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]