Force, mass, weight, moment, pivot (datum), torque, moment arm. These terms are all used in the quest for the Centre of Gravity. So let’s make sure we really understand what they all mean and how they come together to allow us to calculate whether we will take-off at the end of the runway, or not. It is essential to get it the weight and balance right.
To start with some definitions. Mass, (S.I. unit is the kilogram) is a measure of how much ‘stuff’ something has in it. A one kilogram mass has one kilogram of molecules packed into it. If you take it to the moon it still measures one kilogram, and on the sun and anywhere in the universe.
A Force is a push or a pull or twist on an object that will cause it to change its motion (Newton’s First Law). For weight and balance calculations we are concerned with the force of gravity on Earth, which causes an object to accelerate downwards at 10* metres per second every second. The S.I. unit for force is the ‘newton’, and the definition of a newton is the force required to accelerate a mass of one kilogram by one metre per second every second.(kg/s/s or kg/s²).
The Weight of an object is the result of gravity acting on its mass, and can be defined as the mass times the acceleration (m x g) due to gravity. Since weight is a force, its unit is also the newton. We often mix the terms mass and weight in aviation, but it really doesn’t matter, as long as the calculations are meant for planet Earth. Not Jupiter. On the moon your weight would be about 1/6th of your Earth weight, but your mass remains the same.
A Moment is another force, but this time it is a force applied and then felt at a distance; it causes a turning movement, commonly referred to as a Torque. Imagine holding a bag weighing one kilogram out horizontally on your 1 meter** long arm. The bag is exerting a 10 newton (1kg x 10 meter/s/s acceleration) force downwards one meter away from your shoulder. The torque force you are feeling in you shoulder is the ‘moment’, the result of the force exerted at a distance. The units for moment is a newton meter***, or pound inch or pound feet. The ‘moment arm’ is the distance between the shoulder and the bag. Now move the bag up to your elbow. The obvious result is less torque felt at your shoulder because now the moment arm is 0.5 meter away. The moment force felt at you shoulder is now 10 newtons x 0.5 meters, so 5 newton meters. Much less torque than holding it at arm’s length. In weight and balance calculations you are concerned with the torque force around the aircraft’s centre of gravity.
The centre of gravity (C of G)of an object is the point where the total weight of a body can be thought to be concentrated. For an aircraft it is also the centre point through which pitch, yaw and roll will occur. When we do a centre of gravity calculation, we are doing it because the manufacturers and engineers of the aircraft have guaranteed that it will perform well as long as the C of G is within the stated parameters. The forward and aft C of G limits affect the longitudinal stability of an aircraft. As the C of G moves forward it becomes harder to raise the nose, which can cause serious problems at low landing speed if you can’t raise the nose enough, and can’t slow down. With the C of G too far aft, the increased elevator sensitivity in pitch up can lead to an inadvertent stall. This is just the tip of the iceberg for problems caused from being out of the recommended range.
So now the terminology is defined and hopefully understood, it should be easy to grasp the weight and balance calculation. Look at the basic weight and balance form. The columns are for Weight (in pounds lbs), the Arm in inches (meaning the length of the moment arm), and Moment, (which is that force felt at the shoulder) calculated by multiplying the weight and arm together. You might notice that the arms; the position from which the distance is measured, are all aft of a datum. This is a way of making the maths easier and less prone to error. By choosing an arbitrary point somewhere at the front of the aircraft from where to measure the distances, all the calculated moments will be clockwise, so they will all be added. If the datum was the C of G itself the calculation would be more complex and easier to make an error as you would be adding all moments rear of the C of G and subtracting all the anticlockwise moments forward of the C of G.
On the weight and Balance form , the first row is for the Basic Empty Weight. You can get this from the weight and balance schedule for the individual plane you are flying. Rows two and three are calculated for the pilot and passengers. You can see the front seats are 80.5 inches aft and the rear passengers 118.1 inches aft. The fuel tanks’ datum is between the two rows of seats, and the baggage is quite far back. To do the full calculation you have to fill in the weights and multiply them by the arms (as has been done for the pilot and passengers). When that is done, add all the weights together then check it does not exceed the maximum take off weight. There is no need to add all the arms, but do add all the moments (forces) together. Leave that number in your calculator, because you then divide the total moment by the total weight. You have just calculated the new position of the Centre of Gravity. You then take that weight and C of G position and plot it on the C of G Range graph from the aircraft manual (example one below).
This article is not intended to teach you how to do a weight and balance, but rather familiarise you with the terms and show you how it is done. If you have learnt from it great. If there is anything you don’t understand or have any questions do email me and ask, I am always happy to clarify.
- 10m/s/s is close enough for what we need
** You are a spider monkey
*** The unit newton meter is not the same as the newton meter (joule) that refers to work done. A joule is force times distance where the force vector is in the same direction of movement of an object, whereas in moments the newton meter force is a twist, not a straight line.