# 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

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

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

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

½ MR2

R / (2)1/2

½

Solid cylinder

(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

MR2

R

1

Cylindrical shell

##### (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

2/5 MR2

(2/5)1/2 R

2/5

Solid sphere

7/5 MR2

(7/5)1/2 R

7/5

Spherical shell

2/3 MR2

(2/3)1/2 R

2/3

Spherical shell

##### Axis

5/3 MR2

(5/3)1/2 R

5/3

2/5 M [(R25 – R15) / (R23 – R13)]

##### Tangential ML2/12

L / (12)1/2

ML2/3

L / (3)1/2

(M/12)[l2 + b2]

(M/12)[4l2 + b2]

(M/12)[l2 + 4b2]

##### (iii) thickness (y)

(i) M[b2 + t2] / 12

(ii) M[l2 + t2] / 12

(iii) M[b2 + l2] / 12

##### (iii) thickness(z)

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

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

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

M/4 [a2 + b2]

3/10 MR2

Ma2/6

##### Along the edges

(i) Mb2/6

(ii) Ma2/6

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

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