 Ch 7. Centroid/Distributed Loads/Inertia Multimedia Engineering Statics Centroid: Line Area Vol Centroid: Composite Distributed Loads Area Moment of Inertia
 Chapter 1. Basics 2. Vectors 3. Forces 4. Moments 5. Rigid Bodies 6. Structures 7. Centroids/Inertia 8. Internal Loads 9. Friction 10. Work & Energy Appendix Basic Math Units Sections Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Kurt Gramoll ©Kurt Gramoll STATICS - EXAMPLE Area Between Curve and x and y-axis Example 1 Find moment of inertia of the shaded area about a) x axis b) y axis Solution (a) Recall, the moment of inertia is the second moment of the area about a given axis or line. For part a) of this problem, the moment of inertia is about the x-axis. The differential element, dA, is usually broken into two parts, dx and dy (dA = dx dy), which makes integration easier. This also requires the integral be split into integration along the x direction (dx) and along the y direction (dy). The order of integration, dx or dy, is optional, but usually there is an easy way, and a more difficult way. Cross-section Area For this problem, the integration will be done first along the y direction, and then along the x direction. This order is easier since the curve function is given as y is equal to a function of x. The diagram at the left shows the dy going from 0 to the curve, or just y. Thus the limits of integration is 0 to y. The next integration along the x direction goes from 0 to 4. The final integration from is Expanding the bracket by using the formula,      (a-b)3 = a3 - 3 a2 b + 3 a b2 - b3 Solution (b) Similar to the previous solution is part a), the moment of inertia is the second moment of the area about a given axis or line. But in this case, it is about the y-axis, or  Cross-section Area The integral is still split into integration along the x direction (dx) and along the y direction (dy). Again, the integration will be done first along the y direction, and then along the x direction. The diagram at the left shows the dy going from 0 to the curve, or just y. Thus the limits of integration is 0 to y. The next integration along the x direction goes from 0 to 4. The final integration from is Comment The area is more closely distributed about the y-axis than x-axis. Thus, the moment of inertia of the shaded region is less about the y-axis as compared to x-axis. Example 2 Determine the moment of inertia of y = 2 - 2x2 about the x axis. Calculate the moment of inertia in two different ways. First, (a) by taking a differential element, having a thickness dx and second, (b) by using a horizontal element with a thickness, dy. Solution a) The area of the differential element parallel to y axis is dA = ydx. The distance from x axis to the center of the element is namedy.      y = y/2 Using the parallel axis theorem, the moment of inertia of this element about x axis is For a rectangular shape, I is bh3/12. Substituting Ix, dA, and y gives, Performing the integration, gives,  (b) First, the function should be rewritten in terms of y as the independent variable. Due to the x2 term, there is a positive and negative form and it can be expressed as two similar functions mirrored about y axis. The function on the right side of the axis can be expressed as The area of the differential element parallel to x axis is Performing the integration gives, Performing a numerical integration on calculator or by taking t = 2(2 - y) the above integration can be found as, As expected, both methods (a) and (b) provide the same answer.

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