When we make changes to the Thrust by adding or removing power. In what way does it affect the other forces? As Power controls Rate of Climb or Descent of an aircraft, How is it acting on the other aerodynamic forces to make changes to the Rate of Climb and Rate of Descent?
The rate of climb is:
So, increasing thrust (while leaving angle of attack constant) will leave lift coefficient constant, drag constant, airspeed constant, etc. But it will increase the rate of climb.
If the aircraft has the engines mounted with the thrust line substantially above or below the center of gravity, the aircraft may experience a pitch up or pitch down moment with a throttle change. However, this pitching moment is not a primary control mechanism and is not relied on to cause the aircraft to climb or descend.
In general, it is preferred to minimize this kind of effect during the design of the aircraft.
Besides the changes in the thrust force and its follow on effects on lift etc., there are trim changes with power. The changes depend mainly on the location of the thrust line relative to the vertical C of G, and in single engine propeller driven airplanes, the effects of propeller wash on tail surfaces where the surfaces are immersed in the prop stream. To eliminate trim changes with power, the engine(s) have to be mounted so their thrust line passes through the vertical C of G, and have no influence on flying surfaces.
So, engines mounted low, adding power reduces trim speed, and vice versa for engines mounted above the C of G. In jets, this generally this means wing mounted engines pitch up with power, and tail mounted engines pitch down with power.
In a propeller single, the thrust line is usually very close to the vertical C of G, but adding power reduces trim speed by increasing whatever trimming load is being applied to the tail (usually down relative to the wing) via stronger airflow. If you go from cruise to climb power in a Cessna 172 while trimmed at 80 kt, when the plane settles down in the climb you will notice the trim speed as dropped a few knots, and that's almost all from the higher energy propeller wash.
Your figure tell you the story:
If you add trust, the THRUST arrow will elongate, so you have an acceleration, but it is not applied to the center of mass, so you will have some leverage effect (pitch you upward). You will accelerate until the drag will increase (drag increases with speed).
But if you got faster, you get more lift, so you will go up, and also because the center of mass is not at the same axis as the wings, you get also a pitch downward. Weight will not change, so you get such configuration "for ever" (until you trim).
Note: what I wrote is based on your figure. If you have engines on tail (as many executive planes), you get different leverage (note: also the position, they may be installed higher then drag vector). The same about the position of the center of mass.