: / Line 360: @@
 <math>K_c\rho_c^{6/5} \biggl\{1 + 2\cdot 3^4~ \int_0^{\xi_i}
 (3+\xi^2)^{-4} \xi\cdot d\xi\biggr\}</math>
    </td>
 </tr>

Joel2: /* Hydrostatic Balance */

2026-03-13T19:32:51Z

Hydrostatic Balance

← Older revision		Revision as of 15:32, 13 March 2026
Line 351:		Line 351:
	\biggl[1 + \frac{\xi^2}{3}\biggr]^{-4} ~\xi		\biggl[1 + \frac{\xi^2}{3}\biggr]^{-4} ~\xi
	\cdot d\xi</math>		\cdot d\xi</math>
			</td>
			</tr>

			<tr>
			<td align="right"> </td>
			<td align="center"><math>=</math></td>
			<td align="left">
			<math>K_c\rho_c^{6/5} \biggl\{1 + 2\cdot 3^4~ \int_0^{\xi_i}
			(3+\xi^2)^{-4} \xi\cdot d\xi\biggr\}</math>
	</td>		</td>
	</tr>		</tr>

Joel2: /* Hydrostatic Balance */

2026-03-13T19:24:32Z

Hydrostatic Balance

@@ Line 339: / Line 339: @@
 <math>P_c + \frac{K_c^{1 / 2}}{G^{1 / 2}\rho_c^{2 / 5}}\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} ~ \int_0^{\xi_i}
 G^{1 / 2} K_c^{1 / 2} \rho_c^{8/5} \biggl(\frac{2^3\pi}{3}\biggr)^{1 / 2}
 \biggl[1 + \frac{\xi^2}{3}\biggr]^{-4} ~\xi
   \cdot d\xi</math>

Joel2: /* Hydrostatic Balance */

2026-03-13T18:18:02Z

Hydrostatic Balance

@@ Line 330: / Line 330: @@
    <td  align="left">
 <math>\frac{K_c^{1 / 2}}{G^{1 / 2}\rho_c^{2 / 5}}\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} ~ \int_0^{\xi_i} \mathrm{RHS} \cdot d\xi</math>
    </td>
 </tr>