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Why It's Important 
Weight affects the flight performance of an aircraft in many respects. An airplane which is overloaded will be deficient in performance because:

Higher takeoff speed is required.

Longer takeoff run.

Reduced rate of climb performance

Shorter range of flight

Reduced cruise speed

Reduced maneuverability

Higher stall speed

Higher landing speed

Longer landing roll

Balance Principles 
As shown in the figure below, the aircraft is somewhat like a childs "teetertotter" with respect to longitudinal balance. For the plank to be in balance, The sum of the moment s on each side of the pivot point (fulcrum) must be equal. A MOMENT is simply the weight multiplied by a moment arm (distance) from some reference point. In this example, the moments are measured from the center fulcrum point. 

As seen, the plank is in balance because the sum of the moments on each side equals 5000 pound inches. If a weight on either side is moved, or a weight is changed, the plank will no longer be in balance.
An aircraft in flight is very similar. The pivot point (fulcrum) is the located at the Center of Lift of the wing. The load on the left is the total weight of the aircraft located at the Center of Gravity (CG) with a counterbalancing force on the right provided by the elevators.


Note that if the location of the CG or the weight on the left changes, the elevator force must also change in order to maintain the balance. Also note, if the fulcrum (center of wing lift) changes, the elevator force must be changed to maintain a balanced condition. Such an event can occur when the angle of attack and/or engine thrust is changed 
The Center of Gravity 
As previously stated, the weight of an aircraft and its load is distributed throughout the aircraft as shown below by the small downward arrows. All of the small individual weights can be resolved into one single weight acting at the Center of Gravity and shown as the large arrow.. 

From the analogy of the plank above, we can see that if we change either the weight of the aircraft, or the center of gravity, this in turn changes the force (either up or down) that the elevator must produce. 

For each aircraft design, the manufacturer specifies a maximum weight for operation of the aircraft, and also a maximum forward and rearward location of the Center of Gravity (CG). This is called the CG RANGE. For safe operation, the aircraft must be operated within these parameters.
In order to calculate where the center of gravity is located, the manufacturer specifies some point in the aircraft as a reference point (DATUM). In many Cessna 172 type aircraft, the datum is located at the lower firewall of the cabin, just ahead of the rudder pedals. You as a pilot do not need to know where this is located in order to calculate weight and balance, as the manufactirer provided moment arm and/or moment in the weight and balance tables for the aircraft. An aircraft mechanic must know where whis point is, however, if equipment change is made to the aircraft which changes either the aircraft CG or Empty Weight.
An airplane is designed and certified to withstand specified loading on it’s structure. As long as the gross weight and load factor are within limits, the aircraft can be operated safely. Continued operaton of an aircraft in an overloaded condition can cause structural failures. Metal fatigue is hastened, and can lead to stress failures even in normal operating modes.

Effect on Wing Loading 
The location of the CG affects the total load which the wings must sustain. If the CG is at or near the Center of Lift of the wing the elevators do not have to generate much (if any) downward force. If the CG is aft of the center of lift, the elevators must produce an upward force. If the aircraft is nose heavy (forward CG) the load on the wing and elevator surfaces will be greater.
An aft CG location causes the airplane to require more "nose down" elevator for stall recovery. A forward CG enhances stall recovery as the aircraft will naturally want to "nose down".

Definitions 
MOMENT ARM  a horizontal distance of an object measured from a defined “datum” point to the CG of the object, usually measured in inches. A (+) arm means the object is behind the datum. A () arm indicates the object is forward off the datum point.
MOMENT  the product of a moment arm and the associated weight. (Weight x Arm)
EMPTY WEIGHT the combined weight of the aircraft, and permanently mounted equipment. It includes unusable fuel and hydraulic fluid. Most manufactures include the oil in the empty weight.
Center of Gravity  the point at which the airplane will be in balance.
CG Limits  the most forward and most rearward CG points specified by the manufacturer for safe control.
CG Range  the distance from the most forward and rearward CG points as specified for the given aircraft.
DATUM  a point in the aircraft from which all moment arms are measured.
FUEL LOAD  the weight of the useable fuel. It does not include unusable fuel in the tanks and lines.
GROSS WEIGHT Total weight of aircraft, fuel, passengers and baggage.
MAX LANDING WEIGHT  Maximum gross weight allowed for landing.
MAX RAMP WEIGHT  Maximum gross weight prior to taxi and takeoff.
MAX TAKEOFF WEIGHT  the maximum allowable weight at start of takeoff run
USEFUL LOAD  Gross weight minus the empty aircraft weight.
STANDARD WEIGHTS  Gasoline 6 lb. per gal; Oil 7.5 lb. per gal. (US Measure)

Methods of Determining Weight and Balance 
The method of determining weight and balance may vary with aircraft manufacturer and type of aircraft. These methods are:
1. Center of Gravity Calculations
2. Center of Gravity Graphs
3. Center of Gravity Tables
4. Loading Schedules (Placards)

Center of Gravity Calculations 
To determine the location CG, add up all the weights and all the moments. Divide the sum of the moments by the sum of the weights. This is illustrated in the diagram.
All moments aft of the datum are positive numbers. All moments forward of the datum are negative. In the example, the oil is the only negative moment since it is forward of the DATUM.. Calculate the weight of oil at 7.5 pounds per gallon. Calculate fuel at 6 pounds (US) per gallon.


