The math of a 'moment of inertia' in application

[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

 

[jamesrc]

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.

 

[charng]

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!