The Math of Flight

Before every airplane flight, math is used to ensure that the plane is loaded correctly. Airline employees known as flight dispatchers do weight and balance calculations to make sure that the aircraft’s center of gravity is within it’s tested limits. Every aircraft has a published range of where its center of gravity can be. This range is defined as the distance from a “datum” which is an arbitrary point defined by the manufacturer of the aircraft. A common location of the datum is the tip of the nose. The center of gravity location is calculated by dividing the aircraft’s moment weight by its total weight, which reveals where the center of gravity is located relative to the datum. An objects moment is calculated by multiplying the objects weight by its arm, which is its distance away from the datum. An object 20 inches aft of the datum will have a larger moment then an object with the same weight that is 15 inches aft of the datum. Airplane manufacturers publish diagrams like the one shown below to assist with these calculations. Our diagram is from a Cessna 172, a very common light airplane. Let’s do an example! Say the pilot and front passenger weigh 150 lbs. each, there is one rear passenger who weighs 120 lbs., and there are 30 lbs. of baggage in baggage area 1. The airplane’s empty weight and moment is different for every airplane and is given by the manufacturer, so we’ll use a common estimate for this example. The final variable is fuel, which in this airplane has an arm of 47 inches. Aviation fuel weighs 6 lbs./gal, and for this flight we will need 20 gallons. 20 gal * 6 lbs.=120 lbs.    

                           

	Weight(lbs.)	Arm(in.)	Moment(in.-lb.)
Front Seats	300	37	11,100
Rear Seats	120	73	8760
Baggage Area 1	30	95	2850
Empty Weight	1491	39.13	58347
Fuel	120	47	5640
Total	2,061		89,547

Doing the math gives us a total weight of 2,061 pounds and a total moment of 89,547 in.-lbs. Now, we divide the moment by the weight to give us a center of gravity location of 43.4 inches aft of the datum. We can now use a chart like the one below to determine if the center of gravity is located within the airplane’s limitations. First move along the x-axis until you reach 43.4 inches and then move up the y-axis until you reach the total weight of 2,061 pounds. We can see that the intersection of these two lines is within the normal category limits, so our plane is safely loaded to go flying!

This was a fairly simple example using a small airplane that can only seat a few people. Can you imagine doing this for a massive airliner with hundreds of seats and capacity for thousands of pounds of baggage? Modern day flight dispatchers use computer programs to efficiently accomplish this task, but in the early days of commercial aviation, there were no computers, and this essential computation was made by hand. Center of gravity calculations are just one type of math that keeps planes safe.