How To Calculate Lifting Force

Whether you're studying the flight of birds that beat their wings to rise into the sky or the rising of gas from a chimney into the atmosphere, you can study how objects lift themselves against the force of gravity to better learn about these methods of "flight."

For aircraft equipment and drones that soar through the air, flight depends upon overcoming gravity as well as accounting for the force of air against these objects ever since the Wright brothers invented the airplane. Calculating the lifting force can tell you how much force is needed to send these objects airborne.

Objects flying through the air have to deal with the force of air exerted against themselves. When the object moves forward through the air, the drag force is the part of the force that acts parallel to the flow of motion. Lift, by contrast, is the part of the force that is perpendicular to the flow of air or another gas or fluid against the object.

Man-made aircraft such as rockets or planes use the lift force equation of

\(L=\frac{C_L\rho v^2 A}{2}\)

for lift force ​L​, lift coefficient ​_C_L​, density of the material around the object ​ρ​ ("rho"), velocity ​v​ and wing area ​A_​. The lift coefficient sums up the effects of various forces on the airborne object including the viscosity and compressibility of air and the body's angle with respect to the flow making the equation for calculating lift much more simple.

Scientists and engineers typically determine ​_C_L​ experimentally by measuring values of the lift force and comparing them to the object's velocity, the area of the wingspan and the density of the liquid or gas material the object is immersed in. Making a graph of lift vs. the quantity of (​ρ v² A)/2​ would give you a line or set of data points that can be multiplied by the ​C_L_​ to determine the lift force in the lift force equation.

More advanced computational methods can determine more precise values of the lift coefficient.There are theoretical ways of determining the lift coefficient, though. To understand this part of the lift force equation, you can look at the derivation of the lift force formula and how the lift force coefficient is calculated as a result of these airborne forces on an object experiencing lift.

To account for the myriad of forces that affect an object flying through the air, you can define the lift coefficient ​C​_L as

\(C_L=\frac{L}{qS}\)

for lift force ​L​, surface area ​S​ and fluid dynamic pressure ​q​, usually measured in pascals. You can convert the fluid dynamic pressure into its formula

\(q=\frac{\rho u^2}{2}\)

​to get​

\(C_L=\frac{2L}{\rho u^2 S}\)

in which ​ρ​ is the fluid density and ​u​ is the flow speed. From this equation, you can rearrange it to derive the lift force equation​.​

This dynamic fluid pressure and surface area in contact with the air or fluid both also heavily depend on the geometry of the airborne object. For an object that may be approximated as a cylinder such as an airplane, the force should span outward from the body of the object. The surface area, then, would be the circumference of the cylindrical body times the height or length of the object, giving you ​S = C x h​.

You may also interpret the surface area as a product of thickness, a quantity of area divided by length, ​t​ , such that, when you multiply the thickness times the height or length of the object, you get surface area. In this case ​S = t x h​.

The ratio between these variables of surface area lets you graph or experimentally measure how they differ to study the effect of either the force around the circumference of the cylinder or the force that depends on the thickness of the material. Other methods of measuring and studying airborne objects using the lift coefficient exist.

There are many other ways of approximating the lift curve coefficient. Because the lift coefficient needs to comprise many different factors affecting aircraft flight, you can also use it to measure the angle a plane might take with respect to ground. This angle is known as angle of attack (AOA), represented by ​α​ ("alpha"), and you can re-write the lift coefficient

\(C_L=C_{LO}+C_{L\alpha}\alpha\)

With this measure of ​_C_L_​ that has an additional dependency due to AOA α, you can re-write the equation as

\(\alpha = \frac{C_L+C_{LO}}{C_{L\alpha}}\)

and, after experimentally determining the lift force for a single specific AOA, you can calculate the general lift coefficient C_L. Then, you can try measuring different AOAs to determine what values of ​_C_L0​ and ​CL_​_​α​ would fit best fit​.​ This equation assumes that the lift coefficient changes linearly with AOA so there may be some circumstances in which a more accurate coefficient equation may fit better.

To better understand AOA on lift force and lift coefficient, engineers have studied how the AOA changes the way a plane flies. If you graph lift coefficients against AOA, you can calculate the positive value of the slope, which is known as the two-dimensional lift-curve slope. Research has shown, though, that after some value of AOA, the ​C​_​L​ value decreases.

This maximum AOA is known as the stalling point, with a corresponding stalling velocity and maximum ​_C_L_​ value. Research on the thickness and curvature of aircraft material has shown ways of calculating these values when you know the geometry and material of the airborne object.

NASA has an online applet to show how the lift equation impacts the flight of aircraft. This is based off a lift coefficient calculator, and you can use it to set different values of velocity, angle that the airborne object takes with respect to the ground and the surface area that the objects have against the material surrounding the aircraft. The applet even lets you use historical aircraft to show how engineered designs have evolved since the 1900s.

The simulation doesn't account for the change in weight of the airborne object due to changes in the wing area. To determine what effect that would have, you can take measurements of different values of surface areas would have on the lift force and calculate a change in lift force that these surface areas would cause. You can also calculate the gravitational force that different masses would have using W = mg for weight due to gravity W, mass m and gravitational acceleration constant g (9.8 m/s²).

You can also use a "probe" that you can direct around the airborne objects to show the velocity at various points along the simulation. The simulation is also limited that the aircraft is approximated using a flat plate as quick, dirty calculation. You can use this to approximate solutions to the lift force equation.