7.14 List of moment of inertia $(I)$ and radius of gyration $(K)$ of different bodies

Physics > Basics of Rotational Motion > 7.0 Moment of inertia of uniform continious rigid bodies

  Basics of Rotational Motion
    1.0 Rigid body
    2.0 Motion of rigid body
    3.0 Kinematics of a plane motion
    4.0 Moment of inertia
    5.0 Radius of gyration $(K)$
    6.0 Theorems of moment of inertia
    7.0 Moment of inertia of uniform continious rigid bodies
7.14 List of moment of inertia $(I)$ and radius of gyration $(K)$ of different bodies
S. No.BodyAxis of rotationDiagramMoment of inertiaRadius of gyration
1.

Thin rod of length $L$ and mass $M$about an axis passing through its centre and perpendicular to the rod


$$\frac{{M{L^2}}}{{12}}$$$$\frac{L}{{\sqrt {12} }}$$
about an axis passing through one end and perpendicular to the rod


$$\frac{{M{L^2}}}{{3}}$$$$\frac{L}{{\sqrt {3} }}$$
2.


Rectangular lamina of length $a$, breadth $b$ and mass $M$.about an axis passing through centre and parallel to length $a$ in the plane of rectangular lamina


$$\frac{{M{b^2}}}{{12}}$$$$\frac{b}{{\sqrt {12} }}$$
about an axis passing through centre and parallel to breadth $b$ in the plane of rectangular lamina


$$\frac{{M{a^2}}}{{12}}$$$$\frac{l}{{\sqrt {12} }}$$
about an axis passing through centre and perpendicular to the plane of a rectangular lamina


$$\frac{{M\left( {{a^2} + {b^2}} \right)}}{{12}}$$$$\sqrt {\frac{{\left( {{a^2} + {b^2}} \right)}}{{12}}} $$
3.



Circular ring of radius $R$ and mass $M$about an axis passing through centre and perpendicular to the plane of a circular ring


$$M{R^2}$$$$R$$
about an axis along the diameter of the circular ring


$$\frac{{M{R^2}}}{2}$$$$\frac{R}{{\sqrt 2 }}$$
about an axis which is tanget and perpendicular to the plane of circular ring


$$2M{R^2}$$$$\sqrt 2 R$$
about an axis which is tangent in the plane of circular ring


$$\frac{{3M{R^2}}}{2}$$$$\sqrt {\frac{3}{2}} R$$
4.



Circular disc of radius $R$ and mass $M$about an axis passing through centre and perpendicular to the plane of circular disc


$$\frac{{M{R^2}}}{2}$$$$\frac{R}{{\sqrt 2 }}$$
about an axis along the diameter of a circular disc


$$\frac{{M{R^2}}}{4}$$$$\frac{R}{2}$$
about an axis which is tangent and perpendicular to the plane of circular disc


$$\frac{{3M{R^2}}}{2}$$$$\sqrt {\frac{3}{2}} R$$
about an axis which tangent in the plane of circular ring


$$\frac{{5M{R^2}}}{4}$$$$\sqrt {\frac{5}{4}} R$$
5.



Solid cylinder of length $L$, radius $R$ and mass $M$about an axis of the solid cylinder


$$\frac{{M{R^2}}}{2}$$$$\frac{R}{{\sqrt 2 }}$$
about an axis passing through its centre and perpendicular to its own axis


$$M\left( {\frac{{{R^2}}}{4} + \frac{{{L^2}}}{{12}}} \right)$$$$\sqrt {\left( {\frac{{{R^2}}}{4} + \frac{{{L^2}}}{{12}}} \right)} $$
about an axis which is tangent to the circular portion of the cylinder


$$M\left( {\frac{{{R^2}}}{4} + \frac{{{L^2}}}{{3}}} \right)$$$$\sqrt {\left( {\frac{{{R^2}}}{4} + \frac{{{L^2}}}{{3}}} \right)} $$
about an axis which is tangent to the curved portion of the cylinder


$$\frac{{3M{R^2}}}{2}$$$$\sqrt {\frac{3}{2}} R$$
6.

