Example 15.6.6: Finding moments of inertia for a triangular lamina = xy Use the triangular region R with vertices (0, 0), (2, 2), and (2, 0) and with density p(x, y) as in previous examples. Find the moments of inertia. Show solution For regions with constant density Mathematica has a MomentOfIntertia command. For example, if R is the region in the previous Example and p(x, y) = 1 then you could calculate I by specifying the axis through {0, 0} and {1, 0} and doing 00 Moment Of Inertia[Triangle[{{0, 0}, {2, 2}, {2, 0}}], {0, 0}, {1, 0}] Exercise 15.6.6 = Again use the same region R as above and the density function p(x, y) of inertia. 64 IT = 35 Iy 64 35 Io = 128 21 = X cy. Find the moments
Example 15.6.6: Finding moments of inertia for a triangular lamina = xy Use the triangular region R with vertices (0, 0), (2, 2), and (2, 0) and with density p(x, y) as in previous examples. Find the moments of inertia. Show solution For regions with constant density Mathematica has a MomentOfIntertia command. For example, if R is the region in the previous Example and p(x, y) = 1 then you could calculate I by specifying the axis through {0, 0} and {1, 0} and doing 00 Moment Of Inertia[Triangle[{{0, 0}, {2, 2}, {2, 0}}], {0, 0}, {1, 0}] Exercise 15.6.6 = Again use the same region R as above and the density function p(x, y) of inertia. 64 IT = 35 Iy 64 35 Io = 128 21 = X cy. Find the moments
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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find Iy
![### Example 15.6.6: Finding moments of inertia for a triangular lamina
Use the triangular region \( R \) with vertices \((0, 0)\), \((2, 2)\), and \((2, 0)\) and with density \(\rho(x, y) = xy\) as in previous examples. Find the moments of inertia.
**Show solution**
For regions with constant density, Mathematica has a MomentOfInertia command. For example, if \( R \) is the region in the previous Example and \(\rho(x, y) = 1\), then you could calculate \(I_x\) by specifying the axis through \(\{0, 0\}\) and \(\{1, 0\}\) and doing
\[ \text{MomentOfInertia}[\text{Triangle}[{{0, 0}, {2, 2}, {2, 0}}], \{0, 0\}, \{1, 0\}] \]
---
### Exercise 15.6.6
Again, use the same region \( R \) as above and the density function \(\rho(x, y) = \sqrt{xy}\). Find the moments of inertia.
\[ I_x = \frac{64}{35} \,\text{✔} \]
\[ I_y = \frac{64}{35} \,\text{✘} \]
\[ I_0 = \frac{128}{21} \,\text{✔} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2Fcf2122b3-30c0-4ffa-8871-2a48870373b5%2Fvpy4dh7_processed.png&w=3840&q=75)
Transcribed Image Text:### Example 15.6.6: Finding moments of inertia for a triangular lamina
Use the triangular region \( R \) with vertices \((0, 0)\), \((2, 2)\), and \((2, 0)\) and with density \(\rho(x, y) = xy\) as in previous examples. Find the moments of inertia.
**Show solution**
For regions with constant density, Mathematica has a MomentOfInertia command. For example, if \( R \) is the region in the previous Example and \(\rho(x, y) = 1\), then you could calculate \(I_x\) by specifying the axis through \(\{0, 0\}\) and \(\{1, 0\}\) and doing
\[ \text{MomentOfInertia}[\text{Triangle}[{{0, 0}, {2, 2}, {2, 0}}], \{0, 0\}, \{1, 0\}] \]
---
### Exercise 15.6.6
Again, use the same region \( R \) as above and the density function \(\rho(x, y) = \sqrt{xy}\). Find the moments of inertia.
\[ I_x = \frac{64}{35} \,\text{✔} \]
\[ I_y = \frac{64}{35} \,\text{✘} \]
\[ I_0 = \frac{128}{21} \,\text{✔} \]
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