Moment of Inertia Calculator

Moment of Inertia Calculator

Select the geometrical figure and enter the required values. The tool will instantly calculate the moment of inertia, section modulus, area, and centroid.

To Select

Radius along (b):

Width (h)

Distance to height (a)

Height

Moment of Inertia Calculator

This moment of inertia calculator determines the moment of inertia of geometrical figures such as triangles and rectangles. Additionally, you can use this calculator to calculate the area, the centroid of the beam, and the section modulus.

What Is The Moment Of Inertia?

In physical sciences:

“A specific quantity that is responsible for producing the torque in a body about a rotational axis is called the moment of inertia”

moment of inertia

First Moment Of Inertia:

“It represents the spatial distribution of the given shape in relation to its relative axis”

Second Moment Of Inertia:

“This specific property displays the point distribution with respect to the axis”

Moment of Inertia Formula:

$$ I = \frac{L}{?} $$

Where:

L = Angular Momentum

? = Angular Frequency

I = Inertia

More Formulas:

These formulas are valid when the x and y axes pass through the centroid of the shape.

Triangle:
$$ I_x = \frac{(\text{width}) (\text{height})^3}{36} $$
$$ I_y = \frac{(\text{height}) (\text{width})^3 - (\text{height}) a (\text{width})^2 + (\text{width}) (\text{height}) a^2}{36} $$

Rectangle:
$$ I_x = \frac{(\text{width}) (\text{height})^3}{12} $$
$$ I_y = \frac{(\text{height}) (\text{width})^3}{12} $$

Hollow Rectangle:
$$ I_x = \frac{b h^3 - b_1 h_1^3}{12} $$
$$ I_y = \frac{b^3 h - b_1^3 h_1}{12} $$

Circle:
$$ I_x = I_y = \frac{\pi}{4} (\text{radius})^4 $$

Hollow Circle:
$$ I_x = I_y = \frac{\pi}{4} (r_2^4 - r_1^4) $$

Semicircle:
$$ I_x = \left[\frac{\pi}{8} - \frac{8}{9 \pi}\right] (\text{radius})^4 $$
$$ I_y = \frac{\pi}{8} (\text{radius})^4 $$

Ellipse:
$$ I_x = \frac{\pi}{4} (\text{radius}_x) (\text{radius}_y)^3 $$
$$ I_y = \frac{\pi}{4} (\text{radius}_y) (\text{radius}_x)^3 $$

Regular Hexagon:
$$ I_x = I_y = 5 \sqrt{\frac{3}{16}} (\text{side length})^4 $$

I-Beam:
$$ I_x = \frac{H^3 b}{12} + 2\left[\frac{h^3 B}{12} + h B \frac{(H+h)^2}{4}\right] $$
$$ I_y = \frac{b^3 H}{12} + 2\left(\frac{B^3 h}{12}\right) $$

T-Beam:
$$ I_x = \frac{TFw TFt^3}{12} + \frac{Wt Wh^3}{12} + TFw TFt \left(Wh + \frac{TFt}{2} - y_{bot}\right)^2 + Wt Wh \left(\frac{Wh}{2} - y_{bot}\right)^2 $$
$$ I_y = \frac{TFt TFw^3}{12} + \frac{Wh Wt^3}{12} $$

L-Beam / Channel:
$$ I_x = \frac{TFw TFt^3}{12} + \frac{BFw BFt^3}{12} + \frac{Wt h^3}{12} + TFw TFt \left(h - \frac{TFt}{2} - y_{bot}\right)^2 + BFw BFt \left(\frac{BFt}{2} - y_{bot}\right)^2 + Wt h \left(\frac{h}{2} - y_{bot}\right)^2 $$
$$ I_y = \frac{TFt TFw^3}{12} + \frac{BFt BFw^3}{12} + \frac{h Wt^3}{12} + TFt TFw \left(Wt + \frac{TFw}{2} - x_{left}\right)^2 + BFt BFw \left(Wt + \frac{BFw}{2} - x_{left}\right)^2 + h Wt \left(\frac{Wt}{2} - x_{left}\right)^2 $$

As there are many terms involved in these formulas, the free moment of inertia calculator takes a couple of seconds in resolving them and displaying answers.

How To Calculate Moment Of Inertia?

Here we will be solving a couple of examples related to inertial moments. Stay with it!

Example #1: Calculate moment of inertia of an object revolving with angular acceleration of $2 \, \text{rad/s}^2$ with angular torque of about 3Nm. Determine its moment of inertia.

Solution:

We know that:

$$ I = \frac{L}{\text{angular frequency}} $$

$$ = \frac{3}{2} $$

$$ = 1.5 \, \text{kg·m}^2 $$

Example #2: Determine the moment of inertia of the circle in terms of its polar coordinates having a radius of 4cm.

Solution

As we know that:

$$ I_x = I_y $$

$$ = \frac{\pi}{4} (4)^4 $$

$$ = 0.785 \cdot 256 $$

$$ = 200.96 \, \text{kg·m}^2 $$

Here the free polar moment of inertia calculator also shows the same results but in a very short span of seconds, saving your precious time.

How Moment Of Inertia Calculator Works?

Except manual calculations, make use of this second moment of area calculator that generates accurate outputs within a couple of clicks. Let’s find how!

Input:

In the beginning, make a selection of the geometrical figure from the drop down menu for which you want to determine the moment of inertia
After you make a selection, write down the values of the parameter against that selected figure along with the units
Tap the "CALCULATE" button

Output:

Depending upon the input provided, the moi calculator determines either:

Moment of inertia about the x-axis and y-axis
Total area of the given figure
Centroid of the figure in terms of x and y coordinates
Section modulus of the figure in terms of its coordinates

FAQ's:

Why is Newton's first law called inertia?

As the newton’s first law of motion states that:

“A body remains at rest or in continuous motion until or unless acted upon by external force to either move it or stop it”

The above statement defines the inertia of a particular body. This is why the 1st law of motion gives us the definition of inertia.

What is Galileo's law of inertia?

This law states that:

“A particular object remains in the state of motion if no net force acts upon it to stop it."

What is inertia?

A particular property of the matter with the help of which it keeps itself either in the state of continuous motion or rest is known as the inertia.

What does moment of inertia tell?

In actuality, the moment of inertia tells you how difficult it is to rotate a certain object about an axis that could easily be determined by using the free online rotational inertia calculator.

Conclusion:

The moment of inertia is very important to keep the heavy objects in a smooth motion without any damage. The swings like skywheel, discovery, roller coaster, and many others are operated on the moment of inertia. That is why the free mass moment of inertia calculator helps you to determine the moment of inertia to avoid any hurdle before starting such huge swings.

References:

Wikipedia: Moment of Inertia
Khan Academy: Rotational inertia
Lumen Learning: Rotational Kinetic Energy