Lamina Up until now we have been talking about finite systems of point masses which can be handled completely with algebra. Now we will begin considering a flat point of any given shape. Such a shape is often called a lamina. The center of mass of a lamina is often called a centroid. One way to think of the centroid of a lamina is to imagine balancing the plate on top of a pin. The point at which it balances perfectly is the centroid. Instead of being given the mass of a lamina we are often given the density of the plate. If the lamina has a uniform density p then its mass is given by mass = density x area In the case where the lamina can be described as the area between a positive function f(x) and the x-axis, between the lines x = a and x = b we have If the lamina is symmetric about a line I, then when the lamina is reflected about the line of symmetry the centroid would not change. Since the only fixed points of the reflection lie on I we know that the centroid would also lie on l. This is sometimes referred to as the symmetry principle. The symmetry principle can be used to determine the centroid of several standard geometric shapes. Suppose we have a lamina that is built out of multiple non-overlapping plates. If we know the center of mass of each of those plates, then we can find the center of mass of the lamina. Recall that the moments of each of those plates behave the same as a point mass placed at the center of mass. If a lamina R is built out of plates R1, R2, ... , Ry with masses my, m2, ... , my each with centroid (x1. )1). (X2. V2). .-. . (Xn. )n), then as we saw above i = m1 . X1 + .". + mn . Xn y = " m1 ' y1 + .. + mn ' In m, + ..+ my m,+ .+ mn Exercises 3) What is the center of mass of a rectangle? Draw a picture that uses the symmetry principle to justify your answer. 4) Two rectangular plates are welded together. The first plate has vertices (0,0), (1,0), (1,1), and (0,1) with a density of 10. The second plate has a density of 2 and when welded to the first plate its vertices are (1,0), (4,0), (4,5), and (1,5). Find the center of mass of the two plates together. 5) Assuming a constant density of p find the center of mass of the plate to the right which was inspired by the example at the beginning of the lab