- #1
brandon26
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Where is the centre of mass of a semicircular lamina which is uniform? I know it is somewhere along the line of symestry, but where excactly?
Finding Centre of Mass for a Uniform Semicircular Lamina
- Thread starter brandon26
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SUMMARY
The center of mass for a uniform semicircular lamina is located at a distance of \( \frac{4r}{3\pi} \) from the flat edge along the line of symmetry. This conclusion is derived from the formula for the y-coordinate of the centroid, which is calculated using the integral \( \frac{\int y \, dA}{\int dA} \). The area of the region, represented by \( \int dA \), plays a crucial role in determining this centroid. Understanding these concepts is essential for accurately locating the center of mass in similar geometrical shapes.
PREREQUISITES- Understanding of centroid and center of mass concepts
- Familiarity with calculus, specifically integration
- Knowledge of geometric shapes, particularly semicircles
- Basic understanding of uniform density materials
- Study the derivation of the centroid formula for various geometric shapes
- Learn about the applications of center of mass in physics and engineering
- Explore advanced integration techniques for calculating areas and centroids
- Investigate the properties of laminae in different contexts, such as in mechanics
Students of physics and mathematics, educators teaching calculus and geometry, and engineers involved in structural analysis will benefit from this discussion.
- #2
Fermat
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It only takes a few moments to work it out.
com = 4r/3pi
com = 4r/3pi
- #3
brandon26
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Can you be of more help please?
- #4
Fermat
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Of course. Drag your mouse over the answer in my last post.
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HallsofIvy
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Surely, if you have a question like that, you know the basic formulas.
The y-coordinate of the centroid of a region (center of mass assuming uniform density) is \frac{\int y dA}{\int dA}.
\int dA is, of course, the area of the region.
Once, when I was teaching this, a student became fascinated by the word "lamina" (had never seen it before, apparently). As the last question on the final exam, I asked "What is 'lamina' spelled backwards?"
Another student became furious with me because "That question doesn't make any sense!"
The y-coordinate of the centroid of a region (center of mass assuming uniform density) is \frac{\int y dA}{\int dA}.
\int dA is, of course, the area of the region.
Once, when I was teaching this, a student became fascinated by the word "lamina" (had never seen it before, apparently). As the last question on the final exam, I asked "What is 'lamina' spelled backwards?"
Another student became furious with me because "That question doesn't make any sense!"
- #6
brandon26
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Oh sorry. Haha. I didnt realize there was invisible ink on the paper.Fermat said:
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