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Computing centre of gravity and applying a weight shift

Building a loading table from arms and moments, converting the result to percent MAC, and using the weight shift formula to bring an out-of-limits CG back inside.

Weight and balance18 min read

Video scheduled

The lesson video for this topic is in production on The Flight Dispatch Room.

Written lesson

The lesson

Weight and balance is the most mechanical subject on the exam, which makes it the most reliable source of marks. The arithmetic never varies. What varies is which column the question hides the trap in.

The three quantities

  • Weight is the mass of an item, in pounds.
  • Arm is the horizontal distance from the datum to the item, in inches. Aft of the datum is positive, forward of it is negative.
  • Moment is weight multiplied by arm, and is the turning effect of that item about the datum.

The centre of gravity is the total moment divided by the total weight. That is the entire calculation. Everything else in the subject is either bookkeeping or a rearrangement of that relationship.

Percent MAC

Transport aircraft usually express CG limits as a percentage of the mean aerodynamic chord rather than in inches from the datum, because the aerodynamic limits are properly a function of the wing, not the fuselage.

To convert, you need the CG in inches, the leading edge of the MAC as a station in inches, and the length of the MAC in inches. Subtract the leading edge station from the CG to get the distance aft of the leading edge, then express that as a percentage of the MAC length.

Shifting weight instead of removing it

When a computed CG falls outside limits, the usual remedy is not to offload but to move load. Moving weight changes the total moment while leaving the total weight unchanged, so the CG moves and the gross weight does not.

The change in CG equals the weight shifted multiplied by the distance it moved, divided by the total weight. The CG moves in the same direction as the weight was moved.

Examination focus

Key exam concepts

  • Moment = weight x arm. CG = total moment / total weight.
  • Arms are measured from the datum; aft is positive.
  • A moment index is a moment divided by a constant; the divisor must be consistent across the table.
  • Percent MAC = (CG - LEMAC) / MAC length x 100.
  • Weight shift: change in CG = (weight shifted x distance shifted) / total weight.
  • The forward CG limit is generally a control authority limit; the aft limit is a stability limit.

Reference

Important formulas

Moment

Moment = Weight x Arm

Centre of gravity

CG = Total moment / Total weight

Percent MAC

%MAC = ((CG - LEMAC) / MAC length) x 100

LEMAC is the station in inches of the leading edge of the mean aerodynamic chord.

Weight shift

Change in CG = (Weight shifted x Distance shifted) / Total weight

The CG moves in the same direction as the load.

Weight added or removed

Change in CG = (Weight added x (Arm of weight - Old CG)) / New total weight

Use a negative weight for removal.

Method

Worked examples

Compute CG and percent MAC from a loading table

Given

  • Basic operating weight 92,000 lb at arm 640.0 in.
  • Forward cargo 8,000 lb at arm 580.0 in.
  • Aft cargo 6,000 lb at arm 720.0 in.
  • Fuel 24,000 lb at arm 650.0 in.
  • LEMAC is station 600.0 in and the MAC is 200.0 in long.

Steps

  1. 1Basic operating moment: 92,000 x 640.0 = 58,880,000 lb-in.
  2. 2Forward cargo moment: 8,000 x 580.0 = 4,640,000 lb-in.
  3. 3Aft cargo moment: 6,000 x 720.0 = 4,320,000 lb-in.
  4. 4Fuel moment: 24,000 x 650.0 = 15,600,000 lb-in.
  5. 5Total weight: 92,000 + 8,000 + 6,000 + 24,000 = 130,000 lb.
  6. 6Total moment: 58,880,000 + 4,640,000 + 4,320,000 + 15,600,000 = 83,440,000 lb-in.
  7. 7CG: 83,440,000 / 130,000 = 641.8 in.
  8. 8Percent MAC: (641.8 - 600.0) / 200.0 x 100 = 41.8 / 200.0 x 100.

Answer

CG is 641.8 in aft of datum, which is 20.9 percent MAC.

Shift load to move the CG forward

Given

  • Total weight 130,000 lb, current CG 641.8 in.
  • 1,200 lb is moved from the aft cargo hold at arm 720.0 in to the forward hold at arm 580.0 in.

Steps

  1. 1Distance shifted: 720.0 - 580.0 = 140.0 in, forward.
  2. 2Change in CG: (1,200 x 140.0) / 130,000 = 168,000 / 130,000 = 1.29 in.
  3. 3The load moved forward, so the CG moves forward: 641.8 - 1.29.

Answer

The new CG is 640.5 in, which is 20.3 percent MAC. Total weight is unchanged at 130,000 lb because the load was moved rather than removed.

Watch out

Common exam traps

  • Dividing total weight by total moment instead of moment by weight.
  • Mixing figures from tables that use different moment index divisors.
  • Using the MAC length in place of the LEMAC station, or the reverse, in the percent MAC conversion.
  • Assuming a weight shift changes gross weight. It does not; only the CG moves.
  • Moving the CG in the wrong direction. The CG always moves the same way as the load.
  • Forgetting that fuel burn changes the CG in flight, so a compliant takeoff CG does not guarantee a compliant landing CG.

Test yourself

Practice questions

Choose an answer to reveal the discussion. Every option is explained, including why the wrong ones are attractive.

  1. Q01

    An aircraft has a total weight of 80,000 lb and a total moment of 51,200,000 lb-in. What is the CG?

  2. Q02

    An aircraft weighs 60,000 lb with a CG at 500.0 in. 900 lb is moved aft by 200 in. What is the new CG?

  3. Q03

    The CG is 645.0 in, LEMAC is station 615.0 in and the MAC is 150.0 in. What is the CG in percent MAC?

Verify

Official FAA references

  • FAA-H-8083-1Aircraft Weight and Balance Handbook.
  • FAA-H-8083-25Pilot's Handbook of Aeronautical Knowledge — weight and balance.
  • 14 CFR 121.135Manual contents, including loading schedules.

Current regulation text is published on eCFR. Always confirm against the current version before relying on any citation.

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