Kermode.
Bluskis - and friends..
A C Kermode. Mechanics of Flight. Edition 10. Page 229. "Climbing".
First, I need you to imagine a picture - let's, for the sake of continuity, call it "Figure 7.1".
Imagine a picture of an aircraft - side view. Draw on it the four arrows that we are all familiar with. Identify them with the the famous four - L, D, T and, W. For those with bad memories - L, represents Lift and is drawn vertically upwards, W represents Weight and is drawn vertically downwards, T represents Thrust and is drawn horizontally (pointing to the left), D represents drag and is drawn pointing to the right. All the above assumes you have drawn the aircraft the right way up and flying from right to left.
Now - leave the W arrow still pointing vertically downwards, but rotate everything else in the picture - to represent a climb. The amount that you have rotated the picture, measured in degrees, will represent the "climb angle", and will be called "a". Thats a small letter "a" (presumably "alpha").
Imagine now, your thrust arrow is inclined upwards at angle "a", your drag arrow is inclined downwards at angle "a", and the lift arrow is inclined backwards at angle "a". Weight, still vertically downwards. Got it??
Onwards to Kermode....
Climbing
During level flight the power of the engine must produce, via the propeller, jet or rocket, a thrust equal to the drag of the aeroplane at that particular speed of flight. If now the engine has some reserve of power in hand, and if the throttle is further opened, either -
(a) The pilot can put the nose down slightly, and maintain level flight at an increased speed and decreased angle of attack, or
(b) The aeroplane will commence to climb
A consideration of the forces which act upon an aeroplane during a climb is interesting, but slightly more complicated than the other cases which we have considered.
Assuming that the path actually travelled by the aeroplane is in the same direction as the thrust, then the forces will be as shown in Figure 7.1
If "a" is the angle of climb, and if we resolve the forces parallel and at right angles to the direction of flight, we obtain two equations -
(1) T = D + W sin "a"
(2) L = W cos "a"
Translated into non-mathematical language, the first of these equations tells us that during a climb the thrust needed is greater than the drag and increases with the steepness of the climb. This is what we would expect. If a vertical climb were possible, "a" would be 90° and therefore sin "a" would be 1, so the first equation would become T = D + W, which is obviously true because in such an extreme case the thrust would have the opposition of both the weight and the drag. Similarly if "a" = 0° (i.e. there is no climb), sin "a" = 0. Therefore W sin "a" = 0. Therefore T = D, the condition which we have already established for straight and level flight.
The second equation tells us that the lift is less than the weight, which is rather interesting because one often hears it said that an aeroplane climbs when the lift is greater than the weight! One must admit, however, that the misunderstanding is largely due to the rather curious definition which we have assigned to the word 'lift'. Let us consider the second equation under extreme conditions. If the climb were vertical, cos 90° = 0. Therefore L = 0. So that in a vertical climb we have no lift. This simply means that all the real lift is provided by the thrust, the wings doing nothing at all to help. If, on the other hand, "a" = 0, cos "a" = 1, and therefore L = W, which we already know to be the condition of straight and level flight.
(Phew)