Cylindrical shell of length $L$, inner radius $R_1$, outer radius $R_2$ and mass $M$about an axis of the cylindrical shell


$$\frac{{M\left( {R_2^2 + R_1^2} \right)}}{2}$$$$\sqrt {\frac{{\left( {R_2^2 + R_1^2} \right)}}{2}} $$
about an axis passing through its centre and perpendicular to its own axis


$$M\left\{ {\left( {\frac{{R_2^2 + R_1^2}}{4}} \right) + \frac{{{L^2}}}{{12}}} \right\}$$$$\sqrt {\left( {\frac{{R_2^2 + R_1^2}}{4}} \right) + \frac{{{L^2}}}{{12}}} $$
7.

Hollow cylinder of length $L$, radius $R$ and mass $M$about an axis of the hollow cylinder


$$M{R^2}$$$$R$$
about an axis passing through its centre and perpendicular to its own axis


$$M\left( {\frac{{{R^2}}}{2} + \frac{{{L^2}}}{{12}}} \right)$$$$\sqrt {\left( {\frac{{{R^2}}}{2} + \frac{{{L^2}}}{{12}}} \right)} $$
8.

Solid sphere of radius $R$ and mass $M$about an axis along the diameter of a sphere


$$\frac{{2M{R^2}}}{5}$$$$\sqrt {\frac{2}{5}} R$$
about an axis which is tangent to the sphere


$$\frac{{7M{R^2}}}{5}$$$$\sqrt {\frac{7}{5}} R$$
9.

Hollow sphere of radius $R$ and mass $M$about an axis along the diameter of a hollow sphere


$$\frac{{2M{R^2}}}{3}$$$$\sqrt {\frac{2}{3}} R$$
about an axis which is tangent to the hollow sphere


$$\frac{{5M{R^2}}}{3}$$$$\sqrt {\frac{5}{3}} R$$
10.Spherical shell of inner radius $R_1$, outer radius $R_2$ and mass $M$about an axis along the diameter of the hollow sphere


$$\frac{{2M}}{5}\left( {\frac{{R_2^4 + R_2^3{R_1} + R_2^2R_1^2 + {R_2}R_1^3 + R_1^4}}{{R_2^2 + {R_2}{R_1} + R_1^2}}} \right)$$$$\sqrt {\frac{2}{5}\left( {\frac{{R_2^4 + R_2^3{R_1} + R_2^2R_1^2 + {R_2}R_1^3 + R_1^4}}{{R_2^2 + {R_2}{R_1} + R_1^2}}} \right)} $$
11.Solid cone of radius $R$ and height $H$about an axis through its centre and joining its vertex to centre of base


$$\frac{{3M{R^2}}}{10}$$$$\sqrt {\frac{3}{10}} R$$
12.Hollow cone of radius $R$, slant height $l$ and mass $M$about an axis through its centre and joining its vertex to centre of base


$$\frac{{MH{R^2}}}{{2L}}$$$$\sqrt {\frac{{H{R^2}}}{{2L}}} $$
13.


Solid hemisphere of radius $R$ and mass $M$


about an axis passing through centre and perpendicular to the base of the hemisphere


$$\frac{{2M{R^2}}}{5}$$$$\sqrt {\frac{2}{5}} R$$
about an axis along the diameter of the solid hemisphere


$$\frac{{2M{R^2}}}{5}$$$$\sqrt {\frac{2}{5}} R$$
about an axis which is tangent to the solid hemisphere


$$\frac{{7M{R^2}}}{5}$$$$\sqrt {\frac{7}{5}} R$$
14.


Hollow hemisphere of radius $R$ and mass $M$


about an axis passing through centre and perpendicular to base of the hemisphere


$$\frac{{2M{R^2}}}{3}$$$$\sqrt {\frac{2}{3}} R$$
about an axis along the diameter of the hollow hemisphere


$$\frac{{2M{R^2}}}{3}$$$$\sqrt {\frac{2}{3}} R$$
about an axis which is tangent to the hollow hemisphere


$$\frac{{5M{R^2}}}{3}$$$$\sqrt {\frac{5}{3}} R$$
15.Parallelopiped of length $l$, breadth $b$, height $h$ and mass $M$about an axis passing through centre and perpendicular to the plane of the rectangular piped


$$\frac{{M\left( {{l^2} + {b^2}} \right)}}{{12}}$$$$\sqrt {\frac{{\left( {{l^2} + {b^2}} \right)}}{{12}}} $$



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