Thursday, March 6, 2014

Climb of an Airplane Maybe Not Intuitive

During the process of working on an article for the new website on aerodynamics I ran across a couple of simple formulas regarding the climb of an airplane.

Lift = Weight x Cosine Angle of Climb
Thrust = Drag + (Weight x Sine of Angle of Climb)

I proceeded to test the lift formula with a hypothetical model airplane weighing 10 grams for climb angles of 30, 45, and 75 degrees. The results were rather shocking to me as I thought of the four forces of flight and how lift and weight are exactly equal in level flight. It would seem that lift would keep increasing more than weight as the airplane climbs. The cosine of the angle decreases as the angle increases however.  Here are my sample results:

30 degree climb      Lift = 10 grams x .866  = 8.66 grams     Cosine 30 = .866
45 degree climb      Lift = 10 grams x .707 = 7.07 grams    Cosine 45 = .707
75 degree climb      Lift = 10 grams x .259 = 2.59 grams     Cosine 75 = .259 

Sine of an angle increases with greater angles which means the thrust will be increasing as the airplane climbs steeper. That makes sense to me now and then when you think of the situation where the airplane is hovering the airplane will be completely supported by the thrust and there will be no lift from the wing. It also means that the thrust is pulling some of the weight upward, think of vectors.

To validate this I read further articles on the Internet and posed the question on a model airplane listserve.  It is true.

Embrace the Challenge of Learning 

Bill Kuhl

For an explanation of many aspects of simple aerodynamics with math problems check out my article Basic Aerodynamics With a Lesson

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