Question

A lamina with density occupies the region of the plane bounded by (y = a) and (y = b). The mass of lamina is equal to (M). Find the y-coordinate of its center of mass.

a) ( a + b2 )
b) ( 2a + b3 )
c) ( a - b2 )
d) ( a^2 + b^22 )

Answer 1

To find the y-coordinate of the center of mass of the given lamina, we need to consider the mass distribution along the y-axis. By dividing the lamina into infinitesimally small strips, we can find the mass of each strip and integrate to find the y-coordinate of the center of mass.

The correct otpion is not given.

Step-by-step explanation:

To find the y-coordinate of the center of mass of the lamina, we need to consider the mass distribution along the y-axis. Since the density is non-uniform, we can divide the lamina into infinitesimally small strips, each with width dy, and find the mass of each strip.

The mass of each strip is given by the density multiplied by the area of the strip, which is dx times dy. Therefore, the mass of each strip is p(x,y)dxdy.

The y-coordinate of the center of mass can be found by integrating y multiplied by the mass of each strip and dividing by the total mass of the lamina.

Let's denote the position of each strip along the y-axis as y'. Then the total mass of the lamina can be found by integrating p(x, y')dy' from y=a to y=b:

M = ∫(from a to b) p(x, y')dy'

Using the density function p(x, y) = p0xy, where p0 is a constant, and substituting y' with y, we have:

M = ∫(from a to b) p0xydy

Next, we can find the y-coordinate of the center of mass by integrating y multiplied by p0xydy and dividing by the total mass M:

ycm = (1/M)∫(from a to b) yp0xydy

The correct otpion is not given.