Mass Moment of Inertia |
There comes a time in many analyses that the mass moment of inertia is required. This could be the simple comparison of different rims, a computer model of an accelerating kart or an inertial dynamometer. Calculating the mass moment of inertia can be done for simple shapes but can be a real pain for more complex shapes. There's no reason to hurt our heads when simple experimentation will yield accurate results. |
All we need to do is hang a mass from a string wrapped around a known diameter of our test subject. This is simply (crudely?) pictured below. We can see the object that we're going to rotationally accelerate at the top with radius b. The string is wrapped around the object supporting the mass, m. |
Note that the mass will have the weight W which is the mass times the acceleration due to gravity. Most of us have scales that measure in pounds, not slugs (the unit of mass in the Imperial system). All we need to do to get the mass is divide it's weight in pounds by 32.2 feet per second squared. Be careful to keep all the units straight in these calculations. |
| Now all we have to do is measure the distance 's' that the mass will fall. Knowing that, we are ready to begin testing. |
| Now we can repeatedly release the mass and time its descent to the floor. I say repeatedly since we should do this at least three times to insure accurate repeatable results. Be sure that the height is the same each time. |
| Review the results. If any times are drastically different than the others, then discard the out of line times. If there are only a couple times remaining, additional testing should be done to gather more times. Total the remaining results and determine the average. Use this average time in the equation presented below. |
| Always keep accuracy in the back of your head when setting up and performing this experiment. Obviously if the mass reaches the floor in less than a second that will be less accurate than if the mass took three seconds to hit the floor. This is assuming your repsonse time stays plus or minus some value. This value over the total considered should be as low as possible and reasonable. Common sense should be exercised. |
| As a reminder, the variables are defined below: |
| I = the mass moment of inertia. This is not the area moment of inertia usually just described as the moment of inertia used in beam analysis and such. |
| b = radius of the object where the string is wrapped. |
| m = mass of the weight suspended from the string. Found by dividing the weight by 32.2 feet per second squared, (the acceleration due to gravity). |
| g = 32.2 feet per second squared. This is the acceleration due to gravity. |
| s = the height of the mass above the floor. |
| t = the time in seconds that it takes the mass to hit the floor after it's been released. |
Things to Watch Out For |
| Be sure that all your units are in agreement. The acceleration due to gravity provide above uses feet. The radius and height should also be in feet to provide meaningful results. Any units may be uses as long as they are consistent. |
| Just to beat the units horse a little more, your result should have the units pounds times feet times seconds squared, (assuming you used feet and pounds of course). Or if you use slugs and feet, then the units will be feet squared times slugs. Any other form indicates a problem somewhere with the units and therefore the numbers. Do be careful and check your work. |
| Be very careful with the bearings supporting the object under question. Any drag or resistance in the bearings will affect the results of this testing. The bearings must spin freely and have negligible startup resistance. |
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