Mass Moment of Inertia Calculator

The mass moment of inertia calculator is a complex tool that helps estimate the moment of inertia of objects with different shapes. This physical quantity is otherwise known as angular mass or rotational inertia.

The moment of inertia is a characteristic property of a rigid body. It plays a similar role in the dynamics of rotational motion as the normal mass in the dynamics of translational motion. For example, the mass in the kinetic energy formula is replaced by the moment of inertia in the rotational energy equation (see rotational kinetic energy calculator).

You can use the mass moment of inertia calculator right now – just select a figure and enter its parameters. Or read on to learn what is the moment of inertia, what its units are, and how to calculate the moment of inertia. In the following text, we have also prepared a moment of inertia table with about 23 different figures. It contains almost all of the most common object shapes.

Moment of inertia is the measure of the body's rotational inertia relative to a defined, fixed axis of rotation. It determines the torque that is needed for a desired angular acceleration (see torque calculator and angular acceleration calculator). It is just like how mass determines the force needed for a desired acceleration. In other words, the moment of inertia tells us how difficult it is to rotate an object around a specific axis. Remember that the choice of axis is very important; the final moment of inertia value might strongly depend on it!

The physical dimension of the moment of inertia is mass × length². The SI unit of the moment of inertia is kilogram meter squared kg·m², and the imperial or US units are pound-foot second squared lb·ft·s² or pound-foot squared lb·ft². With the mass moment of inertia calculator, you can perform calculations in any of those units you prefer.

The moment of inertia $I$ of a material point is the product of its mass $m$ and the square of the distance $r$ from the axis of rotation. It can be expressed with the following moment of inertia equation:

If you consider a body consisting of $n$ material points, then the total moment of inertia is simply the sum of their moments of inertia:

where:

• $\sum$ – Symbol of the summation, summing all components from $i = 1$ to $i = n$;
• $m_i$ – Mass of $i$'th material point; and
• $r_i$ – Distance of $i$'th material point from the axis of rotation.

However, for bodies with a constant distribution of mass, the summation in the above formula becomes an integral:

where integration takes place over the entire volume $V$ of the body.

Although integration is not always an easy task, there are many ready-made formulas for the moment of inertia of specific solids. You can select the figure from the list in this mass moment of inertia calculator or check the moment of inertia table in the next section.

The mass moment of inertia of a body that we just described and the second moment of area are often confused. Remember that the mass moment of inertia units is kg·m² (lb·ft·s² or lb·ft²), while the second moment of area units is m⁴ (ft⁴).

You have already learned what the moment of inertia is and how you can calculate it from its definition. The table below lists moment of inertia equations for simple objects with a constant mass density that you can select in our mass moment of inertia calculator. When calculating moments of inertia, it is sometimes useful to exploit the parallel axis and perpendicular axis theorems to estimate moments of inertia about different axes.

Data and figures from Wikipedia.

Generally, to calculate the moment of inertia:

1. Measure the masses (m_i) and distances (r_i) from the axis of rotation.
2. Multiply the mass of each particle in the body by the square of its distance from the axis of rotation: m_ir_i².
3. Sum all the products of the particle's mass with the square of its distance:
   I = ∑ m_ir_i².

The SI unit of the moment of inertia is kg⋅m². The formula for a moment of inertia is mr² when the unit of m (mass) is kg, and the unit of r (distance) is m. The imperial or US unit for the moment of inertia is lb·ft·s².

The moment of inertia mostly depends on the position and orientation of the axis of rotation. Other parameters are the figure's size and shape, as well as the distribution of weight around a given axis.

2.94 kg⋅m² or 2.1684 lb·ft·s², assuming that the sphere has a mass of 9 kg and a radius of 0.7 m.

1. Use the moment of inertia formula for a sphere: I = 2/3mr².
2. Enter the mass of 9 kg and radius of 0.7 m:
   I = 2/3 × 9 kg × (0.7 m)² = 2.94 kg⋅m².

The moment of inertia for a point particle is equal to mr², so for a system of equal particles will be I = m ∑ r_i². For example, if you have a system of 3 particles with masses of 0.2 kg and distances of 0.1, 0.3, and 0.5 m from the axis of rotation, the total moment of inertia will be I = 0.2 ∑ [ 0.1² + 0.3² + 0.5²] = 0.07 kg⋅m².

No, the moment of inertia always has positive values. The moment of inertia is the product of mass and distance from the axis of rotation squared. Mass cannot be negative, and the square of any real number is also positive, so the product is always positive.