# The math of a 'moment of inertia' in application

• JohnG
In summary, the conversation discusses determining the moment of inertia of an object using a filament and weight system. The tension in the filament is calculated using F=ma, and the torque is determined to be 12.2in.lbf. However, there is confusion about the units for moment of inertia as it is defined in both lb.in^2 and in.lb.sec^2. The expert clarifies that the units for moment of inertia are mass times length squared, and explains the difference between pound-force and pound-mass. The expert also notes that radians are unitless dimensions.
JohnG
I am trying to detemine the moment of interia of an object, not too accurately.

I have the object mounted in bearings, and on the shaft of the object I have a ring (radius of 2.5in) which I have a filament wrapped around which drops to a weight of 5lbs (mass of .1554 slug). I can release the weight and time the travel over a known distance (approx 24in in 5sec, an acceleration 'a' of 9.6in/sec^2). Angular acceration is 'a'/'r' or 3.84rad/sec^2. I have determined the tension in the filament to be 4.88lbf by F=ma, m=.1554slug * (subtracting 'a'(conv. to ft/sec^2) from g(ft/sec^2)). The torque would then be 4.88lbf * 2.5in or 12.2in.lbf.

My problem has come down to the math. I have seen moment of inertia defined in both lb.in^2 and in.lb.sec^2. I cannot seem to figure out how torque (in.lbf) / angular acceleration (rad/sec^2) can be conveyed to MOI of either version, the units don't seem to cancel out right.

Any input would be greatly appreciated, until then I will have fun putting a sharp stick in my eye.

John

Last edited:
Moment of inertia has units [M]*[L]2 (mass times length squared)
When you see inch*pound*second^2, the "pound" is pound-force.
When you see pound*inch^2. the "pound" is pound-mass.
Also note that radians are unitless dimensions.
I hope that helps.

Dear John,

Thank you for sharing your experiment and calculations regarding the determination of the moment of inertia of an object. The moment of inertia, also known as rotational inertia, is a measure of an object's resistance to changes in its rotational motion. It is an important concept in physics and engineering, particularly in the study of rotational dynamics.

Based on your experiment, it seems that you have correctly identified the necessary variables and equations for calculating the moment of inertia. However, I believe the confusion lies in the units used for torque and angular acceleration.

The moment of inertia is typically expressed in units of mass multiplied by distance squared (kg·m^2 or slug·ft^2). Torque, on the other hand, is typically expressed in units of force multiplied by distance (N·m or lbf·ft). Similarly, angular acceleration is typically expressed in units of radians per second squared (rad/s^2).

To properly calculate the moment of inertia, you will need to ensure that all units are consistent. In your equation, you have used units of in·lbf for torque and rad/s^2 for angular acceleration. To convert these to the appropriate units for moment of inertia, you will need to use the equation I = T/α, where I is the moment of inertia, T is the torque, and α is the angular acceleration.

Using this equation, you would first need to convert the torque from in·lbf to ft·lbf, as 1 in = 0.0833 ft. This would give you a torque of 1.02 ft·lbf. Then, you can plug in the values for torque and angular acceleration to get the moment of inertia in units of slug·ft^2.

Alternatively, you can also use the equation I = m·r^2, where m is the mass of the object and r is the distance from the axis of rotation. In this case, you would need to use units of lbf for force, convert the mass from slugs to lbf·sec^2/ft, and convert the radius from in to ft.

I hope this helps clarify the math behind the moment of inertia calculation. If you have any further questions, please do not hesitate to reach out. Keep up the good work with your experiment!

## 1. What is the moment of inertia and why is it important?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is an important concept in physics and engineering, as it helps us understand how objects behave when subject to rotational forces.

## 2. How is the moment of inertia calculated?

The moment of inertia is calculated by taking the sum of the mass of an object multiplied by the square of its distance from the axis of rotation. The resulting value is dependent on the shape and mass distribution of the object.

## 3. How is the moment of inertia used in real-world applications?

The moment of inertia is used in many real-world applications, such as designing vehicles and machinery, calculating the stability of structures, and predicting the behavior of rotating objects in space.

## 4. How does the moment of inertia affect an object's rotational motion?

The moment of inertia affects an object's rotational motion by determining how much torque is required to change its rotational speed. Objects with a larger moment of inertia require more torque to achieve the same angular acceleration as objects with a smaller moment of inertia.

## 5. How does the moment of inertia differ from the concept of mass in linear motion?

The moment of inertia is similar to mass in linear motion, as they both measure an object's resistance to changes in motion. However, moment of inertia takes into account an object's distribution of mass and its rotational motion, while mass only applies to an object's linear motion.

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