## Weight and Balance

*by H. Clay Gorton*

The Starduster Magazine, Vol. 29, No. 4, Oct. 1999

To maintain the center of gravity of an airplane within acceptable limits is essential to flight. About five years ago a light plane landed at the Skypark Bountiful airport. The plane had been purchased new and this was its first flight. The pilot and two passengers took off from Skypark on a warm afternoon with a full load of fuel and a packed baggage compartment. No checks were made of gross weight or density altitude. This omission cost the lives of three people. After taking off, the plane could not maintain altitude, plunged to the earth and crashed and burned. Whether the plane was nose heavy or over grossed is not known.

A friend of mine who worked as a radio operator for Panagra on flights to South America before the Second World War told of another serious accident related to weight and balance. A cargo DC-3 had been loaded at the Mendoza, Argentina airport, but the loadmaster had failed to secure the cargo. When the plane took off the cargo shifted to the rear causing the plane to nose up. The pilot corrected by pushing the yoke forward. When the plane began to nose down the load shifted to the front of the cargo bay, pushing the center of gravity ahead of the forward limit of the airplane, making it impossible to recover from the dive. The plane was destroyed and all five occupants were killed.

**Center of Gravity**

What is the center of gravity? The c.g. is the point in any body at which all the weight appears to be concentrated. For instance, an irregularly shaped body suspended from the c.g. would be in balance regardless of its position.

Conventional airplanes are considered to be in balance left to right, so the c.g. would be somewhere along a centerline from front to rear. The acceptable limits of the center of gravity for a Starduster Too are from 18 to 27 inches behind the firewall

**Moment Arm**

If a uniform rod were suspended from
the center and a weight were attached to one side at a given distance
from the center, a corresponding weight would need to be attached to the
other side to keep the rod in
balance. However, if the weight to be attached to the other side, were
heavier than the first weight, balance would be achieved if the heavier
weight were attached at a shorter distance from the center than that of
the first weight. The force exerted by a weight at a given distance from
the attach point is called the **moment**, or **torque**. As an
example, a weight of three pounds attached four feet to the left of
center would be balanced by a weight of three pounds attached four feet
to the right of center, or by a weight of 12 pounds, attached 1 foot to
the right of center. The distances to the two weights are defined as the
moment arms of the weights. The torque—3 x 4 ft lbs, or 12 x 1 ft
lbs—would be the same.

**Datum**

The datum is an arbitrary line from which the distances to the various points of interest in weight and balance are measured. The vertical datum is usually taken as the firewall, since the manufacture’s specs for weight and balance are always given in distances from the firewall. However, in practice, distances are usually measured from the front point of the spinner, as it is easier to suspend a plumb bob from that point to the floor, where the measurements are made.

The horizontal datum is conveniently measured along on the top aileron. To do weight and balance the airplane must be in a true horizontal position. This is achieved with a spirit level placed on the top horizontal aileron.

**Weight and Balance Equation**

To perform the weight and balance measurements, the weight of the aircraft under each wheel must be determined, and the distances from center of the point of contact of the wheels with the floor to the datum line must be measured. The procedure to find the center of gravity (c.g.) of the aircraft is a simple one. It comprises dividing the sum of the moments (distance from the datum to the point at which the weight is measured) by the sum of the weights.

The formula is written as follows:

c. g. = | D_{L} x W_{L} +
D_{R} + D_{T} x W_{T} |

W_{L} + W_{R} +
W_{T} |

where D_{L} is the distance from the datum to the center
point of the left wheel,

D_{R} is the distance from the datum to the center point of
the right wheel (these two distances should be equal),

D_{T} is the distance from the datum to the center point of
the tail wheel, and

W_{L}, W_{R}, and W_{T} are the weights measured
under the left, right and tail wheels, respectively.

**Calculating Weight and Balance**

Set-up: For a new aircraft the wt. & bal. measurement must be made with the aircraft completely empty—no oil and no fuel. (The FAA seems to prefer it that way.) However, if the aircraft is not new, it is more convenient to make the measurements with the oil already added, since the aircraft is never intentionally flown without oil.

The wt. & bal. calculations must be made with the aircraft in a horizontal position. First, scales must be placed under each of the three points of contact of the airplane with the ground. Next the aircraft must be brought to a horizontal position by raising the tail wheel until the top longeron is horizontal, measured by a spirit level.

Tare: Any additional weight on the scales, such as chocks under the main gear and the scaffolding under the tail wheel (tare weight) must be subtracted from the scale readings.

Measuring the moment arm: Since the manufacturer's specs
use the firewall as the datum, so shall we. The distance from the datum to
the center of the point of impact of the front wheels with scales must be
measured. It will be assumed that the two landing gear are properly aligned.
Drop a plumb bob from the firewall to the floor. Mark a line through the
point on the floor parallel to the alignment of the two wheels, so that the
distance from the line to each wheel is the same. This distance will be the
moment arm for the weights, W_{L} and W_{R}. Next, drop a
plumb bob from the center of the axle of the tail wheel to the floor. The
distance from this point to the datum will be the moment arm for W_{T}.
Determine the center of gravity of the empty airplane using Table 1.

