# Product of Inertia vs. Moment of Inertia

According to Newton’s Laws, objects will stay in their current state of motion unless something acts upon them. While Newton’s Laws are usually applied to linear motion, they also apply to rotation. A rotating object will continue to rotate unless a force acts on it. How quickly an object rotates depends on its mass properties and the force applied to it.

Moment of inertia (MOI) describes the amount of force, or torque, required to change the rate of rotation of an object. Product of inertia (POI) reveals how an object might be imbalanced. The two are similar but have different overall applications in ensuring the safety and effectiveness of aircraft and spacecraft. Inaccurate measurements can result in system failure, thus machinery is used to enhance accuracy for such critical applications.

## What Is Product of Inertia?

Product of inertia is a characteristic of an object that describes an imbalance relative to a set of coordinate axes. If the object’s mass is evenly distributed on the XY, YZ and ZX planes, there is no imbalance, and the POI is zero. Should the mass be distributed asymmetrically, the result is a non-zero POI, which will cause an imbalance with respect to the defined coordinate axes.

Better understanding of POI can be as simple as thinking of a tire on a motor vehicle. For a vehicle to be safe to drive, the tire must be balanced on the inside and the outside. If a tire is completely balanced, the weight of the wheel is evenly distributed around the tire and between the inside and outside of the rim.

Various factors can cause a tire to become imbalanced, such as uneven wear on the treads or damage to the wheel’s rim. In those instances, the tire is no longer perfectly round. When a person drives a vehicle with imbalanced tires, they are likely to notice vibrations and a bumpy ride when they drive.

POI is used to determine the principal axes of an object in flight. When the principal axes differ from the geometric axes, the flight pattern tends to be “wobbly”, but if you can determine the location of the imbalance and revert the POI back to zero, the “wobble” will disappear.

### How to Calculate POI

When mass is distributed symmetrically, the POI will be zero. It can be calculated using the formula  IXY = ∫ xy dA. In this formula, “A” is the area of the object and “d” is the distance between the x and y axes. Although the target POI is zero, the value can be positive or negative. Whether POI is negative or positive depends largely on the reference axis used.

Often, it’s more accurate to measure the POI of an object rather than calculate it. There are several ways to measure POI and to make adjustments as needed. You can use a vertical two-plane spin balance machine to measure POI directly. With a spin balance machine, the payload spins around a defined axis, potentially producing imbalanced forces.

In some cases, it may not be possible to spin an object. As an example, satellites with solar panels might be damaged if spun. An alternative to spinning is to determine POI using the MOI method.

Using MOI to calculate POI isn’t as accurate as using a spin balance machine. The accuracy of the result depends on factors such as the characteristics of the payload, the quality of the MOI instruments used and the angles between the measurements that are used.

## What Is Moment of Inertia?

Moment of inertia is also known as rotational inertia. MOI tells you how difficult it is to change the rotational velocity of an object on its axis. A baseball player swinging a bat is an example of rotational inertia, as is a ball swinging around a pole while attached by a tether.

The greater the mass of a particular point on the object and the greater the distance from the axis, the greater the MOI.

Stability is a critical element in the design and manufacturing of aerospace craft, and MOI is a property of mass that explains an object’s stability and the force needed to change its motion.

### How to Calculate the Moment of Inertia

The formula to calculate moment of inertia is I=mr2, where “I” is inertia, “m” is mass and “r” is the radius or distance from the axis to the representative point of mass. Once you’ve calculated the moment of inertia formula, you can calculate other statics elements, like an object’s angular momentum and its rotational kinetic energy.

Rotational kinetic energy uses the formula K = Iω2, where “I” is the MOI and “ω” is the angular velocity of the object. Angular momentum uses the formula L = Iω. Another way to write the formula is T = IA, where “T” is torque,  “I” is inertia and “A” is rotational acceleration.

A formula to calculate MOI is often sufficient when the object is simple, such as a wheel or a single sphere. But a simple formula no longer suffices when calculating MOI for a complex object, such as an aircraft engine with multiple moving parts. You’d need to repeat the MOI formulas multiple times for each mass, then add the MOIs together to get an aggregate.

In this case, using an instrument to measure MOI is much more efficient. Raptor Scientific manufactures more than 50 instruments designed to measure MOI on objects ranging in size from less than one gram to more than 10,000 kilograms. Most of our instruments measure MOI using the principle of the inverted torsion pendulum. The object rests on a table and is attached to low-friction bearings. The bearings restrict the object’s motion, allowing only for pure rotation. A digital counter connects to a sensing device to determine the period of oscillation.

## What Is the Difference Between Product of Inertia and Moment of Inertia?

While both POI and MOI are critical in aerospace and engineering applications, there are several notable differences between POI and MOI, the first of which is what they measure. POI refers to the symmetry of mass of an object relative to its coordinate axes. MOI reflects how difficult it is to change the rotational speed of an object about a defined axis. In other words, MOI describes the amount of torque required to change the rate of rotation of an object. POI, on the other hand, reveals how that object is imbalanced in motion.

Another notable difference between POI and MOI is the value of each. POI can be zero, or it can be negative or positive. In contrast, MOI is always positive. One way to remember that MOI is always positive is to remember that the mass of an object is always positive.

