<p>Use equations <img src=\"https://s3.amazonaws.com/answer-board-image/cramster
ID: 1977271 • Letter: #
Question
<p>Use equations <img src="https://s3.amazonaws.com/answer-board-image/cramster-equation-201111131145456345678154572805067173.gif" alt="" align="absmiddle" />, <img src="https://s3.amazonaws.com/answer-board-image/cramster-equation-201111131144316345678147194443967794.gif" alt="" align="absmiddle" /> <br />to calculate the x- and y-coordinates of the center of mass of a semicircular metal plate with uniform density ρ and thickness t. Let the radius of the plate be a. The mass of the plate is thus <img src="https://s3.amazonaws.com/answer-board-image/cramster-equation-201111131147246345678164457496361096.gif" alt="" align="absmiddle" />. Use the coordinate system indicated in the figure .</p><p>Calculate the x- and y-coordinates of the center of mass of a semicircular metal plate.</p>
Explanation / Answer
Since the density is constant, it can be pulled outside the integral. First of all, you need to find an equation for the area of the plate. Then, you can find the bounds of integration for the double integrals you need to calculate the center of mass. I'm assuming that the plate is symmetric about the y axis Let the density be denoted by the letter k. Also, let dint [R] be the double integral over a region R, and int [x1,x2] denote the bounds of integration. For the sake of not completely confusing everyone with notation, I'm going to leave out the actual integration parts and just leave the results. x_com = k/m dint [R] x dA = k/m int [-a,a] int[0, sqrt(a^2-x^2)]x dy dx = k/m int [-a,a] x*sqrt(a^2-x^2) = 0-0 = ___0___ (as expected, the x-coord is at the line of symmetry) y_com = k/m dint [R] y dA = k/m int [-a,a] int [0, sqrt(a^2-x^2)] y dy dx = k/2m int [-a,a] a^2-x^2 dx = k/2m *(4a^3)/3 =___(2*ka^3)/3m___ (this is the y coord for any mass m, uniform density k, and radius a of any such semicircular lamina with symmetry about the y axis)
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