Moment of inertia of a system about axis of rotation is given as
$I=\sum m_i r_i^2$
where $m_i$ is the mass of the ith particle and $r_i$ is its perpendicular distance from the axis of rotation
For a system consisting of collection of discrete particles ,above equation can be used directly for calculating the moment of inertia
For continuous bodies ,moment of inertia about a given line can be obtained using integration technique
SI unit of Moment of inertia is $Kg m^2$
It is a scalar quantity
Some Important formulas
1. Moment of Inertia calculator for a mass m at distance d from axis of Rotation is given by
$I = md^2$
Where
m -> mass
d -> distance d from axis of Rotation
I -> Moment of inertia
2. Moment of Inertia for a thin rectangular rod around perpendicular bisector and perpendicular axis through one end is given by
$I_p = \frac {mL^2}{12}$
$I_e = \frac {mL^2}{3}$
Where
m -> Mass of the thin rectangular rod
L -> Length of the Rod
$I_p$ -> Moment of Inertia around perpendicular bisector
$I_e$ -> Moment of Inertia around perpendicular axis through one end
3. Moment of Inertia for a solid and hollow sphere about the axis through its center is given by
$I_S = \frac {2}{5}mR^2$
$I_H = \frac {2}{3}mR^2$
Where
m -> Mass of the Sphere
R -> Radius of the sphere
$I_S$ -> Moment of Inertia of Solid sphere about the axis through its center
$I_H$ -> Moment of Inertia of Hollow sphere about the axis through its center
4.Moment of Inertia calculator for a thin or solid cylinder/disk is given
Moment of Inertia for Solid Disk or cylider about the central axis
$I_S = \frac {1}{2}mR^2$
Moment of Inertia for thin Disk or cylider about the central axis
$I_H = mR^2$
Moment of Inertia calculator for a mass m at distance d from axis of Rotation
Note
Enter the values of mass and distance
Click on the calculate button.
The formula used for solving the question is
$I = md^2$
Moment of Inertia calculator
Example of Few questions where you can use this Moment of Inertia Solver Question 1
A particle of mass 5 kg is placed at a distance 5 m from the axis of rotation. Find the moment of inertia of particle about the axis of rotation Solution
m = 5 kg, d=5 m, I=?
From the formula
$I = md^2$
$I=5 \times 5^2 = 125 \ kg m^2$
Question 2
A object of mass 2 kg is situated at a distance 10 m from the axis of rotation. Find the moment of inertia of about about the axis of rotation Solution
m = 2 kg, d=10 m, I=?
From the formula
$I = md^2$
$I=2 \times 10^2 = 200 \ kg m^2$
Moment of Inertia for a thin rectangular rod around perpendicular bisector and perpendicular axis through one end
Note
Enter the values of mass and length of the rod
Click on the calculate button.
The formula used for solving the question is
$I_p = \frac {mL^2}{12}$
$I_e = \frac {mL^2}{3}$
Moment of Inertia calculator
Example of Few questions where you can use this Moment of Inertia of thin rectangular rod calculator Question 1
Calculate the Moment of inertia of thin rectangular rod of Mass 2 kg and Length 2 m around the perpendicular bisector and perpendicular axis through one end Solution
m = 2 kg, L=2 m, $I_e$=?,$I_p$=?
Moment of inertia around perpendicular bisector is given by
$I_p = \frac {mL^2}{12}$
$I_p = \frac {2 \times 2^2}{12}= 2/3 \ Kg m^2$br>
Moment of inertia about perpendicular axis through one end is given by
$I_e = \frac {mL^2}{3}$
$I_e = \frac {2 \times 2^2}{3} = 8/3 \ Kg m^2$br>
Question 2
A thin rectangular rod has Mass 1 kg and Length 1 m . Find the moment of inertia around the perpendicular bisector and perpendicular axis through one end Solution
m = 1 kg, L=1 m, $I_e$=?,$I_p$=?
Moment of inertia around perpendicular bisector is given by
$I_p = \frac {mL^2}{12}$
$I_p = \frac {1 \times 1^2}{12}= 1/12 \ Kg m^2$br>
Moment of inertia about perpendicular axis through one end is given by
$I_e = \frac {mL^2}{3}$
$I_e = \frac {1 \times 1^2}{3} = 1/3 \ Kg m^2$br>
Moment of Inertia for a solid and hollow sphere
Note
Enter the values of mass and radius of the sphere
Click on the calculate button.
The formula used for solving the question is
$I_S = \frac {2}{5}mR^2$
$I_H = \frac {2}{3}mR^2$
Moment of Inertia calculator
Example of Few questions where you can use this Moment of Inertia of Sphere calculator Question 1
Calculate the Moment of inertia of hollow sphere of Mass 2 kg and Diameter 6 m about the axis through its center Solution
m = 2 kg, D=6 m, $I$=?
Therefore R=3 m
Moment of inertia about the axis through its center is given by
$I_H = \frac {2}{3}mR^2$
$I_H = \frac {2}{3} \times 2 \times 3^2= 12 \ kg-m^2$
Question 2
A Solid Sphere has Mass 1 kg and Radius 5 m . Find the moment of inertia about the axis through its center Solution
m = 1 kg,R=5 m, I=?
Moment of inertia about the axis through its center is given by
$I_S = \frac {2}{5}mR^2$
$I_S = \frac {2}{5} \times 1 \times 5^2= 10 \ kg-m^2$
Moment of Inertia calculator for a thin or solid cylinder/disk
Note
Enter the values of mass and radius of the sphere
Click on the calculate button.
The formula used for solving the question is
Moment of Inertia for Solid Disk or cylinder about the central axis
$I_S = \frac {1}{2}mR^2$
Moment of Inertia for thin Disk or cylinder about the central axis
$I_H = mR^2$
Moment of Inertia calculator
Example of Few questions where you can use this Moment of Inertia of thin/solid cylinder calculator Question 1
Calculate the Moment of inertia of thin cylinder of Mass 10 kg , Diameter .6 m and Length =1 m about the central axis Solution
m = 10 kg, D=.6 m, L=1 m,$I$=?
Therefore R=.3 m
Moment of Inertia for thin Disk or cylinder about the central axis
$I_H = mR^2$
$I_H = 10 \times .3^2= .9 \ kg-m^2$
Question 2
A Solid disk has Mass 1 kg and Radius .2 m . Find the moment of inertia about the central axis Solution
m = 1 kg,R=.2 m, I=?
Moment of Inertia for Solid Disk or cylinder about the central axis
$I_S = \frac {1}{2}mR^2$
$I_S = \frac {1}{2} \times 1 \times (.2)^2= .02 \ kg-m^2$
