Question

Consider a rigid plane body or "lamina," such as a flat piece of sheet metal, rotating about a point O in the body. If we choose axes so that the lamina lies in the xy plane, which elements of the inertia tensor I are automatically zero? Prove that

Izz=Ixx+Iyy.I_{zz} = I_{xx} + I_{yy}.

Solution

Verified

Step 1

1 of 2

We are considering a rigid plane body or “lamina,” such as a flat piece of sheet metal that lies in xy plane.

Density of the metal sheet can be written as:

ρ(x,y,z)=σ(x,y)δ(z)\rho(x,y,z)=\sigma(x,y)\delta(z)

If we integrate term above, we can get full mass of the sheet:

σ(x,y)δ(z)dxdydz=σ(x,y)dxdyδ(z)dz=σ(x,y)dxdy=μ\int{\sigma(x,y)\delta(z) dxdydz}=\int{\sigma(x,y)dxdy}\int{\delta(z)dz}=\int{\sigma(x,y)dxdy}=\mu

Diagonal inertia tensors are given by:

Ixx=σ(x,y)(y2+z2)δ(z)dxdydz=σ(x,y)y2dxdyI_{xx}=\int{\sigma(x,y) (y^2+z^2)\delta(z)dxdydz}=\int{\sigma(x,y)\cdot y^2dxdy}

Iyy=σ(x,y)(x2+z2)δ(z)dxdydz=σ(x,y)x2dxdyI_{yy}=\int{\sigma(x,y) (x^2+z^2)\delta(z)dxdydz}=\int{\sigma(x,y)\cdot x^2dxdy}

Izz=σ(x,y)(x2+y2)δ(z)dxdydz=σ(x,y)(x2+y2)dxdyI_{zz}=\int{\sigma(x,y) (x^2+y^2)\delta(z)dxdydz}=\int{\sigma(x,y)\cdot (x^2+y^2)dxdy}

We can see that :

Izz=Ixx+Iyy\boxed{I_{zz}=I_{xx}+I_{yy}}

The only non diagonal element that is not equal to zero is:

Ixy=σ(x,y)xyδ(z)dxdydz=σ(x,y)xydxdyI_{xy}=-\int{\sigma(x,y) x\cdot y\delta(z)dxdydz}=-\int{\sigma(x,y)\cdot xy\cdot dxdy}

All other non diagonal elements are equal to zero:

Ixz=Iyz=0\boxed{I_{xz}=I_{yz}=0}

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Create an account to view solutions

By signing up, you accept Quizlet's Terms of Service and Privacy Policy
Continue with GoogleContinue with Facebook

Recommended textbook solutions

Physics for Scientists and Engineers 9th Edition by John W. Jewett, Raymond A. Serway

Physics for Scientists and Engineers

9th EditionJohn W. Jewett, Raymond A. Serway
6,002 solutions
Physics for Scientists and Engineers with Modern Physics 10th Edition by John W. Jewett, Raymond A. Serway

Physics for Scientists and Engineers with Modern Physics

10th EditionJohn W. Jewett, Raymond A. Serway
3,192 solutions
Classical Mechanics 1st Edition by John R. Taylor

Classical Mechanics

1st EditionJohn R. Taylor
741 solutions

Related questions