8.1 Composite Center of Gravity by Moments
Key Takeaways
Every component moment uses its own CG coordinate from the same datum.
Pound-feet divided by pounds gives the CG distance in feet.
Added or removed components change both total weight and moment.
Choose one datum for the calculation
A datum is the reference from which component CG distances are measured. It can be chosen conveniently, provided every distance uses the same reference and direction. Choosing the left end of a skid often avoids negative coordinates, but it is not mathematically mandatory. A consistent signed coordinate system works as well.
For a composite load, multiply each component weight by its own CG distance from the datum. This product is its weight moment. Sum all moments and divide by the total weight:
The result is a distance, not a force or a moment. Units verify the operation: pound-feet divided by pounds gives feet. A result expressed in pound-feet after the final division indicates that the calculation has not been completed or the units have been mislabeled.
Work through a complete inventory
A training skid consists of a uniform 20-ft base weighing 4,000 lb, a 6,000-lb transformer whose CG is 5 ft from the left end, and a 2,000-lb pump whose CG is 16 ft from the left end. The base’s own CG is at 10 ft because the problem explicitly defines it as uniform.
| Component | Weight | CG distance | Moment |
|---|---|---|---|
| Uniform base | 4,000 lb | 10 ft | 40,000 lb-ft |
| Transformer | 6,000 lb | 5 ft | 30,000 lb-ft |
| Pump | 2,000 lb | 16 ft | 32,000 lb-ft |
| Total | 12,000 lb | — | 102,000 lb-ft |
The composite CG is 102,000/12,000 = 8.5 ft from the left end. It lies toward the heavier transformer relative to the base’s midpoint. From the right end, the distance is 20 − 8.5 = 11.5 ft. These are different coordinates describing the same physical point.
The component’s mounting position is not always its CG. If a motor’s mounting feet are centered at 6 ft but its own CG is offset, use the actual CG coordinate. The problem’s supplied component centers represent their weight application points. Do not replace them with the center of the visible case unless uniformity or manufacturer information establishes that equivalence.
Extend the method to other axes
Apply the same weighted-average method independently for transverse and vertical coordinates. For example, two components weigh 3,000 and 1,000 lb, with vertical CG elevations of 2 and 6 ft above the same base datum. Their composite elevation is (3,000 × 2 + 1,000 × 6)/4,000 = 3 ft. That result informs stability, while the longitudinal coordinate informs support reactions.
A three-dimensional coordinate inventory uses each component’s x, y and z positions. The arithmetic does not change, but the physical interpretation does. Do not use an elevation as a horizontal moment arm in a two-support longitudinal reaction problem.
Handle removed or added components
Adding an accessory changes both total weight and total moment. A 1,000-lb accessory added at 18 ft to the 12,000-lb skid increases moment to 120,000 lb-ft and weight to 13,000 lb. The new CG is approximately 9.231 ft from the datum. Reusing 8.5 ft would describe the old assembly.
Removing a component can be represented by subtracting its weight and moment from a known complete inventory. The remaining weight must be positive and the subtraction must refer to the same datum. This bookkeeping is valid only when the original total and the removed component information are reliable. It does not establish the CG of unknown hidden contents.
Check the answer physically
With positive component weights, a coordinate calculated as a weighted average lies between the smallest and largest component CG coordinates. If all component centers lie between 5 and 16 ft, a result at 22 ft cannot follow from that inventory. Check signs, units, omitted weights and moment sums before proceeding.
A hollow or open-frame object can have a CG in empty space. The weighted-average result need not lie within solid material. An L-shaped frame’s balance point may be outside either individual member. This does not permit attaching directly at an unsupported point; it tells where the resultant weight acts.
From CG to rigging
The computed coordinate must be related to actual picks. If the left pick is 2 ft from the datum, the 8.5-ft CG is 6.5 ft from that pick. If the right pick is at 18 ft, it is 9.5 ft from the CG. Those distances, not 8.5 and 11.5 from the load ends, belong in the support-reaction calculation.
Document the component configuration so the calculation remains traceable. Secure movable masses as approved, confirm retained contents and compare the result with manufacturer information where available. A trial lift then checks observed attitude; it cannot make an inaccurate component inventory reliable.
Treat uncertain and shifting contents separately
A partially filled vessel can change its effective mass distribution as it tilts. The fixed-coordinate weighted average describes only the stated condition; it does not establish the CG for every later attitude. Obtain the approved handling information, drain or secure contents where authorized, or have the changing condition assessed before planning rotation.
An inventory also needs attached guards, accessories and the structural base. Leaving the base out because it appears uniform removes both its weight and its moment from the total. In the skid example, omitting the 4,000-lb base would incorrectly give 62,000/8,000 = 7.75 ft. That arithmetic is internally consistent but describes a different incomplete load.
Source: CCO Level II load-characteristic and calculation scope.
A 6,000-lb uniform 24-ft base has CG at 12 ft, and a 10,000-lb compressor has CG at 6 ft. Where is the composite CG from that datum?
6.75 ft
8.25 ft
9.50 ft
12.00 ft
Sections you finish are checked off in the contents.