The procedure to calculate the CG is as follows:
1. Add up all weights, including empty aircraft
2. Multiply each weight by it’s moment arm in inches to get the moment for that item.
3. Add up all the moments
4. Divide the sum of the moments by the total weight to get the CG

A TYPICAL W/B PROBLEM 
ITEM 
WEIGHT lbs 
MOMENT ARM in. 
MOMENT LB IN 
A = AIRCRAFT 
1000 
6 
6000 
P = Pilot 
150 
11 
1650 
B = BAGGAGE 
40 
32 
1280 
O = OIL 
7.5 
4 
30 
F = FUEL 
120 
16 
1920 
TOTAL 
1317.5 
CG? 
10820 
In the example, CG? = 10820/1317.5 = 8.21 in. aft of the datum.

Loading Graph 
Frequently the manufacturer provides a graphical method for determining weight and balance. 

Determine the load moment for each load item using the appropriate line in the graph. For example, for pilot and front seat passenger, total the combined load weight. Go up the Load Weight axis (Y axis) to the pilot & passenger weight. Then go horizontal to the pilot & passenger line (Red line). Then go down to the X axis to find the load moment/1000. Do the same for fuel (blue line), rear passengers (green Line) and baggage (black line). Total up the weight and the Load Moments.
EXAMPLE: Determine the Load and moment from the loading graph above for each of the following load items.

Weight and Balance Using Loading Graphs 

Weight 
**Moment 
Empty Weight 
1,364 
51.7 
Front Seats 
400 
15.0 
Baggage 
120 
11.5 
Fuel (38 gal) 
228 
11.0 
Oil (2 gal) 
15 
0.2 
TOTAL 
2,127 
89.0 
Derive the values as follows: 
1. Empty Weight and moment are values provided by the manufacturer.
2. Pilot & Passenger  Add weight for Pilot and Passenger. In this example 400lb. Go up the left side (Y axis) of the graph to 400 lb. weight, then horizontal to the “Pilot & Passenger” line. Read vertically down from the intersection to the horizontal (X) axis, and read a Moment/1000 value of 15.0.
3. Baggage  The baggage weight is 120 lb. Go up left side of the upper graph to a load of 120 lb. Continue horizontally to intersect the “baggage” line. Go downward from this intersection and read a Moment/1000 value of 11.5.
4. Fuel  38 gallons at 6 pounds per gallon is 228 pounds. Go up the left side of graph to a weight value of 228 lbs., then horizontally to intersect the“fuel line. Go downward to read a Moment/1000 value of 11.0.
5. Oil was not included in the empty weight of this aircraft, therefore it must be entered into the calculation. Two gallons of oil is 15 lb. The moment is 0.2 lbin.

Loading Envelope 
Once the total weight and the total moment/1000 is found, use the load and CG envelope to ascertain that the aircraft is properly loaded.
Go up the left side to a total load of 2127 lbs. Draw a horizontal line. Go along the bottom scale to find the loaded aircraft moment / 100 value of 89.0. Draw a vertical line. The aircraft is properly loaded if the intersection of the horizontal and vertical lines is within the envelope.


According to this envelope, if the weight is greater than 2300 pounds, the aircraft is overloaded. If the intersection is outside the envelope laterally, the loading is out of proper CG range. 
Center of Gravity Tables 
Some manufacturers provide tables instead of graphs or calculation. This method is fairly lengthy, and is used infrequently for small aircraft. Therefore this method will not be covered herein. 
Weight Shift and Change 
The approach to solving both Weight Change and Weight Shift is the same. The simplest method is to REMOVE a changed item, and to ADD the new or shifted item into the new location. 
Weight Change 
Example 1:
An airplane takes off with a Gross Weight of 6230 lb., and a CG of 79.0. The CG of the fuel is at 87.00 aft of datum. What is the New CG location after 50 gallons of fuel is burned?
Procedure:
Subtract the Weight and Moment of the burned fuel from the initial values to arrive at a new set of values. At 6 pounds per gallon, the burned fuel weight is 300 pounds.

WEIGHT CHANGE PROBLEM 

WEIGHT 
CG 
MOMENT 
Initial Weight 
6230 
79.00 
492,170 
Burned Fuel 
300 
87.00 
26,100 
New Weight 
5930 
New CG 
466,070 
NEW CG (after Fuel Burned) = 466,070 / 5930 = 78.59 
Weight Shift 
Example 2:
The gross weight of the aircraft is 3,000 lbs. with a CG of 60 in. Since takeoff, 25 gallons of fuel has been used. The fuel cell CG is 65 in. aft of Datum (Station 65).
Also, a 200 pound passenger moves from Station 50 to Station 90. (Note: Some problems will state the CG location as "Station". The 50 and 90 are CG location in inches aft of datum respectively).
Find New CG.
Procedure:
1. Subtract burned Fuel
2. Subtract Passenger who moves from the old location.
3. Add passenger who moves to the new location.

WEIGHT SHIFT PROBLEM 

Weight 
CG Station 
Moment 
Initial Loading 
300 
60.00 
180,000 
Fuel Burned 
150 
65.00 
9,750 
Passenger Off 
200 
50.00 
10,000 
Passenger On 
+200 
90.00 
+18,000 
New Totals 
2850 
New CG 
178,250 
New Cg = 178,250 / 2850 = Station 62.54 
