<Table> <Tr> <Th> Description </Th> <Th> Figure </Th> <Th> Moment ( s ) of inertia </Th> </Tr> <Tr> <Td> Point mass M at a distance r from the axis of rotation . <P> A point mass does not have a moment of inertia around its own axis , but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved . </P> </Td> <Td> </Td> <Td> I = M r 2 ( \ displaystyle I = Mr ^ ( 2 ) ) </Td> </Tr> <Tr> <Td> Two point masses , m and m , with reduced mass μ and separated by a distance , x about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles . </Td> <Td> </Td> <Td> I = m 1 m 2 m 1 + m 2 x 2 = μ x 2 ( \ displaystyle I = ( \ frac ( m_ ( 1 ) m_ ( 2 ) ) ( m_ ( 1 ) \ ! + \ ! m_ ( 2 ) ) ) x ^ ( 2 ) = \ mu x ^ ( 2 ) ) </Td> </Tr> <Tr> <Td> Rod of length L and mass m , rotating about its center . <P> This expression assumes that the rod is an infinitely thin ( but rigid ) wire . This is a special case of the thin rectangular plate with axis of rotation at the center of the plate , with w = L and h = 0 . </P> </Td> <Td> </Td> <Td> I c e n t e r = 1 12 m L 2 ( \ displaystyle I_ ( \ mathrm ( center ) ) = ( \ frac ( 1 ) ( 12 ) ) mL ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Rod of length L and mass m , rotating about one end . <P> This expression assumes that the rod is an infinitely thin ( but rigid ) wire . This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate , with h = L and w = 0 . </P> </Td> <Td> </Td> <Td> I e n d = 1 3 m L 2 ( \ displaystyle I_ ( \ mathrm ( end ) ) = ( \ frac ( 1 ) ( 3 ) ) mL ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Thin circular hoop of radius r and mass m . <P> This is a special case of a torus for a = 0 ( see below ) , as well as of a thick - walled cylindrical tube with open ends , with r = r and h = 0 . </P> </Td> <Td> </Td> <Td> I z = m r 2 ( \ displaystyle I_ ( z ) = mr ^ ( 2 ) \ ! ) I x = I y = 1 2 m r 2 ( \ displaystyle I_ ( x ) = I_ ( y ) = ( \ frac ( 1 ) ( 2 ) ) mr ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Thin , solid disk of radius r and mass m . <P> This is a special case of the solid cylinder , with h = 0 . That I x = I y = I z 2 ( \ displaystyle I_ ( x ) = I_ ( y ) = ( \ frac ( I_ ( z ) ) ( 2 ) ) \ , ) is a consequence of the perpendicular axis theorem . </P> </Td> <Td> </Td> <Td> I z = 1 2 m r 2 ( \ displaystyle I_ ( z ) = ( \ frac ( 1 ) ( 2 ) ) mr ^ ( 2 ) \ , \ ! ) I x = I y = 1 4 m r 2 ( \ displaystyle I_ ( x ) = I_ ( y ) = ( \ frac ( 1 ) ( 4 ) ) mr ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Thin cylindrical shell with open ends , of radius r and mass m . <P> This expression assumes that the shell thickness is negligible . It is a special case of the thick - walled cylindrical tube for r = r . </P> Also , a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration . <P> </P> </Td>
<Table> <Tr> <Th> Description </Th> <Th> Figure </Th> <Th> Moment ( s ) of inertia </Th> </Tr> <Tr> <Td> Point mass M at a distance r from the axis of rotation . <P> A point mass does not have a moment of inertia around its own axis , but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved . </P> </Td> <Td> </Td> <Td> I = M r 2 ( \ displaystyle I = Mr ^ ( 2 ) ) </Td> </Tr> <Tr> <Td> Two point masses , m and m , with reduced mass μ and separated by a distance , x about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles . </Td> <Td> </Td> <Td> I = m 1 m 2 m 1 + m 2 x 2 = μ x 2 ( \ displaystyle I = ( \ frac ( m_ ( 1 ) m_ ( 2 ) ) ( m_ ( 1 ) \ ! + \ ! m_ ( 2 ) ) ) x ^ ( 2 ) = \ mu x ^ ( 2 ) ) </Td> </Tr> <Tr> <Td> Rod of length L and mass m , rotating about its center . <P> This expression assumes that the rod is an infinitely thin ( but rigid ) wire . This is a special case of the thin rectangular plate with axis of rotation at the center of the plate , with w = L and h = 0 . </P> </Td> <Td> </Td> <Td> I c e n t e r = 1 12 m L 2 ( \ displaystyle I_ ( \ mathrm ( center ) ) = ( \ frac ( 1 ) ( 12 ) ) mL ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Rod of length L and mass m , rotating about one end . <P> This expression assumes that the rod is an infinitely thin ( but rigid ) wire . This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate , with h = L and w = 0 . </P> </Td> <Td> </Td> <Td> I e n d = 1 3 m L 2 ( \ displaystyle I_ ( \ mathrm ( end ) ) = ( \ frac ( 1 ) ( 3 ) ) mL ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Thin circular hoop of radius r and mass m . <P> This is a special case of a torus for a = 0 ( see below ) , as well as of a thick - walled cylindrical tube with open ends , with r = r and h = 0 . </P> </Td> <Td> </Td> <Td> I z = m r 2 ( \ displaystyle I_ ( z ) = mr ^ ( 2 ) \ ! ) I x = I y = 1 2 m r 2 ( \ displaystyle I_ ( x ) = I_ ( y ) = ( \ frac ( 1 ) ( 2 ) ) mr ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Thin , solid disk of radius r and mass m . <P> This is a special case of the solid cylinder , with h = 0 . That I x = I y = I z 2 ( \ displaystyle I_ ( x ) = I_ ( y ) = ( \ frac ( I_ ( z ) ) ( 2 ) ) \ , ) is a consequence of the perpendicular axis theorem . </P> </Td> <Td> </Td> <Td> I z = 1 2 m r 2 ( \ displaystyle I_ ( z ) = ( \ frac ( 1 ) ( 2 ) ) mr ^ ( 2 ) \ , \ ! ) I x = I y = 1 4 m r 2 ( \ displaystyle I_ ( x ) = I_ ( y ) = ( \ frac ( 1 ) ( 4 ) ) mr ^ ( 2 ) \ , \ ! ) </Td> </Tr> <Tr> <Td> Thin cylindrical shell with open ends , of radius r and mass m . <P> This expression assumes that the shell thickness is negligible . It is a special case of the thick - walled cylindrical tube for r = r . </P> Also , a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration . <P> </P> </Td>
mass moment of inertia of cylinder and thin disc
mass moment of inertia of cylinder and thin disc
h = 0
h = 0