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# Mass Moment of Inertia

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

• Moment of Inertia of a body depends on the distribution of mass in the body with respect to the axis of rotation

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 ( kg m2, slug ft2, lbf fts2)

m = mass (kg, slugs)

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

### Example - Moment of Inertia of a Single Mass

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The Moment of Inertia with respect to rotation around the z-axis of a single mass of 1 kg distributed as a thin ring as indicated in the figure above, can be calculated as

Iz = (1 kg) ((1000 mm) (0.001 m/mm))2

= 1 kg m2

.

### Moment of Inertia - Distributed Masses

Point mass is the basis for all other moments of inertia since any object can be "built up" from a collection of point masses.

I = ∑i mi ri 2= m1 r12+ m2 r2 2 + ..... + mn rn 2                    (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 Units for the Moment of Inertia

 Multiply with from to kg m2 g cm2 lbm ft2 lbm in2 slug ft2 slug in2 kg m2 1 1 107 2.37 10 1 3.42 103 7.38 10-1 1.06 102 g cm2 1 10 -7 1 2.37 10-6 3.42 10-4 7.38 10-8 1.06 105 lbm ft2 4.21 10-2 4.21 105 1 1.44 102 3.11 10-2 4.48 lbm in2 2.93 10-4 2.93 103 6.94 10-3 1 2.16 10-4 3.11 10-2 slug ft2 1.36 1.36 107 3.22 10 1 4.63 103 1 1.44 102 slug in2 9.42 10-3 9.42 104 2.23 10-1 3.22 10 1 6.94 10-3 1

### Moment of Inertia - General Formula

A generic expression of the inertia equation is

I = k m r2               (2c)

where

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

### Radius of Gyration (in Mechanics)

The Radius of Gyration is the distance from the rotation axis where a concentrated point mass equals the Moment of Inertia of the actual body. The Radius of Gyration for a body can be expressed as

r g = (I / m)1/2                 (2d)

where

r g = radius of gyration (m, ft)

I = moment of inertia for the body (kg m2, slug ft2)

m = mass of the body (kg, slugs)

.

### Some Typical Bodies and their Moments of Inertia

#### 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 (kg, slugs)

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

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

##### Hollow cylinder

I = 1/2 m (ri 2+ ro 2)                                    (3b)

where

m = mass of hollow (kg, slugs)

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

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

##### Solid cylinder

I = 1/2 m r2                    (3c)

where

m = mass of cylinder (kg, slugs)

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

##### Circular Disk

I = 1/2 m r2                  (3d)

where

m = mass of disk (kg, slugs)

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

#### Sphere

##### Thin-walled hollow sphere

I = 2/3 m r2                  (4a)

where

m = mass of sphere hollow (kg, slugs)

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

##### Solid sphere

I = 2/5 m r2                      (4b)

where

m = mass of sphere (kg, slugs)

r = radius in sphere (m, ft)

#### 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)

## Related Topics

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Motion of bodies and the action of forces in producing or changing their motion - velocity and acceleration, forces and torque.

### • Mechanics

The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

### • Statics

Forces acting on bodies at rest under equilibrium conditions - loads, forces and torque, beams and columns.

## Related Documents

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### Area Moment of Inertia - Typical Cross Sections II

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### Area Moment of Inertia Converter

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