Moment of Inertia

Moment of Inertia (Mass Moment of Inertia) depends on the mass of the object, its shape and its relative point of rotation

Moment of Inertia (Mass Moment of Inertia) - I -  is a measure of an object's resistance to changes in rotation direction. Moment of Inertia has the same relationship to angular acceleration as mass has to linear acceleration.

For a point mass the Moment of Inertia is the mass times the square of perpendicular distance to the rotation reference axis and can be expressed as

I = m r²         (1)

where

I = moment of inertia (lb_m ft², kg m²)

m = mass (lb_m, kg)

r = distance between axis and rotation mass (ft, m)

That point mass relationship are basis for all other moments of inertia since any object can be built up from a collection of point masses.

I = ∑_i m_i r_i² = m₁ r₁² + m₂ r₂² + ..... + m_nr_n²        (2)

For rigid bodies with continuous distribution of adjacent particles, the formula is better expressed as an integral

I = ∫ r² dm         (2b)

where

dm = mass of an infinitesimally small part of the body

Convert between Moment of Inertia Units

Moment of Inertia - General Formula

The Inertia formula may be generally expressed as

I = k m r²         (5)

where

k = inertial constant - depending on the shape of the body

Some Typical Bodies and their Moments of Inertia

Inertia of Cylinder

Thin-walled hollow cylinder:

Moments of Inertia for a thin-walled hollow cylinder is comparable with the point mass (1) and can be expressed as:

I = m r²         (3a)

where

m = mass of the hollow (lb_m, kg)

r = distance between axis and the thin walled hollow (ft, m)

r_o = distance between axis and outside hollow (ft, m)

Hollow cylinder:

I = 1/2 m ( r_i² + r_o²)         (3b)

where

m = mass of hollow (lb_m, kg)

r_i = distance between axis and inside hollow (ft, m)

r_o = distance between axis and outside hollow (ft, m)

Solid cylinder:

I = 1/2 m r² (3c)

where

m = mass of cylinder (lb_m, kg)

r = distance between axis and outside cylinder (ft, m)

Inertia of Sphere

Thin-walled hollow sphere:

I = 2/3 m r²         (4a)

where

m = mass of sphere hollow (lb_m, kg)

r = distance between axis and hollow (ft, m)

Solid sphere:

I = 2/5 m r²         (4b)

where

m = mass of sphere (lb_m, kg)

r = radius in sphere (ft, m)

Rectangular Plane

Moments of Inertia for a rectangular plane with axis through center can be expressed as

I = 1/12 m (a² + b²)        (5)

where

a, b = short and long sides

Moments of Inertia for a rectangular plane with axis along edge can be expressed as

I = 1/3 m a²      (5b)

Slender Rod

Moments of Inertia for a slender rod with axis through center can be expressed as

I = 1/12 m L²        (6)

where

L = length of rod

Moments of Inertia for a slender rod with axis through end can be expressed as

I = 1/3 m L²      (6b)

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