   Chapter 14.6, Problem 58E

Chapter
Section
Textbook Problem

Moments of Inertia In Exercises 57 and 58, verify the moments of inertia for the solid of uniform density. Use a computer algebra system to evaluate the triple integrals. I x = 1 12 m ( a 2 + b 2 ) I y = 1 12 m ( b 2 + c 2 ) I z = 1 12 m ( a 2 + c 2 ) To determine

To prove: Ix=112m(a2+b2),Iy=112m(b2+c2),Iz=112m(a2+c2)

Explanation

Given:

Density of the solid is:

ρ(x,y,z)=k

Formulae used:

m=Qρ(x,y,z)dv

Ixy=Qz2ρ(x,y,z)dv

Ixz=Qy2ρ(x,y,z)dv

Iyz=Qx2ρ(x,y,z)dv

Proof:

Total mass of solid is given by

m=Qρ(x,y,z)dvm=Qkdv=kc/2c/2a/2a/2b/2b/2dzdydx

By following the steps below in computer algebra system, we get total mass of the solid

m=abc(1)

Steps:

i) Search the symbol calculator in the web.

ii) Write down the given triple integral kc/2c/2a/2a/2b/2b/2dzdydx.

iii) Submit the triple integral and write the solution.

The moment of inertia on xy plane is given by

Ixy=Qz2ρ(x,y,z)dvIxy=Qk(z2)dv=kc/2c/2a/2a/2b/2b/2z2dzdydx

By following the steps below in computer algebra system, we get

Steps:

i) Search the symbol calculator in the web.

ii) Write down the given triple integral kc/2c/2a/2a/2b/2b/2z2dzdydx.

iii) Submit the triple integral and write the solution.

Ixy=112b2(abc)=112mb2Ixy=112mb2(fromequation(1))

The moment of inertia on xz plane is given by

Ixz=Qy2ρ(x,y,z)dvIxy=Qk(y2)dv=kc/2c/2a/2a/2b/2b/2y2dzdydx

By following the steps below in computer algebra system, we get

Steps:

i) Search the symbol calculator in the web

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