SUMMARY
The discussion centers on the calculation of the weight of a bar using integrals, specifically addressing the relationship between the integral of the length of the bar and its weight per unit length of 6 lb/ft. The integral is confirmed to represent the total weight of the bar in pounds. Additionally, the term ##\bar{x}## is identified as the x-coordinate of the centroid, which is crucial for understanding the weight distribution along the bar. The participants seek clarity on how these concepts interrelate and the necessity of multiplying by 6.
PREREQUISITES
- Understanding of integral calculus, particularly in relation to physical applications.
- Familiarity with the concept of centroids in mechanics.
- Knowledge of weight distribution and linear density in physics.
- Basic understanding of units of measurement, specifically pounds per foot.
NEXT STEPS
- Study the application of integrals in calculating physical properties, focusing on weight and density.
- Research the concept of centroids and their significance in structural analysis.
- Explore examples of weight distribution in various materials using integral calculus.
- Learn about the implications of linear density in engineering and physics contexts.
USEFUL FOR
Students and professionals in engineering, physics, and mathematics who are involved in structural analysis, weight calculations, and integral applications in real-world scenarios.
![<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}Data\\
\begin{array}{l}\left(dL\right)^2=\left(dx\right)^2+\left(dy\right)^2\\
dL=\sqrt{\left(dx\right)^2+\left(dy\right)^2}\\
dL=\sqrt{\left(\left(\frac{dx}{dy}\right)^2dy^2+\left(1\right)\left(dy\right)^2\right)}\\
dL=\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy\\
y^2=3x\\
\frac{y^2}{3}=x\ →\frac{2y}{3}dy=dx\\
\frac{y^2}{3}=x\ →\frac{2y}{3}=\frac{dx}{dy}\\
\overline{x}=\frac{\int_0^3\tilde{x}dL}{\int_0^3dL}\ →\left[\textcolor{#E94D40}{\ dL=\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy}\right]\\
\overline{x}=\frac{\int_0^3\tilde{x}\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy}{\int_0^3\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy}\ →\left[\textcolor{#E94D40}{\ \frac{dx}{dy}=\frac{2y}{3}}\right]\end{array}\\
\overline{x}=\frac{\int_0^3x\sqrt{\left(\frac{2y}{3}\right)^2+1}dy\ →\ \left[\textcolor{#E94D40}{\frac{y^2}{3}=x}\ \right]\ }{\int_0^3\sqrt{\left(\frac{2y}{3}\right)^2+1}dy}\\
\overline{x}=\frac{\int_0^3\frac{y^2}{3}\sqrt{\left(\frac{2y}{3}\right)^2+1}dy\ \ }{\int_0^3\sqrt{\left(\frac{2y}{3}\right)^2+1}dy}\\
\overline{x}=\frac{5.46}{4.44}p\\
\overline{x}\approx1.23p\\
w=\int_0^36\sqrt{\left(\frac{2y}{3}\right)^2+1}dy\\
w=\textcolor{#28AE61}{\frac{6}{3}}\int_{ }^{ }\sqrt{4y^2+9}dy\ \left(y=\frac{3}{2}\tan\theta,\ dy=\frac{3}{2}\sec^2\theta\ d\theta\right)\\
w=2\int_{ }^{ }\sqrt{9\tan^2\theta+9}\frac{3}{2}\sec^2\theta\ d\theta\\
w=\frac{\cancel{\textcolor{#E94D40}{2}}\cdot3\cdot3}{\cancel{\textcolor{#E94D40}{2}}}\int_0^3\sqrt{\tan^2\theta+1}\sec^2\theta\ d\theta\\
w=9\int_{ }^{ }\sec^3\theta\ d\theta\ \left(u=\sec\theta,\ du=\sec\theta\tan\theta,\ dv=\sec^2\theta,\ v=\tan\theta\right)\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\sec\theta\tan\theta\tan\theta d\theta\right\}\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\sec\theta\tan^2\theta d\theta\right\}\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\left(\sec^3\theta-\sec\theta\right)d\theta\right\}\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\sec^3\theta d\theta+\int_{ }^{ }\sec\theta d\theta\right\}\\
