The discussion revolves around determining the moment of inertia of an area with respect to the y-axis using two different approaches involving rectangular elements. Participants are exploring the mathematical formulation and interpretation of the problem.
The discussion is ongoing, with participants sharing their attempts and questioning the appropriateness of certain equations. There is no explicit consensus yet, but various interpretations and approaches are being explored.
Participants note a potential misinterpretation of variables and are discussing the implications of using different thicknesses for the rectangular elements in their calculations. The original problem statement specifies the need to solve the moment of inertia using two distinct methods.
![<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}I_x=\int_{ }^{ }y^2dA\\
I_x=\int_0^4y^22xdy\ \left[y=4-4x^2,\ \textcolor{#E94D40}{\sqrt{\frac{4-y}{4}}=x}\right]\\
I_x=2\int_0^4y^2\sqrt{\frac{4-y}{4}}dy\ \left(u=\frac{4-y}{4},\ dy=-4du\right)\\
I_x=-8\int_0^4y^2\sqrt{u}du\ \left[u=\frac{4-y}{4},\ \textcolor{#E94D40}{4-4u=y}\right]\\
I_x=-8\int_0^4\left(4-4u\right)^2\sqrt{u}du\\
I_x=-8\int_0^4\left(4^2-2\cdot4\cdot4u+\left(4u\right)^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16-32u+16u^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16\sqrt{u}-32u\sqrt{u}+16u^2\sqrt{u}\right)du\\
I_x=-\frac{32\cdot8}{3}u^{\frac{3}{2}}+\frac{64\cdot8}{5}u^{\frac{5}{2}}-\frac{32\cdot8}{7}u^{\frac{7}{2}}\ \left[\textcolor{#E94D40}{u=\frac{4-y}{4}}\right]\\
I_x=-\frac{32\cdot8}{3}\left(\frac{4-y}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-y}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-y}{4}\right)^{\frac{7}{2}}\begin{bmatrix}4\\
0\end{bmatrix}\\
I_x=\cancel{\textcolor{#E94D40}{-\frac{32\cdot8}{3}\left(\frac{4-4}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-4}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-4}{4}\right)^{\frac{7}{2}}}}\\
-\left(-\frac{32\cdot8}{3}\left(\frac{4-0}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-0}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-0}{4}\right)^{\frac{7}{2}}\right)\\
I_x=19.5\mathit{\text{pulg}}^4\\
\ \end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mrow data-mjx-texclass=ORD/><mrow data-mjx-texclass=ORD/></msubsup><msup><mi>y</mi><mn>2</mn></msup><mi>d</mi><mi>A</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><mn>2</mn><mi>x</mi><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mo>=</mo><mi>x</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mn>2</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mi>d</mi><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>d</mi><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo>=</mo><mi>y</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><msup><mn>4</mn><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>4</mn><mi>u</mi><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>16</mn><mo>−</mo><mn>32</mn><mi>u</mi><mo>+</mo><mn>16</mn><msup><mi>u</mi><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr](https://lh6.googleusercontent.com/w2X_A8frcymArTvwcFv7xwyy1UuPgDt84Kue-NahCFeqIzXBqslUlQUwgsYBbFxP2rcFLIHc1DxyfUiKz6SHshk0V304Zs37Ja-ZcnMUqzQLvw_CsaQl-f8S6uODGM3g2vUCpRFC=s1600 "<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}I_x=\int_{ }^{ }y^2dA\\
I_x=\int_0^4y^22xdy\ \left[y=4-4x^2,\ \textcolor{#E94D40}{\sqrt{\frac{4-y}{4}}=x}\right]\\
I_x=2\int_0^4y^2\sqrt{\frac{4-y}{4}}dy\ \left(u=\frac{4-y}{4},\ dy=-4du\right)\\
I_x=-8\int_0^4y^2\sqrt{u}du\ \left[u=\frac{4-y}{4},\ \textcolor{#E94D40}{4-4u=y}\right]\\
I_x=-8\int_0^4\left(4-4u\right)^2\sqrt{u}du\\
I_x=-8\int_0^4\left(4^2-2\cdot4\cdot4u+\left(4u\right)^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16-32u+16u^2\right)\sqrt{u}du\\
I_x=-8\int_0^4\left(16\sqrt{u}-32u\sqrt{u}+16u^2\sqrt{u}\right)du\\
I_x=-\frac{32\cdot8}{3}u^{\frac{3}{2}}+\frac{64\cdot8}{5}u^{\frac{5}{2}}-\frac{32\cdot8}{7}u^{\frac{7}{2}}\ \left[\textcolor{#E94D40}{u=\frac{4-y}{4}}\right]\\
I_x=-\frac{32\cdot8}{3}\left(\frac{4-y}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-y}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-y}{4}\right)^{\frac{7}{2}}\begin{bmatrix}4\\
0\end{bmatrix}\\
I_x=\cancel{\textcolor{#E94D40}{-\frac{32\cdot8}{3}\left(\frac{4-4}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-4}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-4}{4}\right)^{\frac{7}{2}}}}\\
-\left(-\frac{32\cdot8}{3}\left(\frac{4-0}{4}\right)^{\frac{3}{2}}+\frac{64\cdot8}{5}\left(\frac{4-0}{4}\right)^{\frac{5}{2}}-\frac{32\cdot8}{7}\left(\frac{4-0}{4}\right)^{\frac{7}{2}}\right)\\
I_x=19.5\mathit{\text{pulg}}^4\\
\ \end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mrow data-mjx-texclass=ORD/><mrow data-mjx-texclass=ORD/></msubsup><msup><mi>y</mi><mn>2</mn></msup><mi>d</mi><mi>A</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><mn>2</mn><mi>x</mi><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mo>=</mo><mi>x</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mn>2</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac></msqrt><mi>d</mi><mi>y</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mi>d</mi><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>d</mi><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mi>y</mi><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi><mtext></mtext><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>y</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mtext></mtext><mstyle mathcolor=#E94D40><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo>=</mo><mi>y</mi></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mo>−</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><msup><mn>4</mn><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>⋅</mo><mn>4</mn><mo>⋅</mo><mn>4</mn><mi>u</mi><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>4</mn><mi>u</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msub><mi>I</mi><mi>x</mi></msub><mo>=</mo><mo>−</mo><mn>8</mn><msubsup><mo data-mjx-texclass=OP>∫</mo><mn>0</mn><mn>4</mn></msubsup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>16</mn><mo>−</mo><mn>32</mn><mi>u</mi><mo>+</mo><mn>16</mn><msup><mi>u</mi><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow><msqrt><mi>u</mi></msqrt><mi>d</mi><mi>u</mi></mtd></mtr")
waat, Can you tell me the name of this method?anuttarasammyak said:
This relevant equation seems inappropriate because x should be squared for y-axis rotation inertia.Tapias5000 said: