Help finding mass and center of mass?

Need some help finding the center of mass.?

• A thing metallic plate covers the region in the xy-plane inside the circle centered at the origin with radius π/2, above the x-axis and above the line y= -x/2. Its density at the ...show more

• Answer:

  What a nasty integration! We can deal with area rather than volume as the plate is uniformly thin. The center of mass will be along the line of symmetry (radial arcs have the same density) which is the line y = 4x In its stated position, it is a really ugly integration as any horizontal or vertical strips have densities that do not vary in a linear fashion. To simplify the problem, I'm going to rotate the plate clockwise until the line of symmetry lines up with the positive x axis. This means that the angle above (+A) and below (-A) the x axis are the new positions of the edges of the plate. A will be (pi - tan^-1(1/2))/2 = 1.3389725 rad We will integrate all the radial arcs from r = 0 to pi/2. Each arc is dx wide and 2pi*x*2A/2pi = 2Ax long. Its area is therefore 2Ax dx Its weight is 2Ax*density dx = 2Ax*cos(x^2) dx ......(density = cos(x^2+y^2) = cos(r^2) by rotating the plate this is now cos(x^2) Mass moment from the y axis is 2Axcos(x^2)*xbar dx xbar for a thin radial arc is r*sin(A)/A = xsin(A)/A Mass moment = 2Ax^2cos(x^2)*sin(A)/A dx Now the mass moment of the whole plate is the weight of the plate times the distance from the y axis to the center of mass, aka xbar So xbar * Int 0 to pi/2 [2Axcos(x^2) dx] = Int 0 to pi/2 [2Ax^2cos(x^2)*sin(A)/A dx] xbar = (Int 0 to pi/2 [2Ax^2cos(x^2)*sin(A)/A dx])/(Int 0 to pi/2 [2Axcos(x^2) dx]) xbar = (Int 0 to pi/2 [2x^2cos(x^2)sin(A) dx])/(Int 0 to pi/2 [2Axcos(x^2) dx]) xbar = .148399/.835875 xbar = .1775........seems too small. I thought this answer was wrong until I checked the densities of the plate from x = 1.25 to x = pi/2. The range is from 0 to -.78 (a negative weight out there). The densities from x=0 to x=1.25 range from 1 to 0, so I'm more confident in the answer. Rotating this back into position; xbar = .0431 and ybar = .1722.