p0340 - 290 CHAPTER 3 EFFECTS OF GRAVITY ON PRESSURE 3.40...

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Unformatted text preview: 290 CHAPTER 3. EFFECTS OF GRAVITY ON PRESSURE 3.40 Chapter 3, Problem 40 ....... ....... . .. . . .. . .. . . . . . . h . . ....... ....... . h . . . . . . . .. . .. . . . .. .. . . . . . . . . . 2R . . . . . . . . .. . .. . . ......................... . . ....................... . . . . . . . . . . . . . . . . . . . . . . .. . . ...... . .... ...... . . ... . . ........ .... . . ......... ...... . ...... . . . . . . . . . . . . . . . . . . . . . . . . ....... ... ........ . ............................ . . ...... ...... .... . ... ... . . ... ... ... ... .. .. .. .. .. . . . . . . . . . . .... .. . . . . . ...... . ........ . ... . . ......... ... ....... . . ..... . ... . . . . . . . . . . . .. . .. .. .. .. .. .. ... .. ... ... ..... ..... ........ .......... .... . .. . .. . . . . . . . . . .. .. . . . . .. . . . . . . . . . . . . .. . .. . h/2 R h= R Figure 3.25: square/circle geometry. As shown in Figure 3.25, the centroid of the circle is R below the common vertex, while the centroid of the square is h/2 above it. The moment of inertia of the square and circle, according to the parallel-axis theorem, is equal to the sum of the moment of inertia of the circle shifted by R and the moment of inertia of the square shifted by h/2. So, first note that for the circle, the area and moment of inertia are Ac = R2 Also, for the square, since h = and 1 Ic = R4 4 R, we have and Is = 1 4 2 4 h = R 12 12 As = h2 = R2 Hence, using the parallel-axis theorem for the circle, 1 5 Ioc = Ic + z 2 Ac = R4 + R2 R2 = R4 c 4 4 Similarly, for the square, Ios = Is + z 2 As = s 2 4 2 1 R + R (R2 ) = 2 R4 12 4 3 This, summing these results, the moment of inertia for the square and circle is I= (15 + 4) R4 12 ...
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This note was uploaded on 10/12/2009 for the course AME 309 at USC.

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