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Let Tbe triangular region with vertices (0,0),(1,1),and(1,-1)

If the density at each point in Tis proportional to the pointโ€™s distance from the x-axis, find the center of mass of T.

Short Answer

Expert verified

Mass is k3.

Moment of Mass are My=k4and Mx=k6.

Center of mass are xยฏ=34,yยฏ=12

Step by step solution

01

Given Information

Vertices of triangular region are (0,0),(1,1),and(1,-1)

Density is proportional to point's distance from xaxis.

ฯ(x,y)=ky

02

Mass of Triangular Region

Mass of triangle = Twice the mass of upper triangle.

m=2โˆซ01โˆซ0xkydydxm=โˆฌฮฉฯ(x,y)dA

m=2kโˆซ01y220xdx

m=kโˆซ01x2dx

m=kx3301

m=k3

03

First Moment of Mass about y axis

My=โˆฌฮฉxฯ(x,y)dA

Putting limits

My=โˆซ01โˆซ-xxxkydydx[ฯ(x,y)=ky]

Do half the limit and twice the integral as per the given figure.

My=2โˆซ01โˆซ0xxkydydx

My=2kโˆซ01y220xxdx

My=kโˆซ01x3dx

My=x4401k=k4

04

First Moment of Mass about x axis

Mx=โˆฌฮฉyฯ(x,y)dA

Putting limits

Mx=โˆซ01โˆซ-xxky2dydx[ฯ(x,y)=ky]

Solving inner integral first

Mx=kโˆซ01y33-xxdx

Mx=kโˆซ01(x)3-(-x)33dx

Mx=23kโˆซ01x3dx

Mx=23kx4401=k6

05

Center of Mass

The coordinates are xยฏ=Mym,yยฏ=Mxm

xยฏ=kkk3,yยฏ=6kk3

xยฏ=34,yยฏ=12

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