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Physics Calculators ▶ Moment of Inertia Calculator

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**Table of Content**

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.

In the context of 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”**

**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”**

$$ I = \frac{L}{𝓌} $$

Where:

**L** = Angular Momentum

**𝓌** = Angular Frequency

**I** = Inertia

Finding moments of inertia may involve lots of complex calculations that are not easy to resolve every time. But here we will be using formulas for some of the basic shapes. Keep one thing in mind that these formulas hold only when the x and y axis of the given figure passes from the centroid.

The formulas are as follows:

**Moment Of Inertia Of Triangle:**

$$ I_{x} = \frac{\left(width\right) * \left(height\right)^{3}}{36} $$

$$ I_{y} = \frac{\left(\left(height\right) * \left(width\right)^{3} – \left(height\right) * a * \left(width\right)^{2} + \left(width\right) * \left(height\right) * \left(a\right)^{2}\right)}{36} $$

**Moment Of Inertia Rectangle:**

$$ I_{x} = \frac{\left(width\right) * \left(height\right)^{3}}{12} $$

$$ I_{x} = \frac{\left(height\right) * \left(width\right)^{3}}{12} $$

**Moment Of Inertia Of Hollow Rectangle:**

$$ I_{x} = \frac{b * \left(h\right)^{3} – b_{1} * \left(h_{1}^{3}\right)}{12} $$

$$ I_{y} = \frac{\left(b\right)^3 * h – b_{1}^3 * h_{1}}{12} $$

In case of any hurdle, try using the area moment of inertia calculator for accurate and instant outputs of various parameters that are related to the moment of inertia.

**Moment Of Inertia For Circle:**

$$ I_{x} = I_{y} = \frac{\pi}{4} * \left(radius\right)^{4} $$

**Moment Of Inertia Of Hollow Circle:**

$$ I_{x} = I_{y} = \frac{\pi}{4} \left(r_{2}^4 – r_{1}^4\right) $$

**Moment Of Inertia Of Semicircle:**

$$ I_{x} = [\frac{\pi}{8} – \frac{8}{\left(9 * \pi\right)}] * \left(radius\right)^{4} $$

$$ I_{y} = \frac{\pi}{8} * \left(radius\right)^{4} $$

**Moment Of Inertia Of Ellipse:**

$$ I_{x} = \frac{\pi}{4} * \left(radius\right)_{x} * \left(radius\right)_{y}^{3} $$

$$ I_{y} = \frac{\pi}{4} * \left(radius\right)_{y} * \left(radius\right)_x^{3} $$

**Moment Of Inertia Of Regular Hexagon:**

$$ I_{x} = I_{y} = 5 * \sqrt{\frac{\left(3\right)}{16}} * \left(side length\right)^{4} $

**Moment Of Inertia Of** **I Beam:**

$$ I_{x} = \frac{H^{3} * b}{12} + 2[\frac{h^{3} * B}{12} + h * B * \frac{\left(H + h\right)^{2}}{4}] $$

$$ I_{y} = \frac{b^{3} * H}{12} + 2 * \left(\frac{B^{3} * h}{12}\right) $$

**Moment Of Inertia Of T Beam:**

$$ I_{x} = \frac{TFw * TFt^{3}}{12} + \frac{Wt * Wh^{3}}{12} + TFw * TFt\left(Wh + \frac{TFt}{2} – y_{bot}\right)^{2} + Wt * Wh * \left(\frac{Wh}{2} – y_{bot}\right)^{2} $$

$$ I_{y} = \frac{TFt * TFw^{3}}{12} + \frac{Wh * Wt^{3}}{12} $$

**Moment Of Inertia Of L Beam:**

**Moment Of Inertia Of Channel:**

$$ I_{x} = \frac{TFw * TFt^{3}}{12} + \frac{BFw * BFt^{3}}{12} + \frac{Wt * h}{12} + TFw * TFt * \left(h – \frac{TFt}{2} – y_{bot}\right)^{2} + BFw * BFt * \left(\frac{Bft}{2} – y_{bot}\right) ^{2} + Wt * h * \left(\frac{h}{2} – y_{bot}\right)^{2} $$

$$ I_{y} = \frac{TFt * TFw^{3}}{12} + \frac{BFt * BFw^{3}}{12} + \frac{h * Wt^{3}}{12} +TFt * TFw * \left(Wt + \frac{TFw}{2} – x_{left}\right)^{2} + BFt * BFw * \left(Wt +\frac{BFw}{2} – x_{left}\right)^{2} + h * Wt * \left(\frac{Wt}{2} – x_{left}\right)^{2} $$

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.

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

**Example # 01:**

Calculate moment of inertia of an object revolving with angular acceleration of \(2\frac{rad}{s^{2}}\) with angular torque of about **3Nm**. Determine its moment of inertia.

**Solution:**

We know that:

$$ I = \frac{L}{𝓌} $$

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

$$ I = 1.5kgm^{2} $$

**Example # 02:**

Determine the moment of inertia of the circle in terms of its polar coordinates having a radius of **4cm.**

**Solution:**

As we know that:

$$ I_{x} = I_{y} = \frac{\pi}{4} * \left(radius\right)^{4} $$

$$ I_{x} = I_{y} = \frac{3.14}{4} * \left(4\right)^{4} $$

$$ I_{x} = I_{y} = 0.785 * 256 $$

$$ I_{x} = I_{y} = 200.96kgm^{2} $$

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.

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

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.

This law states that:

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

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.

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.

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.

From the source of Wikipedia: Moment of inertia, Simple pendulum, Compound pendulums, Motion in a fixed plane, Rigid body, Angular momentum, Kinetic energy, Resultant torque, Parallel axis theorem, Scalar moment of inertia in a plane, Inertia tensor, Alternate inertia convention

From the source of Khan Academy: Rotational inertia, More on moment of inertia, Rotational kinetic energy, Moments

From the source of Lumen Learning: Rotational Kinetic Energy, Moment of Inertia, Rotational Energy