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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 point’s distance from the y-axis, find the moments of inertia about the x- and y-axes. Use these answers to find the radii of gyration of Tabout the x- and y-axes.

Short Answer

Expert verified

Radius of gyration is Ry=35andRx=15

Mass of lamina is m=23k

Moments are Ix=215kaboutyaxis and aboutxaxis isIy=25k

Step by step solution

01

Given Information

Density at every point is proportional to point's distance fromyaxis.

02

Moment of Inertia about  axis

xIt is given by

Iy=Ωx2ρ(x,y)dA

Putting limits

Iy=01-xxx2ρ(x,y)dydx

Iy=01-xxx2kxdydx[ρ(x,y)=kx]

Iy=k01-xxx3dydx

Solving inner integral first

Iy=k01[y]-xxx3dx

Iy=k01[x-(-x)]x3dx

Iy=k012x4dx

Integrating wrt x

Iy=2kx5501=25k

03

Moment of Inertia about y axis

Similarly

Ix=Ωy2ρ(x,y)dA

Ix=01-xxy2ρ(x,y)dydx

Ix=01-xxy2kxdydx[ρ(x,y)=kx]

Ix=k01xy33-xxdx

Ix=k01xx3-(-x)33dx

role="math" localid="1653986937609" Ix=k012x43dx

Ix=2x51501k=215k

04

Calculating Mass of Lamina

Mass of lamina is given by m=Ωρ(x,y)dA

m=01-xxkxdydx

m=k01[y]-xxxdx

m=k012x2dx

m=2k3x301

m=23k

05

Find radius of gyration

About both axis, it is given by

Ry=IymandRx=Ixm

Ry=25k23kandRx=215k23k

Ry=35andRx=15

Ry=35andRx=15

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