w=\frac{9}{2}\sec\theta\tan\theta+\frac{9}{2}\ln\left|\sec\theta+\tan\theta\right|\ \left[y=\frac{3}{2}\tan\theta,\ \theta=\arctan\left(\frac{2y}{3}\right)\right]\\
w=\frac{9}{2}\frac{\sqrt{4y^2+9}}{3}\frac{2y}{3}+\frac{9}{2}\ln\left|\frac{\sqrt{4y^2+9}}{3}+\frac{2y}{3}\right|\\
w=y\sqrt{4y^2+9}+\frac{9}{2}\ln\left|\frac{2y+\sqrt{4y^2+9}}{3}\right|\ \begin{bmatrix}3\\
0\end{bmatrix}\\
w=3\sqrt{4\left(3\right)^2+9}+\frac{9}{2}\ln\left|\frac{2\left(3\right)+\sqrt{4\left(3\right)^2+9}}{3}\right|-\left(\cancel{\textcolor{#E94D40}{0\sqrt{4\left(0\right)^2+9}}}+\frac{9}{2}\ln\left|\frac{\cancel{\textcolor{#E94D40}{2\left(0\right)}}+\sqrt{\cancel{\textcolor{#E94D40}{4\left(0\right)^2}}+9}}{3}\right|\right)\\
w=9\sqrt{5}+\frac{9\ln\left(2+\sqrt{5}\right)}{2}-\left(\textcolor{#E94D40}{\cancel{\frac{9}{2}\ln\left|1\right|}}\right)\\
w=26.62lb\end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><mi>D</mi><mi>a</mi><mi>t</mi><mi>a</mi></mtd></mtr><mtr><mtd><mtable columnspacing=1em rowspacing=4pt><mtr><mtd><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>L</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>x</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>y</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>d</mi><mi>L</mi><mo>=</mo><msqrt><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>x</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>y</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup></msqrt></mtd></mtr><mtr><mtd><mi>d</mi><mi>L</mi><mo>=</mo><msqrt><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mi>d</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>1</mn><mo data-mjx-texclass=CLOSE>)</mo></mrow><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>y</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow></msqrt></mtd></mtr><mtr><mtd><mi>d<](https://lh5.googleusercontent.com/VT6GutAdkVXhuDnL3smY7ui5KBx4ORHztbT046vn4TKYK87mqPToV0k9UDzwXCSqOPIsbPT81SXU7WSCLN_3P1G62ZkdnuBK-YbI9SIygPWTUjELey8g4rC-Xdu0jJsvsmHKuaw5=s1600 "<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}Data\\
\begin{array}{l}\left(dL\right)^2=\left(dx\right)^2+\left(dy\right)^2\\
dL=\sqrt{\left(dx\right)^2+\left(dy\right)^2}\\
dL=\sqrt{\left(\left(\frac{dx}{dy}\right)^2dy^2+\left(1\right)\left(dy\right)^2\right)}\\
dL=\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy\\
y^2=3x\\
\frac{y^2}{3}=x\ →\frac{2y}{3}dy=dx\\
\frac{y^2}{3}=x\ →\frac{2y}{3}=\frac{dx}{dy}\\
\overline{x}=\frac{\int_0^3\tilde{x}dL}{\int_0^3dL}\ →\left[\textcolor{#E94D40}{\ dL=\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy}\right]\\
\overline{x}=\frac{\int_0^3\tilde{x}\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy}{\int_0^3\sqrt{\left(\frac{dx}{dy}\right)^2+1}dy}\ →\left[\textcolor{#E94D40}{\ \frac{dx}{dy}=\frac{2y}{3}}\right]\end{array}\\
\overline{x}=\frac{\int_0^3x\sqrt{\left(\frac{2y}{3}\right)^2+1}dy\ →\ \left[\textcolor{#E94D40}{\frac{y^2}{3}=x}\ \right]\ }{\int_0^3\sqrt{\left(\frac{2y}{3}\right)^2+1}dy}\\
\overline{x}=\frac{\int_0^3\frac{y^2}{3}\sqrt{\left(\frac{2y}{3}\right)^2+1}dy\ \ }{\int_0^3\sqrt{\left(\frac{2y}{3}\right)^2+1}dy}\\
\overline{x}=\frac{5.46}{4.44}p\\
\overline{x}\approx1.23p\\
w=\int_0^36\sqrt{\left(\frac{2y}{3}\right)^2+1}dy\\
w=\textcolor{#28AE61}{\frac{6}{3}}\int_{ }^{ }\sqrt{4y^2+9}dy\ \left(y=\frac{3}{2}\tan\theta,\ dy=\frac{3}{2}\sec^2\theta\ d\theta\right)\\
w=2\int_{ }^{ }\sqrt{9\tan^2\theta+9}\frac{3}{2}\sec^2\theta\ d\theta\\
