In this post we'll discuss the weight and balance calculations to determine whether it is safe to fly the aircraft.
It's extremely important in flying to determine the how heavy the aircraft will be when you're fully loaded and the location of the center of gravity (CG). An aircraft in-flight is a lot like a teeter-totter, with the center of pressure (where the lift force is applied) as the center of the teeter-totter, the CG is on one end of the teeter-totter, and the lift force from the tail is on the other end of the teeter-totter, as shown in the picture below. In the picture, the blue arrow is the lift force generated by the wings, acting at the center of pressure, and the front red arrow is the weight of the aircraft and passengers, acting at the center of gravity, and the back red arrow is the lift being generated by the tail. Notice that when the center of gravity is in front of the center of pressure, the tail actually produces a lift force downwards, opposite the lift generated by the wings. This is a desirable situation because if, for some reason, the tail wasn't producing any lift and the center of gravity was behind the center of pressure (opposite the picture), the aircraft would want to pitch up on it's own, and would eventually stall and be uncontrollable. If the CG is in front of the center of pressure, the aircraft will want to pitch down, and the airflow over the control surfaces allow some degree of controllability.
Now, if the center of gravity is too far forward or too far backward, the lift forces generated by the horizontal stabilizer will not be enough to allow the pilot to keep the aircraft level, and the aircraft will either never get off the ground (too far forward), or the aircraft will pitch up too soon (too far backward), and stall on take-off and fall back to the runway. Because of this, it is extremely important to calculate where the CG will be, and whether or not it is within the limits for your aircraft.
One important concept to understand is that of a torque, or a moment. Using the teeter-totter again, imagine that you have a very fat man, sitting very close to the center of the teeter-totter. You could balance him out by having another very fat man sit very close to the teeter-totter, or you could have a skinny man sit very far from the center of the teeter-totter. The torque results when you have the force from the weight of the man acting at a distance from the center of the teeter-totter. So increasing the force applied increases the torque and increasing the distance from the center increases the torque.
THIS might be a better explanation of torques and moments.
The following charts will allow you to easily calculate the important values for
only the 1979 Cessna 172N we're talking about.
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Weight and Balance Sample and Loading GraphThis table is a handy way to calculate how everything will affect the weight and balance of the aircraft. We'll use the values that are used in the sample problem in the left columns. The "weight" column is the column where the weights of the aircraft, fuel, passengers, and luggage are input, and the "moment" column are where the torques, or moments are calculated.
The very first thing that must be done is to calculate the weight of everything going into the aircraft except for the fuel. This includes the pilot, passengers, and luggage. This value is added to the weight of the aircraft to determine how much fuel can be carried. In the sample problem everything (minus the fuel) weighs 2060 lb. Since the maximum take-off weight is 2300 lb, that means that 240 lb of fuel can be added without going over the maximum weight. At 6 lb of fuel per gallon, this is 40 gallons of fuel, and at approximately 8 gallons per hour in flight, this is about 4.5 hours of flight time with VFR reserves.
Now, starting with the first row, the weight of the aircraft with no fuel and full oil is 1454 lb. The moment arm for the weight is already calculated, and fixed for this aircraft, so the moment that results is 57,600 lb-in (note that the values in the second 'moment' column are divided by 1000).
Next, the fuel required was calculated before, and is 240 lb. The moment arm for the fuel is calculated using the "Loading Graph". To use the loading graph, read up the left side to the weight you need, then go across the graph to the line for whatever you are looking for (in this case, the "Fuel" line), and read down to the bottom of the graph to find the moment that results. In this case the resulting moment is 11.5 in-lb (x1000). Input that in to the table.
Next, using the weight of the front seat pilot and passenger, the rear passengers, and the baggage, us the loading graph to find the moment for each.
Then, the total weight and moments can be added together to find the total ramp weight and moment. In the sample, they assume that 7 lb of fuel are used for startup and taxi so the total take-off weight and moment are 2300 lb and 103.6 in-lb.
Next, use the remaining two charts to determine if these values are within the limits specified.
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COG Moment Envelope/COG LimitsThese charts will both tell you whether or not you are within the limits for your aircraft.
To read the COG Moment Envelope chart, you find the weight of the aircraft on the left side of the chart, and the moment you calculated on the bottom of the chart, and see if the point where these two intersect are within the outlined area. If they are, you are safe to fly. If they aren't, you'll have to adjust the loading of the aircraft until they lie within the limits.
To read the COG Limits chart, you use the same method as the COG Moment Envelope chart, except you have to divide the moment you found by the weight to get the moment arm. You then find this value on the bottom of the chart and read up to the weight of the aircraft to see if you are within the outlined envelope.
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Now, most of this is different from the data Brett used earlier for the demo flight, but it should give you an idea of what must be done for a flight to be safely conducted. But, I don't know of any pilot who does all this work on each an every flight they go on. If you fly by yourself, or with one or two other passengers who you know won't put you over the maximum weight, and you don't fly on quarter full tanks to an airport 100 miles away, you will most likely be quite safe in doing so. However, it's important to know this data, and be quire familiar with it in case something happens (you get stuck above the clouds, passengers freak out, etc...), you don't want to be up there trying to remember all this data while you're flying. So, if you're ever (ever, ever, ever) in doubt, just do the calculations quick. It shouldn't take more than five or ten minutes if you're familiar with them.
by Brett_Henderson » Sat Nov 17, 2007 3:49 pm 
Once we get past the 7-step Sim PPL, we'll be getting into other aircraft; other techniques... all kinda fun stuff .. and the more input, the better :)