<P> This article mainly considers symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified . </P> <P> Following are scalar moments of inertia . In general, the moment of inertia is a tensor, see below . </P> <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 m M + m x 2 = μ x 2 (\ displaystyle I = (\ frac (Mm) (M \! + \! m)) 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 . </Td> <Td> </Td> <Td> I ≈ m r 2 (\ displaystyle I \ approx mr ^ (2) \, \!) </Td> </Tr> <Tr> <Td> Solid cylinder of radius r, height h and mass m . <P> This is a special case of the thick - walled cylindrical tube, with r = 0 . </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 12 m (3 r 2 + h 2) (\ displaystyle I_ (x) = I_ (y) = (\ frac (1) (12)) m \ left (3r ^ (2) + h ^ (2) \ right)) </Td> </Tr> <Tr> <Td> Thick - walled cylindrical tube with open ends, of inner radius r, outer radius r, length h and mass m . </Td> <Td> </Td> <Td> <P> I z = 1 2 m (r 2 2 + r 1 2) = m r 2 2 (1 − t + t 2 2) (\ displaystyle I_ (z) = (\ frac (1) (2)) m \ left (r_ (2) ^ (2) + r_ (1) ^ (2) \ right) = mr_ (2) ^ (2) \ left (1 - t+ (\ frac (t ^ (2)) (2)) \ right)) where t = (r--r) / r is a normalized thickness ratio; I x = I y = 1 12 m (3 (r 2 2 + r 1 2) + h 2) (\ displaystyle I_ (x) = I_ (y) = (\ frac (1) (12)) m \ left (3 \ left ((r_ (2)) ^ (2) + (r_ (1)) ^ (2) \ right) + h ^ (2) \ right)) </P> </Td> </Tr> <Tr> <Td> With a density of ρ and the same geometry </Td> <Td> I z = π ρ h 2 (r 2 4 − r 1 4) (\ displaystyle I_ (z) = (\ frac (\ pi \ rho h) (2)) \ left ((r_ (2)) ^ (4) - (r_ (1)) ^ (4) \ right)) <P> I x = I y = π ρ h 12 (3 (r 2 4 − r 1 4) + h 2 (r 2 2 − r 1 2)) (\ displaystyle I_ (x) = I_ (y) = (\ frac (\ pi \ rho h) (12)) \ left (3 ((r_ (2)) ^ (4) - (r_ (1)) ^ (4)) + h ^ (2) ((r_ (2)) ^ (2) - (r_ (1)) ^ (2)) \ right)) </P> </Td> </Tr> <Tr> <Td> Regular tetrahedron of side s and mass m </Td> <Td> </Td> <Td> I s o l i d = 1 20 m s 2 (\ displaystyle I_ (\ mathrm (solid)) = (\ frac (1) (20)) ms ^ (2) \, \!) <P> I h o l l o w = 1 12 m s 2 (\ displaystyle I_ (\ mathrm (hollow)) = (\ frac (1) (12)) ms ^ (2) \, \!) </P> </Td> </Tr> <Tr> <Td> Regular octahedron of side s and mass m </Td> <Td> </Td> <Td> I x, h o l l o w = I y, h o l l o w = I z, h o l l o w = 1 6 m s 2 (\ displaystyle I_ (x, \ mathrm (hollow)) = I_ (y, \ mathrm (hollow)) = I_ (z, \ mathrm (hollow)) = (\ frac (1) (6)) ms ^ (2) \, \!) I x, s o l i d = I y, s o l i d = I z, s o l i d = 1 10 m s 2 (\ displaystyle I_ (x, solid) = I_ (y, solid) = I_ (z, solid) = (\ frac (1) (10)) ms ^ (2) \, \!) </Td> </Tr> <Tr> <Td> Regular dodecahedron of side s and mass m </Td> <Td> </Td> <Td> I x, h o l l o w = I y, h o l l o w = I z, h o l l o w = 39 φ + 28 90 m s 2 (\ displaystyle I_ (x, \ mathrm (hollow)) = I_ (y, \ mathrm (hollow)) = I_ (z, \ mathrm (hollow)) = (\ frac (39 \ phi + 28) (90)) ms ^ (2)) <P> I x, s o l i d = I y, s o l i d = I z, s o l i d = 39 φ + 28 150 m s 2 (\ displaystyle I_ (x, \ mathrm (solid)) = I_ (y, \ mathrm (solid)) = I_ (z, \ mathrm (solid)) = (\ frac (39 \ phi + 28) (150)) ms ^ (2) \, \!) (where φ = 1 + 5 2 (\ displaystyle \ phi = (\ frac (1 + (\ sqrt (5))) (2)))) </P> </Td> </Tr> <Tr> <Td> Regular icosahedron of side s and mass m </Td> <Td> </Td> <Td> I x, h o l l o w = I y, h o l l o w = I z, h o l l o w = φ 2 6 m s 2 (\ displaystyle I_ (x, \ mathrm (hollow)) = I_ (y, \ mathrm (hollow)) = I_ (z, \ mathrm (hollow)) = (\ frac (\ phi ^ (2)) (6)) ms ^ (2)) <P> I x, s o