w=\frac{\cancel{\textcolor{#E94D40}{2}}\cdot3\cdot3}{\cancel{\textcolor{#E94D40}{2}}}\int_0^3\sqrt{\tan^2\theta+1}\sec^2\theta\ d\theta\\
w=9\int_{ }^{ }\sec^3\theta\ d\theta\ \left(u=\sec\theta,\ du=\sec\theta\tan\theta,\ dv=\sec^2\theta,\ v=\tan\theta\right)\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\sec\theta\tan\theta\tan\theta d\theta\right\}\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\sec\theta\tan^2\theta d\theta\right\}\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\left(\sec^3\theta-\sec\theta\right)d\theta\right\}\\
w=9\left\{\sec\theta\tan\theta-\int_{ }^{ }\sec^3\theta d\theta+\int_{ }^{ }\sec\theta d\theta\right\}\\
w=\frac{9}{2}\sec\theta\tan\theta+\frac{9}{2}\ln\left|\sec\theta+\tan\theta\right|\ \left[y=\frac{3}{2}\tan\theta,\ \theta=\arctan\left(\frac{2y}{3}\right)\right]\\
w=\frac{9}{2}\frac{\sqrt{4y^2+9}}{3}\frac{2y}{3}+\frac{9}{2}\ln\left|\frac{\sqrt{4y^2+9}}{3}+\frac{2y}{3}\right|\\
w=y\sqrt{4y^2+9}+\frac{9}{2}\ln\left|\frac{2y+\sqrt{4y^2+9}}{3}\right|\ \begin{bmatrix}3\\
0\end{bmatrix}\\
w=3\sqrt{4\left(3\right)^2+9}+\frac{9}{2}\ln\left|\frac{2\left(3\right)+\sqrt{4\left(3\right)^2+9}}{3}\right|-\left(\cancel{\textcolor{#E94D40}{0\sqrt{4\left(0\right)^2+9}}}+\frac{9}{2}\ln\left|\frac{\cancel{\textcolor{#E94D40}{2\left(0\right)}}+\sqrt{\cancel{\textcolor{#E94D40}{4\left(0\right)^2}}+9}}{3}\right|\right)\\
w=9\sqrt{5}+\frac{9\ln\left(2+\sqrt{5}\right)}{2}-\left(\textcolor{#E94D40}{\cancel{\frac{9}{2}\ln\left|1\right|}}\right)\\
w=26.62lb\end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><mi>D</mi><mi>a</mi><mi>t</mi><mi>a</mi></mtd></mtr><mtr><mtd><mtable columnspacing=1em rowspacing=4pt><mtr><mtd><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>L</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>x</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>y</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>d</mi><mi>L</mi><mo>=</mo><msqrt><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>x</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>y</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup></msqrt></mtd></mtr><mtr><mtd><mi>d</mi><mi>L</mi><mo>=</mo><msqrt><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mi>d</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mn>1</mn><mo data-mjx-texclass=CLOSE>)</mo></mrow><msup><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>(</mo><mi>d</mi><mi>y</mi><mo data-mjx-texclass=CLOSE>)</mo></mrow><mn>2</mn></msup><mo data-mjx-texclass=CLOSE>)</mo></mrow></msqrt></mtd></mtr><mtr><mtd><mi>d<")
![<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}↶_+ΣM_A=0\\
M_A-\overline{x}\cdot26.6=0\\
M_A-1.23\cdot26.6=0\\
\left[\textcolor{#E94D40}{M_A=32.72↶_+}\right]\\
\overrightarrow{+}Σfx=0\\
\left[\textcolor{#E94D40}{A_x=\overrightarrow{0}}\right]\\
_+↑Σfy=0\\
A_y-26.6=0\\
\left[\textcolor{#E94D40}{A_y=26.6_+↑}\right]\\
Solutions\\
\left[\textcolor{#E94D40}{A_x=\overrightarrow{0}}\right]\left[\textcolor{#E94D40}{A_y=26.6_+↑}\right]\\
