Warning (2): file_get_contents(): https:// wrapper is disabled in the server configuration by allow_url_fopen=0 [ROOT/vendors/mathml_to_image.php, line 6]
Warning (2): file_get_contents(https://www.wiris.net/demo/editor/render.png?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3EM%3C%2Fmi%3E%3Cmo%3E%5B%3C%2Fmo%3E%3Cmsubsup%3E%3Cmi%3ER%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmsubsup%3E%3Cmo%3E%26minus%3B%3C%2Fmo%3E%3Cmsubsup%3E%3Cmi%3ER%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmsubsup%3E%3Cmo%3E%5D%3C%2Fmo%3E%3C%2Fmrow%3E%3Cmrow%3E%3Cmn%3E5%3C%2Fmn%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmsubsup%3E%3Cmi%3ER%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmsubsup%3E%3Cmo%3E%26minus%3B%3C%2Fmo%3E%3Cmsubsup%3E%3Cmi%3ER%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmsubsup%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmi%3EM%3C%2Fmi%3E%3Cmsubsup%3E%3Cmi%3ER%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsubsup%3E%3C%2Fmath%3E) [function.file-get-contents]: failed to open stream: no suitable wrapper could be found [ROOT/vendors/mathml_to_image.php, line 6]
MI of Some Standard Bodies|Sureden:Your Education Partner

MI of Some Standard Bodies

Moment of Inertia of Some Standard Bodies About Different Axes

Body

Axis of Rotation

Figure

Moment of Inertia

k

k2/R2

Ring

About an axis passing through C.G. and perpendicular to its plane

MR2

R

1

Ring

About its diameter

½ MR2

R / (2)1/2

½

Ring

About a tangential axis in its own plane

3/2  MR2

(3/2)1/2 R

3/2

Ring

About a tangential axis perpendicular to its own plane

2MR2

(2)1/2 R

2

Disc

About an axis passing through C.G. and perpendicular to its plane

½ MR2

R / (2)1/2

½

Disc

About its Diameter

¼ MR2

R/2

¼

 

Disc

About a tangential axis in its own plane

5/4 MR2

((5)1/2 / 2) R

5/4

Disc

About a tangential axis perpendicular to its own plane

3/2 MR2

(3/2)1/2 R

3/2

Annular disc inner radius = R1 and outer radius = R2

Passing through the centre and perpendicular to the plane

(M/2) [R12 + R22]

-

-

Annular disc

Diameter

(M/4) [R12 + R22]

-

-

Annular disc

Tangential and Parallel to the diameter

(M/4) [5 R12 + R22]

-

-

Annular disc

Tangential and perpendicular to the plane

(M/2) [3 R12 + R22]

-

-

Solid cylinder

About its own axis

½ MR2

R / (2)1/2

½

Solid cylinder

Tangential

(Generator)

3/2 MR2

(3/2)1/2 R2

3/2

Solid cylinder

About an axis passing through its C.G. and perpendicular to its own axis

M[(L2/12) + (R2/4)]

((L2/12) + (R2/4))1/2

 

Solid cylinder

About the diameter of one of faces of the cylinder

M[(L2/3) + (R2/4)]

((L2/3) + (R2/4))1/2

 

Cylindrical shell

About its own axis

MR2

R

1

Cylindrical shell

Tangential
(Generator)

2MR2

(2)1/2 R

2

Cylindrical shell

About an axis passing through its C.G. and perpendicular to its own axis

M[(L2/12) + (R2/2)]

((L2/12) + (R2/2))1/2

 

Cylindrical shell

About the diameter of one of faces of the cylinder

M[(L2/3) + (R2/2)]

((L2/3) + (R2/2))1/2

 

Hollow cylinder with inner radius = R1 and outer radius = R2

Axis of cylinder

(M/2) (R12 + R22)

 

 

Solid Sphere

About its diametric axis
 

2/5 MR2

(2/5)1/2 R

2/5

Solid sphere

About a tangential axis

7/5 MR2

(7/5)1/2 R

7/5

Spherical shell

About its diametric axis

2/3 MR2

(2/3)1/2 R

2/3

Spherical shell

About a tangential
Axis

5/3 MR2

(5/3)1/2 R

5/3

Hollow sphere of inner radius R1 and outer radius R2

 

About its diametric axis

2/5 M [(R25 – R15) / (R23 – R13)]

 

 

Hollow sphere
Tangential

 

 

Long thin rod
About on axis passing through its centre of mass and perpendicular to the rod.

ML2/12

L / (12)1/2

 

Long thin rod
About an axis passing through its edge and perpendicular to the rod

ML2/3

L / (3)1/2

 

Rectangular lamina of length l and breadth b
Passing through the centre of mass and perpendicular to the plane

(M/12)[l2 + b2]

 

 

Rectangular lamina
Tangential perpendicular to the plane and at the mid-point of breadth

(M/12)[4l2 + b2]

 

 

Rectangular lamina
Tangential perpendicular to the plane and at the mid-point of length

(M/12)[l2 + 4b2]

 

 

Rectangular parallelopiped length l, breadth b, thickness t
Passing through centre of mass and parallel to
(i) Length (x)
(ii) breadth (z)
(iii) thickness (y)

(i) M[b2 + t2] / 12

(ii) M[l2 + t2] / 12

(iii) M[b2 + l2] / 12

 

 

Rectangular parallelepiped length l, breath b, thickness t
Tangential and parallel to
(i) length (x)
(ii) breadth (y)
(iii) thickness(z)

(i) (M/12)[3l2 + b2 + t2]

(ii) (M/12) [l2 + 3b2 + t2]

(iii) (M/12) [l2 + b2 + 3t2]

 

 

Elliptical  disc of semimajor axis = a and semiminor axis = b
Passing through CM and perpendicular to the plane

M/4 [a2 + b2]

 

 

Solid cone of radius R and height h
Axis joining the vertex and centre of the base

3/10 MR2

 

 

Equilateral triangular lamina with side a
Passing through CM and perpendicular to the plane

Ma2/6

 

 

Right angled triangular lamina of sides a, b, c
Along the edges

(i) Mb2/6

(ii) Ma2/6

(iii) M/6 [ a2b2 / a2 + b2]

 

 

 

Related Keywords