l i d = I y, s o l i d = I z, s o l i d = φ 2 10 m s 2 (\ displaystyle I_ (x, \ mathrm (solid)) = I_ (y, \ mathrm (solid)) = I_ (z, \ mathrm (solid)) = (\ frac (\ phi ^ (2)) (10)) ms ^ (2) \, \!) </P> </Td> </Tr> <Tr> <Td> Hollow sphere of radius r and mass m . <P> A hollow sphere can be taken to be made up of two stacks of infinitesimally thin, circular hoops, where the radius differs from 0 to r (or a single stack, where the radius differs from - r to r). </P> </Td> <Td> </Td> <Td> I = 2 3 m r 2 (\ displaystyle I = (\ frac (2) (3)) mr ^ (2) \, \!) </Td> </Tr> <Tr> <Td> Solid sphere (ball) of radius r and mass m . <P> A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 to r (or a single stack, where the radius differs from - r to r). </P> </Td> <Td> </Td> <Td> I = 2 5 m r 2 (\ displaystyle I = (\ frac (2) (5)) mr ^ (2) \, \!) </Td> </Tr> <Tr> <Td> Sphere (shell) of radius r and mass m, with centered spherical cavity of radius r . <P> When the cavity radius r = 0, the object is a solid ball (above). </P> <P> When r = r, (r 2 5 − r 1 5 r 2 3 − r 1 3) = 5 3 r 2 2 (\ displaystyle \ left ((\ frac ((r_ (2)) ^ (5) - (r_ (1)) ^ (5)) ((r_ (2)) ^ (3) - (r_ (1)) ^ (3))) \ right) = (\ frac (5) (3)) (r_ (2)) ^ (2)), and the object is a hollow sphere . </P> </Td> <Td> </Td> <Td> I = 2 5 m (r 2 5 − r 1 5 r 2 3 − r 1 3) (\ displaystyle I = (\ frac (2) (5)) m \ left ((\ frac ((r_ (2)) ^ (5) - (r_ (1)) ^ (5)) ((r_ (2)) ^ (3) - (r_ (1)) ^ (3))) \ right) \, \!) </Td> </Tr> <Tr> <Td> Right circular cone with radius r, height h and mass m </Td> <Td> </Td> <Td> I z = 3 10 m r 2 (\ displaystyle I_ (z) = (\ frac (3) (10)) mr ^ (2) \, \!) I x = I y = 3 20 m (r 2 + 4 h 2) (\ displaystyle I_ (x) = I_ (y) = (\ frac (3) (20)) m \ left (r ^ (2) + 4h ^ (2) \ right) \, \!) </Td> </Tr> <Tr> <Td> Right circular hollow cone with radius r, height h and mass m </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 + 2 h 2) (\ displaystyle I_ (x) = I_ (y) = (\ frac (1) (4)) m \ left (r ^ (2) + 2h ^ (2) \ right) \, \!) </Td> </Tr> <Tr> <Td> Torus with minor radius a, major radius b and mass m . </Td> <Td> </Td> <Td> About an axis passing through the center and perpendicular to the diameter: 1 4 m (4 b 2 + 3 a 2) (\ displaystyle (\ frac (1) (4)) m \ left (4b ^ (2) + 3a ^ (2) \ right)) About a diameter: 1 8 m (5 a 2 + 4 b 2) (\ displaystyle (\ frac (1) (8)) m \ left (5a ^ (2) + 4b ^ (2) \ right)) </Td> </Tr> <Tr> <Td> Ellipsoid (solid) of semiaxes a, b, and c with mass m </Td> <Td> </Td> <Td> I a = 1 5 m (b 2 + c 2) (\ displaystyle I_ (a) = (\ frac (1) (5)) m \ left (b ^ (2) + c ^ (2) \ right) \, \!) I b = 1 5 m (a 2 + c 2) (\ displaystyle I_ (b) = (\ frac (1) (5)) m \ left (a ^ (2) + c ^ (2) \ right) \, \!) I c = 1 5 m (a 2 + b 2) (\ displaystyle I_ (c) = (\ frac (1) (5)) m \ left (a ^ (2) + b ^ (2) \ right) \, \!) </Td> </Tr> <Tr> <Td> Thin rectangular plate of height h, width w and mass m (Axis of rotation at the end of the plate) </Td> <Td> </Td> <Td> I e = 1 12 m (4 h 2 + w 2) (\ displaystyle I_ (e) = (\ frac (1) (12)) m \ left (4h ^ (2) + w ^ (2) \ right) \, \!) </Td> </Tr> <Tr> <Td> Thin rectangular plate of height h, width w and mass m (Axis of rotation at the center) </Td> <Td> </Td> <Td> I c = 1 12 m (h 2 + w 2) (\ displaystyle I_ (c) = (\ frac (1) (12)) m \ left (h ^ (2) + w ^ (2) \ right) \, \!) </Td> </Tr> <Tr> <Td> Solid cuboid of height h, width w, and depth d, and mass m . <P> For a similarly oriented cube with sides of length