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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
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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