\left[\textcolor{#E94D40}{M_A=32.72↶_+}\right]\end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><msub><mo>↶</mo><mo>+</mo></msub><mi>Σ</mi><msub><mi>M</mi><mi>A</mi></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>M</mi><mi>A</mi></msub><mo>−</mo><mover><mi>x</mi><mo accent=true>―</mo></mover><mo>⋅</mo><mn>26.6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>M</mi><mi>A</mi></msub><mo>−</mo><mn>1.23</mn><mo>⋅</mo><mn>26.6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>M</mi><mi>A</mi></msub><mo>=</mo><mn>32.72</mn><msub><mo>↶</mo><mo>+</mo></msub></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><mover><mo>+</mo><mo>→</mo></mover><mi>Σ</mi><mi>f</mi><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>x</mi></msub><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi/><mo>+</mo></msub><mo stretchy=false>↑</mo><mi>Σ</mi><mi>f</mi><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>A</mi><mi>y</mi></msub><mo>−</mo><mn>26.6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>y</mi></msub><mo>=</mo><msub><mn>26.6</mn><mo>+</mo></msub><mo stretchy=false>↑</mo></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><mi>S</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>x</mi></msub><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>y</mi></msub><mo>=</mo><msub><mn>26.6</mn><mo>+</mo></msub><mo stretchy=false>↑</mo></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>M</mi><mi>A</mi></msub><mo>=</mo><mn>32.72</mn><msub><mo>↶</mo><mo>+</mo></msub></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr></mtable></math>](https://lh6.googleusercontent.com/eh33mZuwrGdw08zZMT9-XQRNEds6-8zvswUKFxr8ed8vpIAKsCy-QQ-ah63HsRLpYPX4QJuE8nszTzoIGdfGq--DZMZAQqd6-UywrnLtXbqjo8FAzp7d8hiTCFvvqOy-HY23ZUMX=s1600 "<math xmlns=http://www.w3.org/1998/Math/MathML display=block data-is-equatio=1 data-latex=\begin{array}{l}↶_+ΣM_A=0\\
M_A-\overline{x}\cdot26.6=0\\
M_A-1.23\cdot26.6=0\\
\left[\textcolor{#E94D40}{M_A=32.72↶_+}\right]\\
\overrightarrow{+}Σfx=0\\
\left[\textcolor{#E94D40}{A_x=\overrightarrow{0}}\right]\\
_+↑Σfy=0\\
A_y-26.6=0\\
\left[\textcolor{#E94D40}{A_y=26.6_+↑}\right]\\
Solutions\\
\left[\textcolor{#E94D40}{A_x=\overrightarrow{0}}\right]\left[\textcolor{#E94D40}{A_y=26.6_+↑}\right]\\
\left[\textcolor{#E94D40}{M_A=32.72↶_+}\right]\end{array}><mtable columnalign=left columnspacing=1em rowspacing=4pt><mtr><mtd><msub><mo>↶</mo><mo>+</mo></msub><mi>Σ</mi><msub><mi>M</mi><mi>A</mi></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>M</mi><mi>A</mi></msub><mo>−</mo><mover><mi>x</mi><mo accent=true>―</mo></mover><mo>⋅</mo><mn>26.6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>M</mi><mi>A</mi></msub><mo>−</mo><mn>1.23</mn><mo>⋅</mo><mn>26.6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>M</mi><mi>A</mi></msub><mo>=</mo><mn>32.72</mn><msub><mo>↶</mo><mo>+</mo></msub></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><mover><mo>+</mo><mo>→</mo></mover><mi>Σ</mi><mi>f</mi><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>x</mi></msub><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mi/><mo>+</mo></msub><mo stretchy=false>↑</mo><mi>Σ</mi><mi>f</mi><mi>y</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>A</mi><mi>y</mi></msub><mo>−</mo><mn>26.6</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>y</mi></msub><mo>=</mo><msub><mn>26.6</mn><mo>+</mo></msub><mo stretchy=false>↑</mo></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><mi>S</mi><mi>o</mi><mi>l</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>x</mi></msub><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>A</mi><mi>y</mi></msub><mo>=</mo><msub><mn>26.6</mn><mo>+</mo></msub><mo stretchy=false>↑</mo></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr><mtr><mtd><mrow data-mjx-texclass=INNER><mo data-mjx-texclass=OPEN>[</mo><mstyle mathcolor=#E94D40><msub><mi>M</mi><mi>A</mi></msub><mo>=</mo><mn>32.72</mn><msub><mo>↶</mo><mo>+</mo></msub></mstyle><mo data-mjx-texclass=CLOSE>]</mo></mrow></mtd></mtr></mtable></math>")