The maximum forward c.g. limit is determined with the aircraft under the lightest load. (The center point of all the weights to be added except for the oil are located more than 18 inches behind the firewall, and so would increase the tail heaviness of the airplane.) So the max. forward c.g. limited would be calculated with no fuel, no baggage and one light pilot—dead stick landing! The calculation of the max. forward c.g. limit is made using Table 2.

The max. aft. c.g. limit is determined with a fully loaded airplane. Two methods may be used to determine the c.g. of a loaded airplane. In the first method the moment arm to the center of gravity of each component must be measured. This is not always practical. For instance, where is the center of gravity of the oil in the engine, or where is the center of gravity of the pilot seated in the cockpit?

In the practical method, the airplane may be weighed with the components whose centers of gravity are not easily determined placed in the airplane. Baggage weight must also be considered. However, rather than adding baggage to the airplane the baggage could be weighed and the moment arm of the baggage could be measured from the center of the baggage compartment and added to the calculation. The aft c.g. limit would be calculated using Table 3.

Tables 4, 5 and 6 are added as examples of the calculations for the three c.g.’s discussed above. It must be noted that the actual weights and distances will be unique to each aircraft measured.

**Table 1. Empty Weight c.g. Calculations**

WEIGHING POINT |
WEIGHT (-Tare), W, lbs |
MOMENT ARM, D, in |
MOMENT, D x W, in. lbs |

Right Main (R) | |||

Left Main (L) | |||

Tail Wheel (T) | |||

Total |

**Table 2. Max. Forward c.g. Calculations**

WEIGHING POINT |
WEIGHT (-tare), lbs |
MOMENT ARM, in |
MOMENT, in lbs |

Right Main (1) | |||

Left Main (2) | |||

Tail Wheel (3) | |||

Pilot | |||

Oil | |||

Total |

**Table 3. Max. Aft c.g. Calculations**

WEIGHING POINT |
WEIGHT (-tare), lbs |
MOMENT ARM, in |
MOMENT, in lbs |

Right Main (1) | |||

Left Main (2) | |||

Tail Wheel (3) | |||

Pilot | |||

Copilot | |||

Fuel, Main | |||

Fuel, Wing | |||

Baggage | |||

Total |

**Examples**

**Empty Weight c.g. Calculations**

WEIGHING POINT |
WEIGHT (-Tare), W, lbs |
MOMENT ARM, D, in |
MOMENT, D x W, in lbs |

Right Main (R) | 610.0 | 7.5 | 4575.0 |

Left Main (L) | 600.o | 7.5 | 4500.0 |

Tail Wheel (T) | 85.0 | 169.5 | 14407.5 |

Total | 1295.0 | 23482.5 |

c.g. (empty) = 610x7.5 + 600x7.5 + 85x169.5 = 23482.5 = 18.13 in. behind firewall

Note that the specified forward limit is 18 inches behind the firewall. The empty airplane is within spec.

**Max. Forward c.g. Calculations**

WEIGHING POINT |
WEIGHT (-tare), lbs |
MOMENT ARM, in |
MOMENT, in lbs |

Right Main (1) | 610.0 | 7.5 | 4575.0 |

Left Main (2) | 600.0 | 7.5 | 4500.0 |

Tail Wheel (3) | 85.0 | 169.5 | 14407.5 |

Pilot | 175.0 | 70.0 | 12250.0 |

Oil | |||

Total | 1470.0 | 35732.5 |

c.g. (max forward) = 610x7.5 + 600x7.5 + 85x169.5 + 175x70 = 35732.5 = 24.3 in. behind firewall

**Max. Aft c.g. Calculations**

WEIGHING POINT |
WEIGHT (-tare), lbs |
MOMENT ARM, in |
MOMENT, in lbs |

Right Main (1) | 610.0 | 7.5 | 4575.0 |

Left Main (2) | 600.0 | 7.5 | 4500.0 |

Tail Wheel (3) | 85.0 | 169.5 | 14407.5 |

Pilot | 200.0 | 70.0 | 14000.0 |

Copilot | 175.0 | 40.0 | 7000.0 |

Fuel, Main | 150.0 | 9.0 | 1350.0 |

Fuel, Wing | 102.0 | 19.0 | 1938.0 |

Baggage | 20.0 | 96.0 | 1920.0 |

Total | 1942.0 | 49690.0 |

c.g. (max aft) = 49690.0 = 25.58 in. behind firewall

In the above example, the maximum forward and aft c.g.'s of 18.13 in. and 25.58 in. are within the limits of 18 to 27 in. behind the firewall.