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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

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This moment of inertia calculator determines the moment of inertia of geometrical figures such as triangles and rectangles. Additionally, you can 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 determines the torque produced by 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 a given shape relative to its axis.

Second Moment of Inertia: This property measures the distribution of points with respect to the axis.

Moment of Inertia Formula:

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

L = Angular Momentum
? = Angular Frequency
I = Moment of Inertia

Formulas for Common Shapes:

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

These formulas are complex, so using a free moment of inertia calculator provides instant, accurate results.

How To Calculate Moment of Inertia?

Example #1: An object rotates with an angular acceleration of $2 \, \text{rad/s}^2$ and experiences a torque of 3 Nm.

Solution:
$$ I = \frac{L}{\text{angular frequency}} = \frac{3}{2} = 1.5 \, \text{kg·m}^2 $$

Example #2: Moment of inertia of a circle with radius 4 cm.
Solution:
$$ I_x = I_y = \frac{\pi}{4} (4)^4 = 0.785 \cdot 256 = 200.96 \, \text{kg·m}^2 $$

How Moment of Inertia Calculator Works?

Input:

Select the geometrical figure.
Enter relevant parameters and units.
Click calculate.

Output:

Moment of inertia about x and y axes
Total area of the figure
Centroid coordinates
Section modulus

FAQ's:

Why is Newton's first law called inertia?

“A body remains at rest or in uniform motion unless acted upon by an external force.” This defines inertia.

What is Galileo's law of inertia?

“An object continues in its state of motion unless acted upon by a net external force.”

What is inertia?

Inertia is the property of matter that resists changes in its state of motion.

What does moment of inertia tell?

It indicates how difficult it is to rotate an object about a specific axis.

Conclusion:

Moment of inertia is crucial for smooth and safe motion of heavy objects, including Ferris wheels, roller coasters, and swings. A free mass moment of inertia calculator ensures accurate values and prevents potential issues.

References:

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

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