14-CenterOfGravity - Center of Gravity of a Homogeneous...

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Phong Do Center of Gravity of a Homogeneous Lamina Note: Many of you would see this topic again in excruciating details in Engineering Mechanics. Here, we will only give a broad view of this concept. When mentioning “mass” of an object, the layman definition is, “how much stuff it has”. From an engineering point of view, the mass of an object is a measurement of its resistance to motion . To more “massive” it is, the harder it is for it to move. Suppose that you place several objects of masses: m 1 , m 2 , m 3 … on a horizontal bar. Along this bar, you place a fulcrum and you want to have the bar to be nicely balanced. In other words, the bar will remain horizontal. To accomplish this, you would have to slide ” the masses left or right to achieve a perfect balance. When you slide a mass around, you are changing its position from the fulcrum, and we call this position the “ moment arm ”. A mass will generate a moment through this moment arm to cause the bar to rotate up and down about the fulcrum. Perfect balance will be achieved all the moments are zero thus cancelling out the tendency to cause the bar to to rotate. 1 1 2 2 3 3 0 n n m x m x m x m x The point at which this occurs is the “ center of mass ” and it is defined as: i i i m x x m You have 4 masses with mass 10 g, 15 g, 5 g and 10 g and they are arranged as shown. Find the center of mass for this arrangement. (10)( 5) (15)(0) (5)(4) (10)(7) 1 10 15 5 10 i i i m x x m the “green” dot as shown is then the center of mass.
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Phong Do Extending the concept to 2-dimension as shown and you will see that, the position of each mass must be defined as an ordered pairs ( , ) i i x y . With the x-y axis now in place, things are taking on a different meaning. If you form the product, say, 1 1 m x This generates a “ moment ” about the y-axis 1 1 m y This generate a “ moment ” about the x-axis Consequently, when you calculate i i m x you are really calculating the moment generated by all the masses about the y-axis. And this gives you: i i i i i i m x Mass Moment About y Axis x m Total Mass m y Mass Moment About x Axis y m Total Mass ( , ) x y defines the position of the center of mass for the 2-dimensional lay out. At this point in time, the “mass moment” terminology may not mean much to you but for those who are going on to take Engineering Mechanics, it will become more meaningful. As a quick insight: M X is the “area moment” about the x-axis . It defines the resistance to twisting/torsion about the x-axis .
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