A thin rectangular plate with sides of length 'a' and 'b' has mass per unit area varying as `alpha =alpha_(0) ((xy)/(ab))` where `alpha_(0)` is a constant. The co-ordinated of centre of mass of the plate is

Options

A. `((a)/(3),(b)/(3))`
B. `((a)/(2),(b)/(2))`
C. `((4a)/(3),(4b)/(3))`
D. `((2a)/(3),(2b)/(3))`

Answer

D

Explanation

`dm=((alpha_(0)x)/(ab))y(dy)(dx)`

`vec(R )=(x,y)`
`(int vec(R ).dm)/(int dm)=(int(alpha_(0)xy)/(ab)(dxdy)(xhat(i)+yhat(j)))/(int(alpha_(0)xy)/(ab)dx.dy)` `(int(alpha_(0)xy)/(ab)xxdx.dyhat(i))/(int(alpha_(0)xy)/(ab)dx.yd)+(int(alpha_(0)xy)/(ab)dx.dy(y)hat(j))/(int(alpha_(0)xy)/(ab)dx.dy)`
`y=COM" "x=`
`dy=" " dx=`
`(underset(0)overset(a)intx^(2)dx)/(underset(0)overset(a)int xdx)hat(i)+(underset(0)overset(b)int y^(2)dy)/(underset(0)overset(b)int ydx)`
`(2a)/(3)hat(i)+(2b)/(3)hat(j)`