Moment of Inertia

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

monent of inertia bannerMoment 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 Circlemoment of inertia Flat Solid Disk 1

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 Semicirclemoment of inertia Semicircle 1

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 circlemoment of inertia Quarter Circle 1

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 Annulusmoment of inertia Annulus 1

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 Massmoment of inertia Mass 2moment of inertia Mass 1

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 Ellipsemoment of inertia Ellipse 1

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 Cylindermoment of inertia Cylinder 2moment of inertia Cylinder 1

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 Spheremoment of inertia Sphere 2moment of inertia Sphere 1

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 rectanglemoment of inertia Rectangle 1moment of inertia Rec Plane 9

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 Trianglemoment of inertia Triangle 1

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 Cubemoment of inertia Cube 2

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 Rodmoment of inertia Rod 1

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

 

Tags: Equations for Inertia