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Fluid Mechanics Worksheet on First and Second Moments page 1 of 7 Part 1. The "centroid" of an object is the geometric center of an object The "centroid" of an object is independent of the density of the object. For a two dimensional body, the centroid (xc, yc) can be calculated as: 1 1 A A xc x dA yc y dA Part 2. Gaining an intuition for first and second moments The center of mass of a system of particles is a specific point about which the sum of all the linear moments of the particles is zero. It is the point at which all of the system’s mass could be concentrated without affecting its response to gravity. When the density is uniform, the center of mass can be computed only from geometry – it is identical to the centroid. If the density is not uniform, then the center of mass for a 2-dimensional body is given by: 1 My 1 Mx x M x dA M y M y dA M where M is the total mass of the body and Mx and My are called the “first moments” of the body. Mx is the first moment with respect to the x axis. My is the first moment with respect to the y axis. M x y dA and M y x dA and M dA A A A The first moments of a body tell us the torque that it will exert about different axes in a gravitational field. Question/Demonstration 1: Something seems kind of backwards here. Why do you need to sum up all of the “x” distances to get the moment with respect to the y-axis, and sum up all of the “y” distances to get the moment with respect to the x-axis? Fluid Mechanics Worksheet on First and Second Moments page 2 of 7 We are often interested in the energy that is stored in a rotating body, or the amount of energy it would take to get a body to rotate. In this case, we need to calculate the second moments (moments of inertia) of the body, given by: I y 2 dA x A I x 2 dA y A The energy to rotate an object depends a lot on whether we're rotating it about the x-axis or the y-axis. Demonstration 2: Compare I x A y dA with 2 I y x 2 dA A Demonstration 3: Shifting the axis has a big effect on the moment of inertia To account for this, we can use the parallel axis theorem, which states that the moment of inertia about any axis parallel to the axis through the center of mass is given by Iparallel axis = Icenter of mass + Md2 Some important points : Moments are only meaningful with respect to some axis The axis doesn’t have to be the x or y axis, it can be some arbitrary axis Fluid Mechanics Worksheet on First and Second Moments page 3 of 7 Part 3: The term “moment of inertia” means different things to different people. Quantity Engineers Physicists and mathematicians tend to call the quantity tend to call the quantity First moment First area moment OR M x y dA with respect to the x axis First moment of the area A with respect to the x axis First moment First area moment OR M y x dA with respect to the y axis First moment of the area A with respect to the y axis First mass moment First moment M x y dA with respect to the x axis with respect to the x axis A First mass moment First moment M y x dA with respect to the y axis with respect to the y axis A Moment of inertia OR Area moment of inertia OR I x y 2 dA Second moment Second moment of the area A with respect to the x axis with respect to the x axis Moment of inertia OR Area moment of inertia OR I y x 2 dA Second moment Second moment of the area A with respect to the y axis with respect to the y axis Mass moment of inertia OR Moment of inertia OR I x y 2 dA Second mass moment Second moment A with respect to the x axis with respect to the x axis Mass moment of inertia OR Moment of inertia OR I y x 2 dA Second mass moment Second moment A with respect to the y axis with respect to the y axis Part 4: What terms will we use, in order to avoid confusion? Answer: To avoid ambiguity, in this class we will always include the word “mass” when the integral includes density, and we will always include the word “area” when the integral does not include density. This means that our terms will be: First area moment and First mass moment Area moment of inertia and Mass moment of inertia When you go out in the world – to industry, to graduate school, to a non-profit, or to consulting – be sure that you are sharing the same definitions as the other people you are working with. Fluid Mechanics Worksheet on First and Second Moments page 4 of 7 Part 5: Quiz time. You need to know this for the exam Write down the integrals that express: A) First area moment with respect to the x axis B) Mass moment of inertia with respect to the y axis C) First mass moment with respect to the x axis D) First area moment with respect to the y axis E) Area moment of inertia with respect to the x axis F) Mass moment of inertia with respect to the x axis Fluid Mechanics Worksheet on First and Second Moments page 5 of 7 Part 6: Problems. You need to know this for the exam You will be provided with the table of integrals that appears on the last page of this handout. Problem 1: Find the area moment of inertia with respect to the y-axis that goes through the centroid of a rectangle of length A and height B. Problem 2: Find the area moment of inertia with respect to the x-axis that passes through the centroid of a circle of radius R. Fluid Mechanics Worksheet on First and Second Moments page 6 of 7 Problem 3: Find the mass moment of inertia with respect to the y-axis that passes through the centroid of a circle of radius R. Assume constant density over the region. Problem 4: Find the mass moment of inertia with respect to the x-axis for a half circle of radius R that lies in the upper half plane (y>0). Assume constant density over the region. Problem 5: Find the mass moment of inertia about the line parallel to the x-axis that passes through the centroid of a half circle of radius R. Assume constant density over the region. Homework: Practice finding the area moments of inertia on page 61 of Munson et al. Fluid Mechanics Worksheet on First and Second Moments page 7 of 7 Possibly useful integrals:

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posted: | 11/27/2011 |

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