Question

Find the center of the mass of a thin plate of constant density (delta) covering the region bounded by the x-axis and the curve y=5/2 cos x, -pi/4

Answer 1

To find the center of mass of a thin plate of constant density covering the region bounded by the x-axis and the curve y=5/2 cos x, we can use the concept of density and integration. The x-coordinate (xcm) and y-coordinate (ycm) of the center of mass can be found separately by integrating x*(sigma)dA and y*(sigma)dA, respectively, over the region. The center of mass is then given by (xcm, ycm).

Step-by-step explanation:

The center of mass of a thin plate can be found by using the concept of density and integration. In this case, the density is constant, so we can use the surface mass density (sigma) to represent it. To find the center of mass, we need to find the x-coordinate (xcm) and y-coordinate (ycm) separately and then combine them.

1. To find xcm, we integrate x*(sigma)dA over the region bounded by the x-axis and the curve y=5/2 cos x. We can rewrite the curve equation as x = arccos(2y/5). So, xcm = integral from 0 to pi of x*(sigma)dx = integral from 0 to pi of (arccos(2y/5))*(sigma)dx.
2. To find ycm, we integrate y*(sigma)dA over the same region. In this case, y = 5/2 cos x. So, ycm = integral from 0 to pi of (5/2 cos x)*(sigma)dx.
3. Finally, the center of mass is given by (xcm, ycm).