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Let Rbe rectangle with coordinates (0,0),(b,0),(0,h),and(b,h)

If the density at each point in Ris proportional to the square of the point’s distance from the y-axis, find the center of mass of R.

Short Answer

Expert verified

The center of mass isx¯=34b,y¯=h2.

Step by step solution

01

Given Information

The vertices of rectangular region is (0,0),(b,0),(0,h)and(b,h).

ρ(x,y)=kx2

02

Calculation of x-

The formula is x¯=Ωxρ(x,y)dAΩρ(x,y)dAandy¯=Ωyρ(x,y)dAΩρ(x,y)dA

Also ρ(x,y)=kx2

x¯=0b0hxkx2dydx0b0hkx2dydx

x¯=0b0hkx3dydx0b0hkx2dydx

x¯=0bkx3[y]0hdx0bkx2[y]0hdx=0bkhx3dx0bkhx2dx=kh0bx3dxkh0bx2dx

x¯=x440bx330b=b44b33

x¯=34b

03

Calculation of y-

The formula is y¯=Ωyρ(x,y)dAΩρ(x,y)dA

y¯=0b0hykx2dydx0b0hkx2dydx

y¯=0bkx2y220hdx0bkx2[y]0hdx=0bkx2h22dx0bkhx2dx=kh20bx2dx2kh0bx2dx

y¯=hx220b2x220b=hb222b22

y¯=h2

Hence,x¯=34b,y¯=h2

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