s (\ displaystyle s), I C M = 1 6 m s 2 (\ displaystyle I_ (CM) = (\ frac (1) (6)) ms ^ (2) \, \!) </P> </Td> <Td> </Td> <Td> I h = 1 12 m (w 2 + d 2) (\ displaystyle I_ (h) = (\ frac (1) (12)) m \ left (w ^ (2) + d ^ (2) \ right)) I w = 1 12 m (d 2 + h 2) (\ displaystyle I_ (w) = (\ frac (1) (12)) m \ left (d ^ (2) + h ^ (2) \ right)) I d = 1 12 m (w 2 + h 2) (\ displaystyle I_ (d) = (\ frac (1) (12)) m \ left (w ^ (2) + h ^ (2) \ right)) </Td> </Tr> <Tr> <Td> Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal . <P> For a cube with sides s (\ displaystyle s), I = 1 6 m s 2 (\ displaystyle I = (\ frac (1) (6)) ms ^ (2) \, \!). </P> </Td> <Td> </Td> <Td> I = 1 6 m (W 2 D 2 + D 2 L 2 + W 2 L 2 W 2 + D 2 + L 2) (\ displaystyle I = (\ frac (1) (6)) m \ left ((\ frac (W ^ (2) D ^ (2) + D ^ (2) L ^ (2) + W ^ (2) L ^ (2)) (W ^ (2) + D ^ (2) + L ^ (2))) \ right)) </Td> </Tr> <Tr> <Td> Triangle with vertices at the origin and at P and Q, with mass m, rotating about an axis perpendicular to the plane and passing through the origin . </Td> <Td> </Td> <Td> I = 1 6 m (P ⋅ P + P ⋅ Q + Q ⋅ Q) (\ displaystyle I = (\ frac (1) (6)) m (\ mathbf (P) \ cdot \ mathbf (P) + \ mathbf (P) \ cdot \ mathbf (Q) + \ mathbf (Q) \ cdot \ mathbf (Q))) </Td> </Tr> <Tr> <Td> Plane polygon with vertices P, P, P,..., P and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin . </Td> <Td> </Td> <Td> I = 1 6 m (∑ n = 1 N ‖ P n + 1 × P n ‖ ((P n ⋅ P n) + (P n ⋅ P n + 1) + (P n + 1 ⋅ P n + 1)) ∑ n = 1 N ‖ P n + 1 × P n ‖) (\ displaystyle I = (\ frac (1) (6)) m \ left ((\ frac (\ sum \ limits _ (n = 1) ^ (N) \ \ mathbf (P) _ (n + 1) \ times \ mathbf (P) _ (n) \ \ left (\ left (\ mathbf (P) _ (n) \ cdot \ mathbf (P) _ (n) \ right) + \ left (\ mathbf (P) _ (n) \ cdot \ mathbf (P) _ (n + 1) \ right) + \ left (\ mathbf (P) _ (n + 1) \ cdot \ mathbf (P) _ (n + 1) \ right) \ right)) (\ sum \ limits _ (n = 1) ^ (N) \ \ mathbf (P) _ (n + 1) \ times \ mathbf (P) _ (n) \)) \ right)) </Td> </Tr> <Tr> <Td> Plane regular polygon with n - vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through its barycenter . R is the radius of the circumscribed circle . </Td> <Td> </Td> <Td> I = 1 2 m R 2 (1 − 2 3 sin 2 ⁡ (π n)) (\ displaystyle I = (\ frac (1) (2)) mR ^ (2) \ left (1 - (\ frac (2) (3)) \ sin ^ (2) \ left ((\ tfrac (\ pi) (n)) \ right) \ right)) </Td> </Tr> <Tr> <Td> An isosceles triangle of mass M, vertex angle 2β and common - side length L (axis through tip, perpendicular to plane) </Td> <Td> </Td> <Td> I = 1 2 m L 2 (1 − 2 3 sin 2 ⁡ (β)) (\ displaystyle I = (\ frac (1) (2)) mL ^ (2) \ left (1 - (\ frac (2) (3)) \ sin ^ (2) \ left (\ beta \ right) \ right)) </Td> </Tr> <Tr> <Td> Infinite disk with mass normally distributed on two axes around the axis of rotation with mass - density as a function of x and y: <Dl> <Dd> ρ (x, y) = m 2 π a b e − ((x / a) 2 + (y / b) 2) / 2, (\ displaystyle \ rho (x, y) = (\ tfrac (m) (2 \ pi ab)) \, e ^ (- ((x / a) ^ (2) + (y / b) ^ (2)) / 2) \,,) </Dd> </Dl> </Td> <Td> </Td> <Td> I = m (a 2 + b 2) (\ displaystyle I = m (a ^ (2) + b ^ (2)) \, \!) </Td> </Tr> <Tr> <Td> Uniform disk about an axis perpendicular to its edge . </Td> <Td> </Td> <Td> I = 3 2 m r 2 (\ displaystyle I = (\ frac (3) (2)) mr ^ (2)) </Td> </Tr> </Table> <Tr> <Th> Description </Th> <Th> Figure </Th> <Th> Moment (s) of inertia </Th> </Tr>

Moment of inertia for a rod spinning about its center
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