Moment of Inertia

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

Moment of Inertia

Moment of inertia ( $$I$$ ) (also called rotational inertia) measures the resists or change an object has to rotational acceleration about an axis.  The larger the moment of inretia the more difficult to try to get an object moving and the smaller the moment of inretia the easier of relatively easier to get an object moving.

Inertia is the tendency of an object in motion to remain in motion or an object at rest to remain at rest unless acted upon by a force.  The larger the mass of an object the more resistance to change in motion than objects of a lesser mass.  It is the tendency of objects to keep moving in a straight line at a constant velocity and direction forever unless acted upon by gravity or another force.

Moment of Inertia of a Circle

Circle, Hollow Plane formula

$$\large{ I_z = m r^2 }$$

Circle, Solid Plane formula

$$\large{ I_z = \frac {1}{2} m r^2 }$$

$$\large{ I_z = \frac {1}{2} \pi r^4 }$$

$$\large{ I_x = I_y = \frac {1}{4} m r^4 }$$

$$\large{ I_x = I_y = \frac {1}{4} \pi r^4 }$$

$$\large{ I_x = I_y = \frac {1}{64} d^4 }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ d }$$ = diameter

$$\large{ m }$$ = mass

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

Moment of Inertia of a Semicircle

Semicircle Solid Plane formula

$$\large{ I_x = I_y = \frac {\pi r^4}{8} }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

Moment of Inertia of a Quarter circle

Quarter circle, Solid Plane formula

$$\large{ I_x = I_y = \frac {\pi r^4}{16} }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

Moment of Inertia of an Annulus

Annulus are two circles that have the same center.

Annulus, Solid Plane formula

$$\large{ I_z = \frac {\pi}{2} \left( r_2{^4} - r_1{^4} \right) }$$

$$\large{ I_x = I_y = \frac {\pi}{4} \left( r_2{^4} - r_1{^4} \right) }$$

$$\large{ I_x = I_y = \frac {\pi}{64} D^4 - \frac {\pi}{64} d^4 }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ d }$$ = diameter

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

Moment of Inertia of a Mass

Single Mass formula

$$\large{ I = m r^2 }$$

Multiple Masses formula

$$\large{ I = m_1 r_1{^2} + m_2 r_2{^2} + m_3 r_3{^2} }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ m }$$ = mass

$$\large{ r }$$ = radius

Moment of Inertia of an Ellipse

Ellipse, Solid Plane formula

$$\large{ I_x = \frac {\pi}{4} lw^3 }$$

$$\large{ I_y = \frac {\pi}{4} l^3w }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

Moment of Inertia of a Cylinder

Hollow Cylinder formula

$$\large{ I_z = m r^2 }$$

Solid Cylinder formula

$$\large{ I_z = \frac {1}{2} m r^2 }$$

$$\large{ I_x = I_y = \frac {1}{12} m \left( 3r{^2} + l^2 \right) }$$

Hollow Core Cylinder formula

$$\large{ I_z = \frac {1}{12} m \left( r_1{^2} + r_2{^2} \right) }$$

$$\large{ I_x = I_y = \frac {1}{12} m \left( 3 \left( r_2{^2} + r_1{^2} \right) + l^2 \right) }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ m }$$ = mass

$$\large{ r }$$ = radius

Moment of Inertia of a Sphere

Hollow Sphere formula

$$\large{ I = \frac {2}{3} m r^2 }$$

Solid Sphere formula

$$\large{ I = \frac {2}{5} m r^2 }$$

Hollow Core Sphere formula

$$\large{ I = \frac {2}{5} m \left( \frac { r_2^5 - r_1{^5} } { r_2{^3} - r_1{^3} } \right) }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ m }$$ = mass

$$\large{ r }$$ = radius

Moment of Inertia of a rectangle

rectangle, Solid Plane, z Axis formula

$$\large{ I_z = \frac {1}{12} m \left( l^2 + w^2 \right) }$$

$$\large{ I_z = \frac {1}{12} lw \left( l^2 + w^2 \right) }$$

$$\large{ I_{z1} = \frac {1}{12} m \left( 4l^2 + w^2 \right) }$$

rectangle, Solid Plane, x Axis formula

$$\large{ I_x = \frac {1}{12} lw^3 }$$

$$\large{ I_x = \frac {1}{12} m l^2 }$$

$$\large{ I_{x1} = \frac {1}{3} lw^3 }$$

$$\large{ I_{x1} = \frac {1}{3} m w^2 }$$

rectangle, Solid Plane, y Axis formula

$$\large{ I_y = \frac {1}{12} l^3w }$$

$$\large{ I_{y1} = \frac {1}{3} l^3w }$$

rectangle, Hollow Core Plane, x Axis formula

$$\large{ I_x = \frac {lw^3}{12} - \frac {l_1w_1{^3} }{12} }$$

rectangle, Hollow Core Plane, Y Axis formula

$$\large{ I_y = \frac {l^3w}{12} - \frac {l_1{^3} w_1}{12} }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ l }$$ = length

$$\large{ m }$$ = mass

$$\large{ w }$$ = width

Moment of Inertia of a Triangle

Triangle, Solid Plane formula

$$\large{ I_x = \frac {l w^3}{12} }$$

$$\large{ I_y = \frac {l^3 w}{12} }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ l }$$ = length

$$\large{ w }$$ = width

Moment of Inertia of a Cube

Cube, Solid, Center axis formula

$$\large{ I_h = \frac {1}{12} m \left( l^2 + w^2 \right) }$$

$$\large{ I_l = \frac {1}{12} m \left( h^2 + w^2 \right) }$$

$$\large{ I_w = \frac {1}{12} m \left( l^2 + h^2 \right) }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ h }$$ = height

$$\large{ l }$$ = length

$$\large{ m }$$ = mass

$$\large{ w }$$ = width

Moment of Inertia of a Thin Rod

Thin Rod, End Axis formula

$$\large{ I = \frac {1}{3} m l^2 }$$

Thin Rod, Center Axis formula

$$\large{ I = \frac {1}{12} m l^2 }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ l }$$ = length

$$\large{ m }$$ = mass