Remember, MOI is referenced to an axis while POI is referenced to a plane, such as XY, YZ or ZX.

### Applications of POI vs. MOI

One of the uses of MOI is to determine how a mass will behave in response to a known torque. Torque is the measurement of the force needed to make an object rotate on an axis. It’s a vector quantity and is only used to measure rotation. Torque is calculated by multiplying force times distance.

When you know the MOI of an object and the torque, you can divide torque into the MOI to find the angular acceleration.

Understanding POI allows you to correct asymmetry or imbalance in an object. To improve symmetry and ensure a smoother flight or ride, your goal is to get POI to zero.

## Instruments Used to Measure MOI

A very crude method of measuring MOI is to hang the object from a wire, then oscillate it. While the object oscillates, measure the time it takes for one oscillation.  Many variables can affect the results. The object is likely to swing back and forth or bounce up and down, affecting the accuracy of the timing. Furthermore, large or unusually shaped objects might be difficult to suspend.

Fortunately, multiple instruments are available to measure MOI accurately and efficiently. An inverted torsion pendulum allows you to get an exact measurement of the oscillation period of the object. When using the instrument, you rest the object on a rotary table at the top of the device. Low-friction air bearings support the table and object. These instruments allow for far more accurate and reliable measurements of MOI, especially for professional applications.

Among the benefits of using an inverted torsion pendulum to measure MOI are minimal fixturing, a well-defined axis and minimal computational techniques.

Using an inverted torsion pendulum is simple:

• The object is attached to the table and oscillated. A digital timer provides the time period of oscillation. The total moment of inertia is then determined by multiplying the oscillation time by the machine’s calibration constant.
• The object is taken off of the table. The table is then oscillated on its own to determine its tare moment of inertia.
• The tare moment of inertia is subtracted from the total moment of inertia with the object attached. The difference is the MOI of the object alone.

Raptor Scientific produces several MOI instruments that use an inverted torsion pendulum:

• XKR Series: The XKR series measures the MOI of objects ranging from 0.1 to 2.3 kilograms (kg). They are extremely accurate, with an accuracy of 0.1%.
• XR Series: Measure the MOI of objects up to 120 kg. They have an accuracy of 0.25% and are designed for general use.
• GB Series: Measure the MOI of heavier objects up to 6,800 kg. They are ideal for use in critical military and space applications. They have an accuracy of 0.1%.
• MP Series: Measure the MOI and center of gravity, as well as the weight, of objects up to 3,000 kg. They have an accuracy of 0.25%.
• KSR Series: Measure the MOI and center of gravity of objects up to 11,500 kg, with an accuracy of 0.1%.
• POI Series: Measure all types of mass properties, including MOI, of objects up to 10,500 kg. They have an accuracy of 0.1%.

## Benefits of Measuring MOI

Using an instrument to measure MOI is often much faster than trying to calculate MOI. There are other reasons to measure rather than calculate MOI, such as :

• Lowered costs: Using an instrument to measure MOI takes less time than calculating, so your team of engineers can spend less time on rote calculations and more time on the things that matter most to your company.
• Increased accuracy: Instruments are often much more accurate than calculations, meaning you’ll enjoy fewer errors and instances of needing to return to the design process to fix issues.
• Improved quality control: Flight vehicles must have a certain MOI to ensure high performance. The greater accuracy of measuring MOI is likely to mean fewer quality control issues.

## Instruments Used to Measure POI

A spin balance machine can measure POI on certain objects and is often the most commonly used instrument for measuring POI. Spin balance machines rotate the object at a set speed and then measure the reaction forces on the upper and lower bearings. The machines have a computer that calculates POI automatically, using the height of the center of gravity of the object and the spacing between the two bearings.

There are several benefits and drawbacks of using a spin balance machine for POI. The machines can be very sensitive, minimizing air turbulence and improving results. A drawback of this machine is that it can’t measure the POI of objects, such as satellites with large solar panels or control fins. In those cases, it might be more effective to use the MOI method.

### Can You Use MOI to Measure POI?

Sometimes it’s not possible to use a spin balance machine to measure POI directly. These cases are usually due to the shape or size of the object. Some objects can’t be spun at all. An alternative is to use MOI measurements to calculate POI. Using MOI for POI isn’t as accurate as measuring POI directly, but is often the best alternative.

To use MOI to measure POI, you need to measure the object in six positions on a torsion pendulum. Once you have all six measurements, you can calculate the POI using rotational angles.

While this method lets you calculate POI on objects you otherwise wouldn’t be able to, it has some drawbacks. The process is long and arduous. It can take many hours to perform, adding to its overall cost. On the other hand, it often costs less than a spin-balancing machine and puts less stress on the object you’re measuring.

## Measure POI and MOI With Raptor Scientific

The product of inertia and the moment of inertia are critical calculations engineers make in many industries. Raptor Scientific has more than five decades of experience with mass properties measurement. We manufacture POI and MOI instruments and offer mass properties measurement services in our state-of-the-art laboratory.

Safety is of paramount importance in aerospace applications, and Raptor Scientific has the products to help ensure aerospace safety. To learn more, contact us today for a quote.