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→Total energy in gliding: a more interesting quantity
Note that the sign conventions are correct: \(V_2>V_1\) while \(H_1>H_2\).
Now suppose there is a mysterious instrument "Anyometer" on the aeroplane that is a pressure instrument. The Anyometer uses a pressure source at location X on the aeroplane as its input. Suppose that, at location X, the pressure coefficient is \(C_{pX}\). By rearranging the definition presented before, we have the following relationships:
\[ p_{X1}=p_1+C_{pX} \frac{1}{2} \rho V_1^2 \]
\[ p_{X2}=p_1+C_{pX} \frac{1}{2} \rho V_2^2 \]
Now the Anyometer is designed so that, when the aeroplane descends from \(H_1\) to \(H_2\) as outlined above, its reading does not change, in other words:
\[ p_{X1} = p_{X2} \]
Substituting the two equations for \(p_{X1}\) and \(p_{X2}\) and rearranging slightly, we have:
\[ p_1 - p_2 = C_{pX} \times \frac{1}{2} \rho (V_2^2 - V_1^2) \]
Where \(p_1-p_2\) is the static (atmospheric) pressure difference at \(H_1\) and \(H_2\) respectively. Because \(H_2\) is lower than \(H_1\), \(p_2\) is higher than \(p_1\). Explicitly, the following relationship from hydrostatics holds:
\[ p_1 - p_2 = -\rho g (H_1 - H_2) \]
Therefore:
\[ -\rho g (H_1 - H_2) = C_{pX} \times \frac{1}{2} \rho (V_2^2 - V_1^2) \]
We now substitute in the relationship \(V_2^2 - V_1^2 = 2g \times (H_1-H_2)\) as derived before, and everything will just cancel out magically. What is left is:
\[ C_{pX} = -1 \]
This is a remarkable result. This implies that if we can monitor the pressure at a location X on the aeroplane such that \( C_{pX} = -1 \), we can track the total mechanical energy change of the aeroplane. This is the fundamental working principle of a "total energy compensated variometer". When installed on a glider, this is known without ambiguity as the '''variometer'''.
==== How to measure total energy ====
