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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Assume that the solid has constant density k.

44. Find the moments of inertia for a rectangular brick with dimensions a, b, and c and mass M if the center of the brick is situated at the origin and the edges arc parallel to the coordinate axes.

To determine

To find: The moment of inertia of the rectangular brick if the solid has constant density k.

Explanation

Given:

The rectangular brick has dimensioned as a, b and c. which is length, breadth and height respectively.

The mass is M and the center is situated at origin and also parallel to the coordinate axes.

Calculation:

The edges of the brick parallel to the coordinate axes and also center is at the origin. Therefore x varies from a2 to a2 , y varies from b2 to b2 and z varies from c2 to c2 .

density=massvolume=Mabc=k

Initially, calculate the moment of inertia over the x-axis. Therefore, the integration will be over (x2+y2)ρ(x,y,z) but here the density is constant.

The moment of inertia over the x-axis is, Ix=E(y2+z2)ρ(x,y,z)dV

Ix=kc2c2b2b2a2a2(y2+z2)dxdydz

Integrate the above integral with respect to x and apply the limit of it.

Ix=kc2c2b2b2[y2x+z2x]a2a2dydz=kc2c2b2b2[(y2(a2)+z2(a2))(y2(a2)+z2(a2))]dydz=kc2c2b2b2[y2(a2)+z2(a2)+y2(a2)+z2(a2)]dydz=kc2c2b2b2[2y2(a2)+2z2(a2)]dydz

On further simplification the above integral becomes,

Ix=kac2c2b2b2(y2+z2)dydz

Integrate the above integral with respect to y and apply the limit of it.

Ix=kac2c2[y33+z2y]b2b2dz=kac2c2[((b2)33+z2(b2))((b2)33+z2(b2))]dz=kac2c2[(b2)33+z2(b2)+(b2)33+z2(b2)]dz=kac2c2[2(b2)33+2z2(b2)]dz

On further simplification the above integral becomes,

Ix=kac2c2[b312+z2b]dz

Integrate the above integral with respect to z and apply the limit of it.

Ix=ka[112b3z+bz33]c2c2=ka[(112b3(c2)+b(c2)33)(112b3(c2)+b(c2)33)]=ka[(124b3c+124bc3)(124b3c124bc3)]=ka[124b3c+124bc3+124b3c+124bc3]

On further simplification the above integral becomes,

Ix=ka[224b3c+224bc3]=kabc12(b2+c2)

Thus, the moment of inertia of the rectangular brick is abck12

Thus, the moment of inertia of the cylindrical brick about the x-axis is kabc12(b2+c2) .

Calculate the moment of inertia over the y-axis. Therefore, the integration will be over (z2+x2)ρ(x,y,z) but here the density is constant.

The moment of inertia over they y-axis is, Iy=E(z2+x2)ρ(x,y,z)dV

Iy=kc2c2b2b2a2a2(z2+x2)dxdydz

Integrate the above integral with respect to x and apply the limit of it.

Iy=kc2c2b2b2[z2x+x33]a2a2dydz=kc2c2b2b2[(z2(a2)+(a2)33)(z2(a2)+(a2)33)]dydz=kc2c2b2b2[(z2(a2)+(a2)33)+z2(a2)(a2)33]dydz=kc2c2b2b2[az2+a312]dydz

Integrate the above integral with respect to y and apply the limit of it.

Iy=kc2c2[az2y+a312y]b2b2dz=kc2c2[(az2(b2)+a312(b2))(az2(b2)+a312(b2))]dz=kc2c2[az2(b2)+a312(b2)+az2(b2)+a312(b2)]dz=kc2c2[abz22+a3b24+abz22+a3b24]dz

On further simplification the above integral becomes,

Iy=kc2c2[2abz22+2a3b24]dz=kc2c2[abz2+a3b12]dz

Integrate the above integral with respect to z- and apply the limit of it

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