# Why Is Moment of Inertia Important?

The moment of inertia (MOI) is one of the many mass properties that describe an object’s stability and the forces needed to alter its motion. For aerospace engineering, **stability is a crucial element** in designing and manufacturing air and spacecraft. Knowing the MOI about various axes is vital to determining how a device can hold up to external and internal forces. Accuracy in measuring this value will ensure a better-produced craft or equipment to meet the specific needs of various projects in the stringent air and space industry.

## What Is Mass Moment of Inertia?

The mass moment of inertia reflects an object’s amount of resistance to any change in its rotational rate about an axis. Mass MOI derives from Newton’s first law of motion regarding inertia. This law says that objects at rest will resist being put into motion, and objects in motion will resist stopping without external forces acting upon them. For linear motion, the force needed to move something is the product of its mass and acceleration (F=ma). Higher masses require more force to change their motion from standing still or moving.

Applying this concept to rotational motion results in the mass MOI correlating to the linear mass. The torque required comes from multiplying the mass moment of inertia by the angular acceleration. Therefore, torque in rotational motion is similar to linear force. Angular acceleration is the rotational movement equivalent of linear acceleration. Similar to linear inertia, **greater mass MOI requires more torque** to reach a specific angular acceleration.

The mass MOI has different forms depending on the object’s shape and the distribution of mass around a given rotational axis. For example, considering a single-point mass, the MOI is the product of the mass times the distance from the axis squared. This type of MOI works with evenly weighted ring shapes with a rotational axis perpendicular to the ring. Point mass MOI can also help calculate the value for an object with distributed masses by using the point mass for each distribution.

Since most objects don’t have a single, evenly distributed mass around the edge, this measurement alone will not suffice. Calculating MOI in objects with distributed mass requires a different approach. For distributed masses, the MOI is the sum of each of the point masses.

## Why Does Mass MOI Matter?

The MOI of an object determines how much torque an object needs to reach a specific angular acceleration. When calculating torque, or rotational force, you need to know the mass MOI. Multiplying the MOI by the required angular acceleration will give you the required torque value necessary to achieve that angular acceleration. Larger MOI values correspond to greater torque required to achieve a given acceleration.

The ability to precisely determine these values allows the designer to properly size their components while achieving the high performance demands the aerospace and defense industries need. The designer can optimize the trade-off between size, weight and power to achieve the mission goals. In other cases, the measurement of MOI can be used to verify that manufacturing and assembly tolerances and processes goals are nominal. In yet a third example, higher performance can be achieved by minimizing MOI while not sacrificing strength.

## How to Calculate Mass Moment of Inertia

The mass moment of inertia has different means of calculating it, depending on the shape, distribution of masses and the rotational axis. Simple objects require using the point mass or multiple point masses to find the mass MOI.

To find the point mass, multiply the squared distance between the rotational axis and the mass by the mass at the given point. Some objects with a single mass evenly distributed around the axis will have the mass MOI equal to this point mass.

Shape also makes a difference when finding the MOI. The **general formula for mass MOI is I=Mk^2**, where M is the mass of the object and k is the distance between the rotational axis and the radius of gyration.

For calculating the mass moment of inertia for more complex objects, the summation of all the point masses of the piece gives the mass MOI. However, instead of calculating several point masses, calculus could help by using the integral of the squared radius of the mass of an infinitesimally small area of the object (dm).

Calculating the mass MOI presents some issues, especially in highly complex designs. However, considering this value’s importance, engineers and manufacturers of any aerospace craft should know it. Another way of determining the mass moment of inertia is through measurement.

## How to Measure Moment of Inertia

Measuring the mass moment of inertia can take much less time than calculating it. Various equipment can take this measurement. When selecting measurement instruments, consider the object’s mass, the types of additional measurements needed and the required accuracy. Some series only provide measurements for the MOI, while others include the center of gravity and other mass properties. All series use the inverted torsion pendulum method to determine the mass moment of inertia about a known rotational axis. The payload does not suspend from a pendulum but sits on a table with low friction. Sensors measure the object’s oscillation period, which helps calculate the moment of inertia.

We offer the following mass MOI measuring instruments.

### XKR Series

For small objects weighing from 0.1 kg to 2.3 kg (0.2 lbs. to 5 lbs.), the XKR series uses air bearings to provide extreme MOI measurement accuracy. Measurement accuracy of better than 0.1% is an option for instruments in this series.

### XR Series

Designed for general use, the XR series has a load capacity of up to 115 kg (250 lbs.). Accuracies for measurements of MOI with this series go up to 0.25%.

### GB Series

The GB series tackles the heaviest of objects. It can handle pieces weighing between 68 kg to 6,000 kg (150 lbs. and 13,000 lbs.). Due to its exceptional ability to measure MOI on large objects, the military and aerospace industries frequently use instruments in the GB series. It can produce measurements with an accuracy of up to 0.1%.

### KSR Series

The KSR series combines the center of gravity and MOI measurement capability. Instruments in this series can hold objects up to 9,070 kg (20,000 lbs.). Additionally, accuracy is extremely high, with options of up to 0.1% measurement accuracy for both the center of gravity and moment of inertia.

### MP Series

The MP series is a set of general-use instruments that can accommodate heavier loads than the XR series. Instruments in this series can measure payloads up to 4,500 kg (10,000 lbs.). This series goes beyond the XR series by also measuring the center of gravity and weight with the moment of inertia. Accuracy for this series is 0.25%.

### POI Series

The POI instrument series combines center of gravity, MOI and product of inertia (POI) measurements with dynamic balancing capability. POI instruments offer the highest measurement accuracy available. This series holds payloads measuring up to 10,500 kg (23,000 lbs.) Products in this series take all measurements of mass, including the moment of inertia, the center of gravity, dynamic imbalance and the product of inertia. The instruments in the POI series are highly accurate, up to 0.1%.

## Calculating vs. Measuring the Moment of Inertia

Finding the mass MOI through calculations alone could cause problems with the final results. Calculating the MOI may not be the best option when accuracy is essential, as in the air and space industry.

Calculations, especially those based on sums of point masses, are only as reliable as the components used to find the result. Therefore, a single miscalculation in finding a point mass could cause a significant error in finding an object’s total mass moment of inertia, magnifying any mistakes in initial measurements.

Using calculations could cause errors of 30% compared to the measured MOI value. Such a drastic difference in the calculated versus measured values could create problems during manufacturing or practical use of the object.

Another significant problem with calculating values comes from the cost wasted. Calculating the MOI, finding out the value is incorrect and returning to engineers for another calculation squanders time. **Doing the math could take more time than many instruments require** for finding a correct measurement for the MOI. With highly accurate results of up to 0.1%, devices save time and increase precision over calculating.

Lastly, determining accurate MOI values is essential for many aerospace or military projects. Correct measurements of the MOI and other mass values ensure the project’s quality. Knowing the mass moment of inertia from measurements provides predictability of flight characteristics for air or spacecraft or performance metrics for other devices.

## Accurately Measure the Mass Moment of Inertia With Instruments From Raptor Scientific

Accuracy of mass measurements, including the moment of inertia, is crucial for industries that demand such information. While calculating the moment of inertia is possible, measuring it gives more exact values. At Raptor Scientific, we have a range of moment of inertia measurement instruments for measuring devices between 1 gram and 10,000 kg. These devices provide precision for mass MOI up to 0.1% to satisfy military and aerospace applications. Explore our range of measurement instruments online or contact us at Raptor Scientific for a free quote on any of our instruments.