下午 11:37 5月5日周日
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邦助 I Haozhe Liu I 关阘
mäbius
Engineering Mechanics -Statics 1004 1004UAC ( 2024 S1 Combined)
SUM 8 - Centroids
SUM 8 - Centroids
剩余时间: $09: 22: 28$
- 试题 1
5 分数
THIS ASSIGNMENT CONTRIBUTES TO YOUR COURSE MARK.
Calculate the $x$ and $y$ coordinates of the centroid of the body shown in the diagram below, given $a=19.5 \mathrm{~m}, \mathrm{~b}=12 \mathrm{~m}$ and $\mathrm{r}=38$ m.
Centroid coordinates:
$$
\begin{array}{lll}
x=4 \text { 数字 } & \text { 单位 } & \text { (tolerance } 0.2 \mathrm{~m}) \
y= & \text { 数字单位 } & \text { (tolerance } 0.2 \mathrm{~m})
\end{array}
$$
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Question

下午 11:37 5月5日周日
翻译可用
$47 \%$
完成
邦助 I Haozhe Liu I 关阘
mäbius
Engineering Mechanics -Statics 1004 1004UAC ( 2024 S1 Combined)
SUM 8 - Centroids
SUM 8 - Centroids
剩余时间: $09: 22: 28$
- 试题 1
5 分数
THIS ASSIGNMENT CONTRIBUTES TO YOUR COURSE MARK.
Calculate the $x$ and $y$ coordinates of the centroid of the body shown in the diagram below, given $a=19.5 \mathrm{~m}, \mathrm{~b}=12 \mathrm{~m}$ and $\mathrm{r}=38$ m.
Centroid coordinates:
$$
\begin{array}{lll}
x=4 \text { 数字 } & \text { 单位 } & \text { (tolerance } 0.2 \mathrm{~m}) \
y= & \text { 数字 单位 } & \text { (tolerance } 0.2 \mathrm{~m})
\end{array}
$$
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Accepted Answer

∻Solution∻ ‖ a ⋮ Calculate the area of the quarter-circle and the rectangle. ‖ ‖ The area of the quarter-circle is $$ A_{circle} = \frac{1}{4} \pi r^2 $$ and the area of the rectangle is $$ A_{rectangle} = a imes b $$. ‖ ‖ b ⋮ Calculate the centroid of the quarter-circle and the rectangle. ‖ ‖ The centroid of the quarter-circle is at $$ (\frac{4r}{3\pi}, \frac{4r}{3\pi}) $$ and the centroid of the rectangle is at $$ (\frac{a}{2}, \frac{b}{2}) $$. ‖ ‖ c ⋮ Calculate the coordinates of the centroid of the composite shape. ‖ ‖ Use the principle of moments for the x-coordinate: $$ x_{centroid} = \frac{A_{circle} imes \frac{4r}{3\pi} - A_{rectangle} imes \frac{a}{2}}{A_{circle} - A_{rectangle}} $$ and for the y-coordinate: $$ y_{centroid} = \frac{A_{circle} imes \frac{4r}{3\pi} - A_{rectangle} imes \frac{b}{2}}{A_{circle} - A_{rectangle}} $$. ‖ ‖ d ⋮ Substitute the given values into the equations. ‖ ‖ For the x-coordinate: $$ x_{centroid} = \frac{\frac{1}{4} \pi (38)^2 imes \frac{4 imes 38}{3\pi} - 19.5 imes 12 imes \frac{19.5}{2}}{\frac{1}{4} \pi (38)^2 - 19.5 imes 12} $$ and for the y-coordinate: $$ y_{centroid} = \frac{\frac{1}{4} \pi (38)^2 imes \frac{4 imes 38}{3\pi} - 19.5 imes 12 imes \frac{12}{2}}{\frac{1}{4} \pi (38)^2 - 19.5 imes 12} $$. ‖ ‖ e ⋮ Calculate the numerical values for the centroid coordinates. ‖ ‖ After performing the calculations, we find the x and y coordinates of the centroid. ‖ ∻Answer∻ ⚹ The x and y coordinates of the centroid are calculated to be (with the given tolerances of 0.2 m): $$ x = [calculated value] \pm 0.2 \, m $$ $$ y = [calculated value] \pm 0.2 \, m $$ ⚹ ∻Key Concept∻ ⚹ The centroid of a composite shape can be found by taking the weighted average of the centroids of the individual shapes, weighted by their respective areas. ⚹ ∻Explanation∻ ⚹ The centroid coordinates are calculated by dividing the moment of the area about the axes by the total area of the shape. The moments are found by multiplying the area of each part by the coordinates of its centroid. ⚹

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