A thin plate covers the triangular region bounded by the x-axis and the lines x=1 and y=2x in the first quadrant. The plate density at the point (x,y) is rho (x,y) = 6x+6y+6. find the plates, mass, first moment, centre of mass about the coordinate axes.?

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1 Answer
Shiva Prakash M V
Feb 6, 2018

mass is #m=30t#

first moment about y axis is #My=16t#

center of mass about y axis is #x'=8/15#

first moment about x axis is #Mx=34t#

center of mass about x axis is #y'=17/15#

Explanation:

At any point (x,y); the element in the thin plate has the dimensions dx and dy.
The area of the element is dx.dy
If #rho(x,y)# is the density given by
#rho(x,y)=6x+6y+6#.

Since the plate has uniform thickness t, which is small;
volume of the element #dV=(t)(dx)(dy)#
mass of the element # dm# is given by #rho.dV#
#dm=rho.t.dxdy#
#rho=6x+6y+6#
#dm=t(6x+6y+6)dxdy#
taking 6 common,
#dm=6t(x+y+1)dxdy#
#y=2x#
for x=1, #y=2(1)=2#
Integrating wrt x from #(0,1)# along x axis
and from #(0,2)# along y axis
#intdm=6tint({int(x+y+1)dx)}dy#
#m=6tint(1/2x^2+yx+x)dy#

Simplifying
#=6tint(1/2(1^2-0^2)+y(1-0)+(1-0))dy#
#=6tint(1/2(1)+y(1)+1)dy#
#=6tint(1/2+y+1)dy#
#=6tint(y+3/2)dy#
#=6t(1/2y^2+3/2y)#
#=6t(1/2(2^2-0^2)+3/2(2-0))#
#=6t(1/2(4)+3/2(2))#
#=6t(2+3)#
#=6t(5)#
#m=30t#

First moment about y-axis
The mass element is at a distance of x from y axis
Hence, moment
#dMy=6tx(x+y+1)dxdy#
Integrating from x=0 to x=1 along x axis
and from#(0,2)# along y axis
#intdMy=int(6tx(x+y+1)dxdy#
#=6tint(int(x^2+xy+x)dx)dy#
#=6tint(1/3x^3+y/2x^2+1/2x^2)dy#
#=6tint(1/3(1^3-0^3)+y/2(1^2-0^2)+1/2(1^2-0^2))dy#
#=6tint(1/3(1)+y/2(1)+1/2(1))dy#
#=6tint(1/3+1/2+y/2)dy#
#=6tint(1/2y+5/6)dy#
#=6t(1/2(1/2y^2)+5/6y)#
#=6t(1/4y^2+5/6y)#
#=6t(1/4(2^2-0^2)+5/6(2-0))#
#=6t(1/4(4)+5/6(2))#
#=6t(1+5/3)#
#=6t(3/3+5/3)#
#=6t(8/3)#
#My=16t#

center of mass about y axis
#x'=(My)/m#
#=(16t)/30t#
#x'=8/15#
first moment about x axis
The mass element is at a distance of y from x axis
Hence, moment
#dMx=6ty(x+y+1)dxdy#

Integrating from x=0 to x=1 along x axis
and from#(0,2)# along y axis
#intdMy=int(6ty(x+y+1)dxdy#
#=6tint(int(yx+y^2+y)dx)dy#
#=6tint(y/2x^2+y^2x+yx))dy#
#=6tint(y/2(1^2-0^2)+y^2(1-0)+y(1-0))dy#
#=6tint(y/2(1-0)+y^2(1)+y(1))dy#
#=6tint(y/2+y^2+y)dy#
#=6tint((1/2)1/2y^2+1/3y^3+1/2y^2)dy#
#=6t(1/4y^2+1/3y^3+1/2y^2)#
#=6t(3/4y^2+1/3y^3)#
#=6t(3/4(2^2-0^2)+1/3(2^3-0^3))#
#=6t(3/4(4)+1/3(8))#
#=6t(3+8/3)#
#=6t(9/3+8/3)#
#=6t(17/3)#
#Mx=34t#

center of mass about x axis
#y'=(Mx)/m#
#=(34t)/30t#
#y'=17/15#

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