Measurement - Find the Center Of Mass

Motion free of constraint is centered at one point called the center of mass. Forces act about this special point pushing and twisting. When the motion of an object is limited by touching other objects, forces act where they make contact. These forces also act back at the contact points providing the limits on the objects motion. Knowing where the center of mass is, helps to determine the magnitude of these interaction forces and their effect on the object in motion.

The center of mass is a point in 3-dimensional space. With simple means, like balancing, we can find it along each of the three spacial axies. That means three measurements. Using a side view on graph paper, though, we can locate the center of mass in 2-dimensions at a time, so only 2 measurements are needed. But many times, we are really only interested in the balance point on one axis.

The actual location of the center of mass may surprize you. Depending on the shape and distribution of matter, an object's center of mass may actually lie outside of it. For example, a the center of mass of a wheel lies in the center bore of the wheel which is usually hollow. Also, consider a high-jumper. His center of mass is outside his body and can actually be under the pole! Just bend over, and yours will be outside your body too.

A balancing act

The easiest way to find the center of mass of an object is just to balance it on a suitable fulcrum along each of its axies, or just the ones of interest. It may not be as easy as it seems.

If the object has an odd shape that is not level over the fulcrum, it may slide off instead of balance. A remedy that works many times is to put something sticky on the object at the point where you think it should balance.

Sometimes the object is not very long along the axis you're balancing on so that it is impossible to balance it. In this case, find where it has an equal tendancy to fall to one side or the other.

If the fulcrum is too wide, you may not be able to get close enough to the location of the center of mass. Here, you need to find where the object falls to one side, mark it, then find where it falls to the other side and mark it. Most often the center of mass is right in the middle of those two points.

Picture the center of mass

How do you find the center of mass of an object you haven't built yet? You start with an acurate picture! Below is a worksheet to help you find the center of mass of your custom vehicle from a drawing on graph paper. In the table, the three axies of your vehicle are X, length; Y, height; Z, width. Of course, if you don't need to know the center of mass on all three axies, just use the table for those you need.

You will need two pictures to find the center of mass in 3-dimensions. Each picture has two axies. A side view has a length axis, X, and a height axis, Y. A top view has a length axis, X, and a width axis, Z. The front view has a width axis, Z, and a height axis, Y. If your drawings are acurate, you should get the same locations for the center of mass on the axis that is duplicated in different views.

Depending on what your vehicle is, there are certain parts you will almost always want to include in your measurement. Free weight is the only "part" that most vehicles have in common, so it is already entered in the worksheet. Note, if you don't know how much free weight you need, you can designate a desired center of mass and solve for the amount and placement of free weight you need.

Center Of Mass Worksheet

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    Part       Wi (___)     Xi (___)       WixXi      Yi (___)       WixYi      Zi (___)       WixZi
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Reference                 ____________              ____________              ____________
Free Weight  ____________ ____________ ____________ ____________ ____________ ____________ ____________
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 Total       ____________              ____________              ____________              ____________
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CMx = Total WixXi/Total Wi = ____________ (___) with respect to the X ref. line
CMy = Total WixYi/Total Wi = ____________ (___) with respect to the Y ref. line
CMz = Total WixZi/Total Wi = ____________ (___) with respect to the Z ref. line

Follow these instructions to fill out the chart.

1. Draw a baseline at right angles to each of the x (length), y (height) and z (width) axies. At least one of the axies will be in a separate drawing if you want to find all three. The location of the base line along its axis is not all that important, but making it at an edge of the vehicle will help some.
2. Identify each part of your vehicle that can be weighed. Write it in the table in an empty row.
3. Weigh each part or use an alternative method of estimation. Put the weight in the part's row in the column labeled "Wi". There is a space at the top of the column to record the units (like ounces) you measured in. It is assumed all your weight measurements will be in the same units.
4. For each axis, measure the distance along the axis from the baseline to the center of the part. It is better to use the location of the part's center of mass rather than just its center. Put these measurements in the part's row in the column "Xi", "Yi" and "Zi". Put the units of the measurements in the spaces provided at the top of the columns. It is assumed all your distance measurements will be in the same units.
5. For each part, multiply its weight (Wi) by its distance along each axis. Record the result in the column labeled "WixXi" for the X axis, "WixYi" for the Y axis and "WixZi" for the Z axis.
6. At the bottom of the chart, there are spaces for four totals. Add up all the numbers in the "Wi" column and put the total in the space below that column. Likewise for the "WixXi", "WixYi" and "WixZi" columns.
7. At the very bottom of the chart, there are three spaces for the results of three calculations (and their units). Calculate CMx = Total WixXi/Total Wi using the values from the "Total" row and put the result with its units in the space beside the formula. Complete the result for CMy = Total WixYi/Total Wi and CMz = Total WixZi/Total Wi in the same way. The coordinate (CMx,CMy,CMz) is a good estimate of the center of mass of your vehicle.