Question

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.

D is the triangular region with vertices (0, 0), (2, 1), (0, 3); ρ (x, y) = x + y

Answer 1

Step 1:

The density function of Lamina is .

Region bounded by a triangle with vertices .

The lamina mass can be defined as .

Region bounded:

First graph the vertices to find the region.

Graph :

(1) Draw the coordinate plane.

(2) Plot the vertices .

(3) Connect the plotted vertices to a smooth triangle.

Observe the graph :

The x-bounds are .

The line passing (0,0) and (2,1) :

Using two points form of a line equation is .

The line passing (0,3) and (2,1) :

Therefore y-bounds are .

Region bounded by the density function is and .

Step 2:

Evaluate the mass of lamina  .

                     

The mass of the lamina is .

Answer 2

Contd.....

Step 3:

Centre of mass of the lamina :

Centre mass of the lamina can be defined as

Where  ,

           ,

           and is mass of lamina : .

Step 4:

Step 5:

Step 6:

Centre of mass of the lamina :  .

Solution:

The mass of the lamina is .

Centre of mass of the lamina :  .