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A solution to find the moment of inertia for a green shape about the x and y axes. The shape is represented by the area between two curves, and the calculation involves integrating the moment of inertia for each elemental area da with respect to the x and y axes. The document also mentions the use of the parallel axis theorem and the centroid. Useful for students studying mechanics, calculus, or physics, particularly those focusing on moments of inertia and integral calculus.
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Find the moment of inertia for the green shape about the x and y axes. The units shown are in inches; the drawing is not to scale.
Solution:
The vertical dA is used to find Iy. The height of this rectangle is y 1 – y 2 or 3x/4 - x^2 /2. The width is dx.
The horizontal dA can be used to find Ix. The width of this rectangle is x 2 – x 1 or sqrt(2y) – 4y/3. The height is dy.
Note: we had to solve each of the functions for x in terms of y to be able to use this integral. Sometimes this is difficult. It would be great to be able to use the vertical element for finding Ix. To do that, integrate dIx. That is, find the moment of inertia for the blue vertical rectangle about the x axis.
The blue rectangle is just a rectangle. So we can use the parallel axis theorem plus 1/12 bh^3 to express the total moment of inertia for the rectangle about the x axis. The parallel axis theorem says that the moment of inertia is the centroidal value + Ad^2 where d is the distance from the axis to the centroid (measured perpendicularly from the axis.)
Warning: sometimes students get creative here in trying to figure out how to use the integral y^2 dA. Please avoid this.
