Definition Polar moment of inertia

note: while has become common find term moment of inertia used describe polar , planar second moments of area, construct of engineering fields. term moment of inertia, within physics , mathematics fields, strictly mass moment of inertia, or second moment of mass, used describe massive object s resistance rotational motion, not resistance torsional deformation. while polar , planar second moments of inertia integrated on infinitesimal elements of given area in two-dimensional cross-section, mass moment of inertia integrated on infinitesimal elements of mass in three-dimensional space occupied object. put, polar , planar second moments of inertia indication of rigidity, , mass moment of inertia rotational motion resistance of massive object.

the equation describing polar moment of inertia multiple integral on cross-sectional area,



a


{\displaystyle a}

, of object.

j
=

∬

a



ρ

2


d
a


{\displaystyle j=\iint \limits _{a}\rho ^{2}da}

where,



ρ


{\displaystyle \rho }

distance element



d
a


{\displaystyle da}

.

substituting



x


{\displaystyle x}

,



y


{\displaystyle y}

components, using pythagorean theorem:

j
=

∬

a


(

x

2


+

y

2


)
d
x
d
y


{\displaystyle j=\iint \limits _{a}(x^{2}+y^{2})dxdy}








j
=

∬

a



x

2


d
x
d
y
+

∬

a



y

2


d
x
d
y


{\displaystyle j=\iint \limits _{a}x^{2}dxdy+\iint \limits _{a}y^{2}dxdy}

given planar second moments of area equations, where:

i

x


=

∬

a



x

2


d
x
d
y


{\displaystyle i_{x}=\iint \limits _{a}x^{2}dxdy}







i

y


=

∬

a



y

2


d
x
d
y


{\displaystyle i_{y}=\iint \limits _{a}y^{2}dxdy}

it shown polar moment of inertia can described summation of



x


{\displaystyle x}

,



y


{\displaystyle y}

planar moments of inertia,




i

x




{\displaystyle i_{x}}

,




i

y




{\displaystyle i_{y}}

∴
j
=

i

z


=

i

x


+

i

y




{\displaystyle \therefore j=i_{z}=i_{x}+i_{y}}

this shown in perpendicular axis theorem. objects have rotational symmetry, such cylinder or hollow tube, equation can simplified to:

j
=
2

i

x




{\displaystyle j=2i_{x}}

or



j
=
2

i

y




{\displaystyle j=2i_{y}}

for circular section radius r:

i

z


=

∫

0


2
π



∫

0


r



ρ

2


ρ

d
ρ

d
ϕ
=



π

r

4



2




{\displaystyle i_{z}=\int _{0}^{2\pi }\int _{0}^{r}\rho ^{2}\rho \,d\rho \,d\phi ={\frac {\pi r^{4}}{2}}}






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