All these expressions can be written in polar coordinates by substituting zr cos 0, y = r sin, and dA = r dr dº. For example, Io = √²p(r cos 0, r sin 8) r dr dê. 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 p(x, y) — zy as in previous examples. Find the moments of inertia. Show solution
All these expressions can be written in polar coordinates by substituting zr cos 0, y = r sin, and dA = r dr dº. For example, Io = √²p(r cos 0, r sin 8) r dr dê. 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 p(x, y) — zy as in previous examples. Find the moments of inertia. Show solution
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![All these expressions can be written in polar coordinates by substituting = r cos 0, y = r sin 0, and
dA = r dr de. For example, Io = √²p(r cos 6, r sin 6) r dr de.
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 p(x, y) = xy 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
OO
MomentOfInertia [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) = √y. Find the moments of
inertia.
IT
Iy
Io
||
||
| 5
64
64
128
21](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2F3a71e60e-43a3-4399-916d-553b74977686%2Ftc6771m_processed.png&w=3840&q=75)
Transcribed Image Text:All these expressions can be written in polar coordinates by substituting = r cos 0, y = r sin 0, and
dA = r dr de. For example, Io = √²p(r cos 6, r sin 6) r dr de.
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 p(x, y) = xy 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
OO
MomentOfInertia [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) = √y. Find the moments of
inertia.
IT
Iy
Io
||
||
| 5
64
64
128
21
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